From a3090080ca4197517493e39a78321b4e4afccdbb Mon Sep 17 00:00:00 2001 From: axiom-marbell Date: Tue, 18 Aug 2026 11:19:32 +0200 Subject: [PATCH 1/2] harvest: exact-mass one-loop regulator atoms Advances #76 without promoting scientific claims.\n\nCo-authored-by: prime-agent --- src/substrate_framework/__init__.py | 24 + .../scalar_one_loop_mass.py | 513 ++++++++++++++++++ tests/test_scalar_one_loop_mass.py | 441 +++++++++++++++ 3 files changed, 978 insertions(+) create mode 100644 src/substrate_framework/scalar_one_loop_mass.py create mode 100644 tests/test_scalar_one_loop_mass.py diff --git a/src/substrate_framework/__init__.py b/src/substrate_framework/__init__.py index 12815fdd..db86dc0d 100644 --- a/src/substrate_framework/__init__.py +++ b/src/substrate_framework/__init__.py @@ -749,6 +749,19 @@ leading_scalar_newton_shift_coefficient, scalar_heat_kernel_a2, ) +from .scalar_one_loop_mass import ( + KNOWN_ONE_LOOP_REGULATORS, + SMOOTH_PROPER_TIME_REGULATOR, + ZETA_POWER_SUBTRACTED_REGULATOR, + ExactMassNewtonShift, + ExactMassVacuumShift, + RegulatorSchemeLedger, + curvature_proper_time_integral, + exact_mass_inverse_newton_shift, + exact_mass_vacuum_density_shift, + regulator_scheme_ledger, + vacuum_proper_time_integral, +) from .covariant_sine_gordon_action import ( CovariantSineGordonAction, CovariantSineGordonFluctuation, @@ -2176,6 +2189,17 @@ "metric_ricci_scalar", "metric_ricci_tensor", "worldline_reparametrization_residual", + "KNOWN_ONE_LOOP_REGULATORS", + "SMOOTH_PROPER_TIME_REGULATOR", + "ZETA_POWER_SUBTRACTED_REGULATOR", + "ExactMassNewtonShift", + "ExactMassVacuumShift", + "RegulatorSchemeLedger", + "curvature_proper_time_integral", + "exact_mass_inverse_newton_shift", + "exact_mass_vacuum_density_shift", + "regulator_scheme_ledger", + "vacuum_proper_time_integral", ] __version__ = "0.0.0" diff --git a/src/substrate_framework/scalar_one_loop_mass.py b/src/substrate_framework/scalar_one_loop_mass.py new file mode 100644 index 00000000..dcf08151 --- /dev/null +++ b/src/substrate_framework/scalar_one_loop_mass.py @@ -0,0 +1,513 @@ +"""Exact-mass one-loop proper-time coefficients for preregistered regulators. + +This is the Route 1 (induced / Sakharov gravity) rung that follows +``covariant_sine_gordon_action``. The landed action rung hands the +fluctuation operator + + D_E = -nabla_E**2 + xi*R_E + m**2, m**2 = V''(phi_bg), + +to the determinant. This module evaluates the two leading proper-time +integral classes of ``Gamma_E = (1/2)*ln(det(D_E))`` with the mass retained +exactly --- no expansion of ``exp(-tau*m**2)`` inside the regulated integral +--- for three preregistered regulators: + +``SHARP_PROPER_TIME_REGULATOR`` (reused from ``scalar_induced_newton``) + A sharp lower proper-time cutoff ``tau_0 = Lambda**-2``. Writing + ``z = m**2/Lambda**2``, the curvature class integral is + + I_2 = integral_{tau_0}^{infty} tau**-2 * exp(-m**2*tau) dtau + = Lambda**2 * (exp(-z) - z*E1(z)), + + with ``E1 = expint(1, .)``. The factor ``exp(-z) - z*E1(z)`` is the + load-bearing finite-mass correction named by the PR #13 review + (approximately 0.1485 at z = 1): a leading UV-asymptotic ``Lambda**2`` + term alone does not establish the finite-cutoff result. + +``SMOOTH_PROPER_TIME_REGULATOR`` + A smooth essential-singularity proper-time weight ``exp(-1/(Lambda**2 + * tau))``. All power-divergent classes become convergent single + integrals; the closed forms are modified Bessel functions: + + I_2 = 2*Lambda**2*sqrt(z)*BesselK_1(2*sqrt(z)), + I_3 = 2*Lambda**4*z*BesselK_2(2*sqrt(z)). + +``ZETA_POWER_SUBTRACTED_REGULATOR`` + The Mellin finite-part scheme in which the power divergences are + subtracted; it requires a declared renormalization scale ``mu`` (the + cutoff argument is rejected) and yields the pure logarithmic running + + I_2 = m**2*(ln(m**2/mu**2) + EulerGamma - 1), + I_3 = -(m**4/2)*(ln(m**2/mu**2) + EulerGamma - 3/2). + +Authority note: ``scalar_induced_newton`` and +``covariant_sine_gordon_action`` are landed conditional (unpromoted) +prior-work APIs -- the PR #14 and PR #25 harvests promoted no claims -- and +the accepted ``C-GRV-001`` supplies only the conditional dimensional and +additive-baseline ledger, leaving the coefficient, field content, and +regulator as premises. Every public symbol in this module is likewise +conditional, unpromoted infrastructure linked to open goal #76. + +Composition with that landed conditional scheme factor (one real scalar, +determinant weight 1/2, heat-kernel prefactor ``(4*pi)**-2``, +Einstein-Hilbert matching factor ``16*pi``) gives the exact-mass induced +inverse-Newton shift + + Delta(1/G) = N * coefficient_per_field(xi) * I_2(regulator), + +where ``coefficient_per_field`` is taken from the landed conditional +``scalar_induced_newton.leading_scalar_newton_shift_coefficient`` API, so +the massless sharp limit reproduces that conditional API's +``s*Lambda**2 = N*(1-6*xi)*Lambda**2/(12*pi)`` exactly. The vacuum sector +of the same mass-resummed expansion is + + Delta(rho_Gamma) = -(N/2)*(4*pi)**-2 * I_3(m**2) + +per scheme. The ``-m**2`` entry of the landed conditional +``scalar_heat_kernel_a2`` weights belongs to the *unresummed* organization, +in which the exponential is expanded and the mass survives only as that +coefficient; applying it on top of the resummed ``I_3`` double-counts the +mass. The two organizations agree to first order through the exact +derivative identity ``d I_3/d m**2 = -I_2`` (which holds for all three +schemes and is tested), i.e. ``I_3(m**2) = I_3(0) - m**2*I_2(0) + O(m**4)``. + +Declared derivation oracles (all exact, SymPy): + +* sharp ``I_2``: ``d/dtau_0 I_2(tau_0) = -tau_0**-2*exp(-m**2*tau_0)`` plus + the tail decay, so the closed form is the tail integral by uniqueness; +* sharp ``I_3``: the integration-by-parts identity + ``I_3 = exp(-z)*Lambda**4/2 - (m**2/2)*I_2``; +* smooth closed forms: the exact differential recurrence + ``d/dz J_p = -J_{p-1}`` of the integrals + ``J_p(z) = integral_0^{infty} t**-p exp(-z*t - 1/t) dt`` together with the + boundary limits ``J_2(0+) = 1`` and ``J_3(0+) = 1`` (verified against + SymPy Bessel derivatives), plus independent high-precision quadrature; +* zeta finite parts: the cutoff-subtraction limits + ``lim_{tau_0 -> 0} [I(tau_0) - power terms - log terms]`` evaluated + symbolically. + +This module does NOT identify the cutoff with any substrate scale, choose a +renormalization condition among the three schemes, derive a total Newton +constant (the accepted C-GRV-001 additive baseline remains independent), +evaluate the tau**-1 class (curvature-squared and m**2*R structures), or +confront any empirical comparator. Those are later rungs of issue #76. +Scheme differences are the deliverable: at z = 1 the smooth regulator +induces about 1.88 times the sharp curvature coefficient, and the +power-subtracted scheme changes both the parametric structure and the sign +at small log; any later usable-normalization claim must cite this spread. + +``ExactMassNewtonShift.curvature_weight_sign`` describes only the weight +``1/6 - xi`` and carries no total-shift verdict; ``value_sign`` is the +decidable sign of the full returned value (``None`` when symbolic inputs +leave it undecidable). For the two cutoff schemes ``I_2 > 0`` always, so +the two agree there; for the power-subtracted scheme ``I_2`` itself changes +sign with ``m**2/mu**2`` and the two fields genuinely differ. +""" + +from __future__ import annotations + +from dataclasses import dataclass +from typing import Any + +import sympy as sp + +from .exact_symbolic import exact_real as _exact_real +from .exact_symbolic import positive_exact as _positive_exact +from .scalar_induced_newton import SHARP_PROPER_TIME_REGULATOR +from .scalar_induced_newton import leading_scalar_newton_shift_coefficient # conditional landed API + +SMOOTH_PROPER_TIME_REGULATOR = "proper_time_smooth_essential" +ZETA_POWER_SUBTRACTED_REGULATOR = "zeta_power_subtracted" + +KNOWN_ONE_LOOP_REGULATORS = ( + SHARP_PROPER_TIME_REGULATOR, + SMOOTH_PROPER_TIME_REGULATOR, + ZETA_POWER_SUBTRACTED_REGULATOR, +) + +_FOUR_DIMENSIONAL_HEAT_KERNEL_PREFACTOR = (4 * sp.pi) ** -2 + + +def _nonnegative_exact(value: Any, name: str) -> sp.Expr: + expression = _exact_real(value, name) + if expression.is_nonnegative is not True: + raise ValueError(f"{name} must be provably nonnegative") + return expression + + +def _positive_integer(value: Any, name: str) -> sp.Expr: + expression = _positive_exact(value, name) + if expression.is_integer is not True: + raise ValueError(f"{name} must be a positive integer field count") + return expression + + +def _resolve_regulator( + regulator: Any, + cutoff: Any, + renormalization_scale: Any, +) -> tuple[sp.Expr | None, sp.Expr | None]: + """Validate the scheme contract and return (cutoff, scale) exact values.""" + + if regulator not in KNOWN_ONE_LOOP_REGULATORS: + raise ValueError( + f"unknown regulator {regulator!r}; pass one of " + f"{KNOWN_ONE_LOOP_REGULATORS}" + ) + if regulator == ZETA_POWER_SUBTRACTED_REGULATOR: + if cutoff is not None: + raise ValueError( + "the power-subtracted scheme has no cutoff; pass " + "renormalization_scale only" + ) + if renormalization_scale is None: + raise ValueError( + "the power-subtracted scheme requires an explicit " + "renormalization_scale; it must not default" + ) + return None, _positive_exact(renormalization_scale, "renormalization_scale") + if renormalization_scale is not None: + raise ValueError( + "renormalization_scale belongs to the power-subtracted scheme " + "only; a cutoff scheme must not declare one" + ) + if cutoff is None: + raise ValueError("this regulator requires an explicit positive cutoff") + return _positive_exact(cutoff, "cutoff"), None + + +def _sharp_curvature_integral(cutoff: sp.Expr, mass_squared: sp.Expr) -> sp.Expr: + if mass_squared.is_zero is True: + # Massless limit: the tail integral of tau^-2 is exactly Lambda^2. + return cutoff**2 + z = mass_squared / cutoff**2 + return sp.simplify(cutoff**2 * (sp.exp(-z) - z * sp.expint(1, z))) + + +def _smooth_curvature_integral(cutoff: sp.Expr, mass_squared: sp.Expr) -> sp.Expr: + if mass_squared.is_zero is True: + # Massless limit: 2*sqrt(z)*K_1(2*sqrt(z)) -> 1 as z -> 0+. + return cutoff**2 + z = mass_squared / cutoff**2 + return sp.simplify( + 2 * cutoff**2 * sp.sqrt(z) * sp.besselk(1, 2 * sp.sqrt(z)) + ) + + +def _zeta_curvature_integral( + renormalization_scale: sp.Expr, mass_squared: sp.Expr +) -> sp.Expr: + if mass_squared.is_zero is True: + # Power divergences subtracted: no induced curvature term at m = 0. + return sp.Integer(0) + return sp.simplify( + mass_squared + * (sp.log(mass_squared / renormalization_scale**2) + sp.EulerGamma - 1) + ) + + +def _sharp_vacuum_integral(cutoff: sp.Expr, mass_squared: sp.Expr) -> sp.Expr: + if mass_squared.is_zero is True: + # Massless limit: the tail integral of tau^-3 is Lambda^4/2. + return cutoff**4 / 2 + z = mass_squared / cutoff**2 + return sp.simplify( + sp.exp(-z) * cutoff**4 / 2 + - (mass_squared / 2) * _sharp_curvature_integral(cutoff, mass_squared) + ) + + +def _smooth_vacuum_integral(cutoff: sp.Expr, mass_squared: sp.Expr) -> sp.Expr: + if mass_squared.is_zero is True: + # Massless limit: integral_0^infty t^-3 exp(-1/t) dt = 1, so Lambda^4. + return cutoff**4 + z = mass_squared / cutoff**2 + return sp.simplify(2 * cutoff**4 * z * sp.besselk(2, 2 * sp.sqrt(z))) + + +def _zeta_vacuum_integral( + renormalization_scale: sp.Expr, mass_squared: sp.Expr +) -> sp.Expr: + if mass_squared.is_zero is True: + return sp.Integer(0) + return sp.simplify( + -(mass_squared**2 / 2) + * ( + sp.log(mass_squared / renormalization_scale**2) + + sp.EulerGamma + - sp.Rational(3, 2) + ) + ) + + +def curvature_proper_time_integral( + regulator: Any, + *, + cutoff: Any = None, + mass_squared: Any = 0, + renormalization_scale: Any = None, +) -> sp.Expr: + """Return the exact tau**-2-class proper-time integral for one scheme. + + This is the integral that multiplies the declared heat-kernel weights + ``((1/6 - xi)*R_E - m**2)`` in ``Gamma_E``; with the landed conditional + scheme factor it also multiplies the induced inverse-Newton shift. + """ + + resolved_cutoff, scale = _resolve_regulator(regulator, cutoff, renormalization_scale) + mass = _nonnegative_exact(mass_squared, "mass_squared") + if regulator == SHARP_PROPER_TIME_REGULATOR: + return _sharp_curvature_integral(resolved_cutoff, mass) + if regulator == SMOOTH_PROPER_TIME_REGULATOR: + return _smooth_curvature_integral(resolved_cutoff, mass) + return _zeta_curvature_integral(scale, mass) + + +def vacuum_proper_time_integral( + regulator: Any, + *, + cutoff: Any = None, + mass_squared: Any = 0, + renormalization_scale: Any = None, +) -> sp.Expr: + """Return the exact tau**-3-class proper-time integral for one scheme. + + This is the integral that multiplies the heat-kernel weight ``1`` (the + cosmological/vacuum sector of the one-loop action). + """ + + resolved_cutoff, scale = _resolve_regulator(regulator, cutoff, renormalization_scale) + mass = _nonnegative_exact(mass_squared, "mass_squared") + if regulator == SHARP_PROPER_TIME_REGULATOR: + return _sharp_vacuum_integral(resolved_cutoff, mass) + if regulator == SMOOTH_PROPER_TIME_REGULATOR: + return _smooth_vacuum_integral(resolved_cutoff, mass) + return _zeta_vacuum_integral(scale, mass) + + +@dataclass(frozen=True) +class ExactMassNewtonShift: + """Exact-mass induced inverse-Newton shift data for one scheme.""" + + regulator: str + field_count: sp.Expr + non_minimal_coupling: sp.Expr + mass_squared: sp.Expr + cutoff: sp.Expr | None + renormalization_scale: sp.Expr | None + proper_time_value: sp.Expr + coefficient_per_field: sp.Expr + value: sp.Expr + massless_leading_value: sp.Expr + finite_mass_factor: sp.Expr | None + curvature_weight_sign: int + value_sign: int | None + + +def exact_mass_inverse_newton_shift( + field_count: Any, + non_minimal_coupling: Any, + *, + regulator: Any, + cutoff: Any = None, + mass_squared: Any = 0, + renormalization_scale: Any = None, +) -> ExactMassNewtonShift: + """Return ``Delta(1/G) = N * coefficient_per_field * I_2`` exactly. + + ``coefficient_per_field`` is the per-field scheme factor + ``scheme_factor*(1/6 - xi)`` read from the landed conditional + ``scalar_induced_newton.leading_scalar_newton_shift_coefficient`` API + (its massless sharp regulator tag is used only to read that factor, + which is regulator-independent). The massless sharp limit reproduces + that module's ``N*(1-6*xi)*Lambda**2/(12*pi)`` exactly. + + ``curvature_weight_sign`` is the decidable sign of ``1/6 - xi`` only. + ``value_sign`` is the decidable sign of the full returned value, or + ``None`` when symbolic inputs leave it undecidable; the two coincide + for the cutoff schemes (``I_2 > 0``) but differ for the + power-subtracted scheme whenever ``I_2 < 0``. + """ + + resolved_cutoff, scale = _resolve_regulator(regulator, cutoff, renormalization_scale) + mass = _nonnegative_exact(mass_squared, "mass_squared") + count = _positive_integer(field_count, "field_count") + xi = _exact_real(non_minimal_coupling, "non_minimal_coupling") + + landed = leading_scalar_newton_shift_coefficient( + 1, xi, regulator=SHARP_PROPER_TIME_REGULATOR + ) + coefficient_per_field = landed.coefficient_per_field + + if regulator == SHARP_PROPER_TIME_REGULATOR: + proper_time_value = _sharp_curvature_integral(resolved_cutoff, mass) + massless_leading_value = sp.simplify( + count * coefficient_per_field * resolved_cutoff**2 + ) + finite_mass_factor = sp.simplify(proper_time_value / resolved_cutoff**2) + elif regulator == SMOOTH_PROPER_TIME_REGULATOR: + proper_time_value = _smooth_curvature_integral(resolved_cutoff, mass) + massless_leading_value = sp.simplify( + count * coefficient_per_field * resolved_cutoff**2 + ) + finite_mass_factor = sp.simplify(proper_time_value / resolved_cutoff**2) + else: + proper_time_value = _zeta_curvature_integral(scale, mass) + massless_leading_value = sp.Integer(0) + finite_mass_factor = None + + value = sp.simplify(count * coefficient_per_field * proper_time_value) + curvature_weight = sp.simplify(sp.Rational(1, 6) - xi) + if curvature_weight.is_positive is True: + curvature_weight_sign = 1 + elif curvature_weight.is_zero is True: + curvature_weight_sign = 0 + elif curvature_weight.is_negative is True: + curvature_weight_sign = -1 + else: + raise ValueError( + "non_minimal_coupling must have a decidable relation to the " + "four-dimensional conformal value 1/6" + ) + if value.is_positive is True: + value_sign: int | None = 1 + elif value.is_zero is True: + value_sign = 0 + elif value.is_negative is True: + value_sign = -1 + else: + value_sign = None + + return ExactMassNewtonShift( + regulator=regulator, + field_count=count, + non_minimal_coupling=xi, + mass_squared=mass, + cutoff=resolved_cutoff, + renormalization_scale=scale, + proper_time_value=proper_time_value, + coefficient_per_field=coefficient_per_field, + value=value, + massless_leading_value=massless_leading_value, + finite_mass_factor=finite_mass_factor, + curvature_weight_sign=curvature_weight_sign, + value_sign=value_sign, + ) + + +@dataclass(frozen=True) +class ExactMassVacuumShift: + """Exact-mass one-loop vacuum-sector data for one scheme.""" + + regulator: str + field_count: sp.Expr + mass_squared: sp.Expr + cutoff: sp.Expr | None + renormalization_scale: sp.Expr | None + tau_minus_two_value: sp.Expr + tau_minus_three_value: sp.Expr + value: sp.Expr + # tau_minus_two_value is exposed for reference and for the derivative + # identity d I_3 / d m**2 = -I_2; the returned value composes only the + # mass-resummed tau**-3 class. + + +def exact_mass_vacuum_density_shift( + field_count: Any, + *, + regulator: Any, + cutoff: Any = None, + mass_squared: Any = 0, + renormalization_scale: Any = None, +) -> ExactMassVacuumShift: + """Return ``Delta(rho) = -(N/2)*(4*pi)**-2*I_3(m**2)`` exactly. + + This is the vacuum (cosmological-sector) coefficient of the one-loop + action density in the mass-resummed organization: the trace integrand is + ``exp(-tau*m**2)*[tau**-2 + tau**-1*(1/6-xi)*R_E + ...]``, so the vacuum + sector is the tau**-3 class with the exponential retained. The + ``-m**2`` heat-kernel weight of the landed conditional + ``scalar_heat_kernel_a2`` is the first-order remnant of that same + exponential in the unresummed organization and must not be added again; + the exact bridge is the derivative identity ``d I_3/d m**2 = -I_2``. + The sector is exhibited rather than omitted whenever the curvature-sector + shift is quoted. + """ + + resolved_cutoff, scale = _resolve_regulator(regulator, cutoff, renormalization_scale) + mass = _nonnegative_exact(mass_squared, "mass_squared") + count = _positive_integer(field_count, "field_count") + + tau_minus_two = curvature_proper_time_integral( + regulator, + cutoff=resolved_cutoff, + mass_squared=mass, + renormalization_scale=scale, + ) + tau_minus_three = vacuum_proper_time_integral( + regulator, + cutoff=resolved_cutoff, + mass_squared=mass, + renormalization_scale=scale, + ) + value = sp.simplify( + -count + * sp.Rational(1, 2) + * _FOUR_DIMENSIONAL_HEAT_KERNEL_PREFACTOR + * tau_minus_three + ) + return ExactMassVacuumShift( + regulator=regulator, + field_count=count, + mass_squared=mass, + cutoff=resolved_cutoff, + renormalization_scale=scale, + tau_minus_two_value=tau_minus_two, + tau_minus_three_value=tau_minus_three, + value=value, + ) + + +@dataclass(frozen=True) +class RegulatorSchemeLedger: + """Exact scheme spread of the curvature-class integral at common scales.""" + + cutoff: sp.Expr + mass_squared: sp.Expr + renormalization_scale: sp.Expr + sharp_value: sp.Expr + smooth_value: sp.Expr + zeta_value: sp.Expr + sharp_over_smooth: sp.Expr + sharp_minus_zeta: sp.Expr + + +def regulator_scheme_ledger( + cutoff: Any, + mass_squared: Any, + renormalization_scale: Any, +) -> RegulatorSchemeLedger: + """Return the three curvature-class values and their exact contrasts. + + The sharp and smooth schemes are evaluated at the declared cutoff; the + power-subtracted scheme at the declared renormalization scale. The + contrasts are exact expressions, not numeric fits, so a later + renormalization-condition rung can cite them as reviewable provenance. + """ + + resolved_cutoff = _positive_exact(cutoff, "cutoff") + mass = _nonnegative_exact(mass_squared, "mass_squared") + scale = _positive_exact(renormalization_scale, "renormalization_scale") + + sharp_value = _sharp_curvature_integral(resolved_cutoff, mass) + smooth_value = _smooth_curvature_integral(resolved_cutoff, mass) + zeta_value = _zeta_curvature_integral(scale, mass) + return RegulatorSchemeLedger( + cutoff=resolved_cutoff, + mass_squared=mass, + renormalization_scale=scale, + sharp_value=sharp_value, + smooth_value=smooth_value, + zeta_value=zeta_value, + sharp_over_smooth=sp.simplify(sharp_value / smooth_value), + sharp_minus_zeta=sp.simplify(sharp_value - zeta_value), + ) diff --git a/tests/test_scalar_one_loop_mass.py b/tests/test_scalar_one_loop_mass.py new file mode 100644 index 00000000..69bef40d --- /dev/null +++ b/tests/test_scalar_one_loop_mass.py @@ -0,0 +1,441 @@ +"""Exact-mass one-loop proper-time coefficient tests (P230, advances #76). + +Oracles, per ``.agents/skills/physics-erdos-loop/references/oracles.md``: + +* exact identities are verified symbolically (derivative/tail uniqueness for + the sharp cutoff scheme, the integration-by-parts identity for its vacuum + class, the differential recurrence plus boundary limits for the smooth + Bessel family, and cutoff-subtraction limits for the power-subtracted + scheme); +* the special-function closed forms are independently corroborated by + high-precision mpmath quadrature; +* load-bearing mutations (wrong prefactor, wrong Bessel order, wrong + exponential-integral branch, wrong xi sign, unknown regulator, defaulted + scales) must each break a relevant check. +""" + +import mpmath as mp +import pytest +import sympy as sp + +import substrate_framework as framework +from substrate_framework.scalar_induced_newton import ( + SHARP_PROPER_TIME_REGULATOR, + leading_scalar_newton_shift_coefficient, +) +from substrate_framework.scalar_one_loop_mass import ( + KNOWN_ONE_LOOP_REGULATORS, + SMOOTH_PROPER_TIME_REGULATOR, + ZETA_POWER_SUBTRACTED_REGULATOR, + curvature_proper_time_integral, + exact_mass_inverse_newton_shift, + exact_mass_vacuum_density_shift, + regulator_scheme_ledger, + vacuum_proper_time_integral, +) + +Lam, mu, m2 = sp.symbols("Lambda mu m2", positive=True) +tau0, t = sp.symbols("tau0 t", positive=True) +z = sp.symbols("z", positive=True) + + +def test_one_loop_mass_api_is_exported_from_package() -> None: + assert framework.SMOOTH_PROPER_TIME_REGULATOR == SMOOTH_PROPER_TIME_REGULATOR + assert framework.ZETA_POWER_SUBTRACTED_REGULATOR == ZETA_POWER_SUBTRACTED_REGULATOR + assert framework.curvature_proper_time_integral is curvature_proper_time_integral + assert framework.vacuum_proper_time_integral is vacuum_proper_time_integral + assert framework.exact_mass_inverse_newton_shift is exact_mass_inverse_newton_shift + assert framework.exact_mass_vacuum_density_shift is exact_mass_vacuum_density_shift + assert framework.regulator_scheme_ledger is regulator_scheme_ledger + + +def test_sharp_curvature_closed_form_is_the_tail_integral_exactly() -> None: + # I(tau0) = Lam^2*(exp(-z)-z*E1(z)) with z = m2*tau0, Lam^2 = 1/tau0. + closed = (sp.exp(-m2 * tau0) - m2 * tau0 * sp.expint(1, m2 * tau0)) / tau0 + derivative_residual = sp.simplify( + sp.diff(closed, tau0) + tau0 ** -2 * sp.exp(-m2 * tau0) + ) + assert derivative_residual == 0 + # Tail decay pins the integration constant. E1(x) is by definition the + # tail integral integral_x^infty exp(-u)/u du; for u >= x > 0 the + # integrand is bounded by exp(-u)/x, whose tail integral sympy evaluates + # exactly, so 0 <= E1(x) <= exp(-x)/x -> 0. The explicit exponential + # piece vanishes directly. + assert sp.limit(sp.exp(-m2 * tau0) / tau0, tau0, sp.oo) == 0 + x, u = sp.symbols("x u", positive=True) + assert sp.integrate(sp.exp(-u) / x, (u, x, sp.oo)) == sp.exp(-x) / x + mp.mp.dps = 30 + for xf in (mp.mpf(30), mp.mpf(40)): + assert mp.e1(xf) < mp.e ** (-xf) / xf + assert mp.e1(xf) * xf * mp.e ** xf < 1 + # The module's expression is this closed form. + value = curvature_proper_time_integral( + SHARP_PROPER_TIME_REGULATOR, cutoff=Lam, mass_squared=m2 + ) + assert sp.simplify(value - Lam**2 * (sp.exp(-m2 / Lam**2) - (m2 / Lam**2) * sp.expint(1, m2 / Lam**2))) == 0 + + +def test_sharp_vacuum_closed_form_satisfies_the_by_parts_identity() -> None: + sharp2 = curvature_proper_time_integral( + SHARP_PROPER_TIME_REGULATOR, cutoff=Lam, mass_squared=m2 + ) + sharp3 = vacuum_proper_time_integral( + SHARP_PROPER_TIME_REGULATOR, cutoff=Lam, mass_squared=m2 + ) + expected = sp.exp(-m2 / Lam**2) * Lam**4 / 2 - (m2 / 2) * sharp2 + assert sp.simplify(sharp3 - expected) == 0 + # Independent derivation by parts from the tail-integral form. + by_parts = sp.integrate(sp.exp(-m2 * t) / t**3, (t, tau0, sp.oo)) + target = (sp.exp(-m2 * tau0) / (2 * tau0**2)) - (m2 / 2) * ( + (sp.exp(-m2 * tau0) - m2 * tau0 * sp.expint(1, m2 * tau0)) / tau0 + ) + # sympy may leave Ei-branch structure; compare via the derivative oracle. + residual = sp.diff(by_parts - target, tau0).replace( + sp.exp_polar, lambda arg: sp.exp(arg) + ) + assert sp.simplify(residual) == 0 + + +def test_smooth_closed_forms_satisfy_the_differential_recurrence() -> None: + # J_p(z) = integral_0^infty t^-p exp(-z t - 1/t) dt obeys dJ_p/dz = -J_{p-1}. + J1 = 2 * sp.besselk(0, 2 * sp.sqrt(z)) + J2 = 2 * sp.sqrt(z) * sp.besselk(1, 2 * sp.sqrt(z)) + J3 = 2 * z * sp.besselk(2, 2 * sp.sqrt(z)) + J0 = 2 * z ** sp.Rational(-1, 2) * sp.besselk(1, 2 * sp.sqrt(z)) + assert sp.simplify(sp.diff(J2, z) + J1) == 0 + assert sp.simplify(sp.diff(J3, z) + J2) == 0 + assert sp.simplify(sp.diff(J1, z) + J0) == 0 + # Boundary values pin the solutions of the recurrence. + assert sp.limit(J2, z, 0, "+") == 1 + assert sp.limit(J3, z, 0, "+") == 1 + # The module returns these forms at z = m2/Lam^2. + value2 = curvature_proper_time_integral( + SMOOTH_PROPER_TIME_REGULATOR, cutoff=Lam, mass_squared=m2 + ) + value3 = vacuum_proper_time_integral( + SMOOTH_PROPER_TIME_REGULATOR, cutoff=Lam, mass_squared=m2 + ) + assert sp.simplify(value2 - Lam**2 * J2.subs(z, m2 / Lam**2)) == 0 + assert sp.simplify(value3 - Lam**4 * J3.subs(z, m2 / Lam**2)) == 0 + + +@pytest.mark.parametrize("zf_str", ["0.1", "1.0", "2.5"]) +def test_smooth_closed_forms_match_high_precision_quadrature(zf_str) -> None: + mp.mp.dps = 40 + zf = mp.mpf(zf_str) + num2 = mp.quad(lambda tt: mp.e ** (-zf * tt - 1 / tt) / tt**2, [0, 1, mp.inf]) + closed2 = 2 * mp.sqrt(zf) * mp.besselk(1, 2 * mp.sqrt(zf)) + assert abs(num2 - closed2) < mp.mpf("1e-30") * abs(closed2) + num3 = mp.quad(lambda tt: mp.e ** (-zf * tt - 1 / tt) / tt**3, [0, 1, mp.inf]) + closed3 = 2 * zf * mp.besselk(2, 2 * mp.sqrt(zf)) + assert abs(num3 - closed3) < mp.mpf("1e-30") * abs(closed3) + + +@pytest.mark.parametrize("zf_str", ["0.1", "1.0", "2.5"]) +def test_sharp_closed_form_matches_quadrature(zf_str) -> None: + mp.mp.dps = 40 + zf = mp.mpf(zf_str) + num = mp.quad(lambda tt: mp.e ** (-zf * tt) / tt**2, [1.0, mp.inf]) + closed = mp.e ** (-zf) - zf * mp.e1(zf) + assert abs(num - closed) < mp.mpf("1e-30") * abs(closed) + + +def test_zeta_closed_forms_are_the_power_subtraction_limits() -> None: + sharp2 = (sp.exp(-m2 * tau0) - m2 * tau0 * sp.expint(1, m2 * tau0)) / tau0 + lim2 = sp.limit(sharp2 - 1 / tau0 - m2 * sp.log(tau0), tau0, 0, "+") + zeta2 = curvature_proper_time_integral( + ZETA_POWER_SUBTRACTED_REGULATOR, mass_squared=m2, renormalization_scale=mu + ) + assert sp.simplify(lim2 - m2 * (sp.log(m2) + sp.EulerGamma - 1)) == 0 + assert sp.simplify(zeta2 - m2 * (sp.log(m2 / mu**2) + sp.EulerGamma - 1)) == 0 + + sharp3 = sp.exp(-m2 * tau0) / (2 * tau0**2) - (m2 / 2) * sharp2 + lim3 = sp.limit( + sharp3 - 1 / (2 * tau0**2) + m2 / tau0 + (m2**2 / 2) * sp.log(tau0), + tau0, + 0, + "+", + ) + zeta3 = vacuum_proper_time_integral( + ZETA_POWER_SUBTRACTED_REGULATOR, mass_squared=m2, renormalization_scale=mu + ) + expected3 = -(m2**2 / 2) * (sp.log(m2) + sp.EulerGamma - sp.Rational(3, 2)) + assert sp.simplify(lim3 - expected3) == 0 + assert sp.simplify(zeta3 + (m2**2 / 2) * (sp.log(m2 / mu**2) + sp.EulerGamma - sp.Rational(3, 2))) == 0 + + +def test_massless_sharp_limit_reproduces_the_landed_conditional_shift() -> None: + N, xi = sp.Integer(3), sp.Rational(1, 12) + shift = exact_mass_inverse_newton_shift( + N, + xi, + regulator=SHARP_PROPER_TIME_REGULATOR, + cutoff=Lam, + mass_squared=0, + ) + landed = leading_scalar_newton_shift_coefficient( + N, xi, regulator=SHARP_PROPER_TIME_REGULATOR + ) + assert sp.simplify(shift.value - landed.coefficient * Lam**2) == 0 + assert sp.simplify(shift.finite_mass_factor - 1) == 0 + + smooth = exact_mass_inverse_newton_shift( + N, + xi, + regulator=SMOOTH_PROPER_TIME_REGULATOR, + cutoff=Lam, + mass_squared=0, + ) + # The smooth weight also integrates to exactly Lambda^2 when massless. + assert sp.simplify(smooth.value - landed.coefficient * Lam**2) == 0 + + zeta = exact_mass_inverse_newton_shift( + N, + xi, + regulator=ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=0, + renormalization_scale=mu, + ) + # Power divergences subtracted: no induced shift survives at m = 0. + assert sp.simplify(zeta.value) == 0 + + +def test_small_z_series_recovers_the_exact_logarithmic_structure() -> None: + F = sp.exp(-z) - z * sp.expint(1, z) + series = sp.series(F, z, 0, 2).removeO() + assert sp.simplify(series - (1 + z * (sp.log(z) + sp.EulerGamma - 1))) == 0 + + +def test_scheme_ledger_reports_exact_contrasts() -> None: + ledger = regulator_scheme_ledger(Lam, m2, mu) + assert sp.simplify( + ledger.sharp_over_smooth + - ( + (sp.exp(-m2 / Lam**2) - (m2 / Lam**2) * sp.expint(1, m2 / Lam**2)) + / (2 * sp.sqrt(m2 / Lam**2) * sp.besselk(1, 2 * sp.sqrt(m2 / Lam**2))) + ) + ) == 0 + # At z = 1 the smooth scheme induces about 1.88 times the sharp value. + one = regulator_scheme_ledger(1, 1, 1) + ratio = float(sp.N(one.sharp_over_smooth, 12)) + assert abs(ratio - 0.5308496410) < 1e-8 + # The PR #13 review's load-bearing factor is reproduced at z = 1. + factor = sp.N( + sp.exp(-1) - sp.expint(1, 1), + 10, + ) + assert abs(factor - sp.Float("0.1484955068", 10)) < sp.Float("1e-9") + + +def test_exact_mass_shift_composes_the_landed_conditional_scheme_factor() -> None: + N, xi = sp.Integer(2), sp.Rational(1, 12) + shift = exact_mass_inverse_newton_shift( + N, + xi, + regulator=SHARP_PROPER_TIME_REGULATOR, + cutoff=Lam, + mass_squared=m2, + ) + landed = leading_scalar_newton_shift_coefficient( + 1, xi, regulator=SHARP_PROPER_TIME_REGULATOR + ) + assert sp.simplify(shift.coefficient_per_field - landed.coefficient_per_field) == 0 + assert sp.simplify(shift.value - N * landed.coefficient_per_field * shift.proper_time_value) == 0 + + +def test_vacuum_shift_is_the_mass_resummed_determinant_integrand() -> None: + # In the mass-resummed organization the trace integrand is + # exp(-tau*m^2)*[tau^-2 + tau^-1*(1/6-xi)*R + ...], so the vacuum sector + # is the tau^-3 class alone. The value must compose ONLY I_3. + N = sp.Integer(4) + for regulator, kwargs in ( + (SHARP_PROPER_TIME_REGULATOR, {"cutoff": Lam, "mass_squared": m2}), + (SMOOTH_PROPER_TIME_REGULATOR, {"cutoff": Lam, "mass_squared": m2}), + ( + ZETA_POWER_SUBTRACTED_REGULATOR, + {"mass_squared": m2, "renormalization_scale": mu}, + ), + ): + vac = exact_mass_vacuum_density_shift(N, regulator=regulator, **kwargs) + expected = -N * sp.Rational(1, 2) * (4 * sp.pi) ** -2 * vac.tau_minus_three_value + assert sp.simplify(vac.value - expected) == 0 + # Massless sharp vacuum sector: I_3 = Lambda^4/2. + vac0 = exact_mass_vacuum_density_shift( + 1, regulator=SHARP_PROPER_TIME_REGULATOR, cutoff=Lam, mass_squared=0 + ) + assert sp.simplify(vac0.value + Lam**4 / (4 * (4 * sp.pi) ** 2)) == 0 + + +def test_double_counted_vacuum_composition_is_rejected_by_the_oracle() -> None: + # The reviewed defect: composing the resummed I_3 with the unresummed + # -m^2 weight (i.e. I_3 + m^2*I_2) differs from the exact factored + # determinant by the generally nonzero term -(N/2)*(4*pi)^-2*m^2*I_2. + N = sp.Integer(1) + vac = exact_mass_vacuum_density_shift( + N, regulator=SHARP_PROPER_TIME_REGULATOR, cutoff=Lam, mass_squared=m2 + ) + double_counted = -N * sp.Rational(1, 2) * (4 * sp.pi) ** -2 * ( + vac.tau_minus_three_value + m2 * vac.tau_minus_two_value + ) + assert sp.simplify(double_counted - vac.value - m2 * vac.tau_minus_two_value * (-N * sp.Rational(1, 2) * (4 * sp.pi) ** -2)) == 0 + # ... and the spurious term is nonzero for m^2 > 0 (it vanishes only at + # m = 0, which is why the massless regression could not catch it). + assert sp.simplify(m2 * vac.tau_minus_two_value) != 0 + + +@pytest.mark.parametrize( + "regulator,kwargs", + [ + (SHARP_PROPER_TIME_REGULATOR, {"cutoff": Lam, "mass_squared": m2}), + (SMOOTH_PROPER_TIME_REGULATOR, {"cutoff": Lam, "mass_squared": m2}), + ( + ZETA_POWER_SUBTRACTED_REGULATOR, + {"mass_squared": m2, "renormalization_scale": mu}, + ), + ], +) +def test_derivative_identity_links_the_two_integral_classes(regulator, kwargs) -> None: + # d I_3 / d m^2 = -I_2 holds exactly in every scheme; it is the exact + # bridge showing the -m^2 heat-kernel weight is the first-order remnant + # of the retained exponential, not an additive piece of I_3. + i2 = curvature_proper_time_integral(regulator, **kwargs) + i3 = vacuum_proper_time_integral(regulator, **kwargs) + assert sp.simplify(sp.diff(i3, m2) + i2) == 0 + + +def test_mutations_break_the_oracles() -> None: + # Wrong prefactor: doubling the sharp closed form fails the tail identity. + doubled = 2 * (sp.exp(-m2 * tau0) - m2 * tau0 * sp.expint(1, m2 * tau0)) / tau0 + residual = sp.simplify(sp.diff(doubled, tau0) + tau0 ** -2 * sp.exp(-m2 * tau0)) + assert residual != 0 + # Wrong exponential-integral branch: Ei instead of E1 fails it too. + wrong_branch = (sp.exp(-m2 * tau0) - m2 * tau0 * sp.Ei(m2 * tau0)) / tau0 + residual_branch = sp.simplify( + sp.diff(wrong_branch, tau0) + tau0 ** -2 * sp.exp(-m2 * tau0) + ) + assert residual_branch != 0 + # Wrong Bessel order: K_2 in the curvature class breaks the recurrence. + wrong_order = 2 * sp.sqrt(z) * sp.besselk(2, 2 * sp.sqrt(z)) + J1 = 2 * sp.besselk(0, 2 * sp.sqrt(z)) + assert sp.simplify(sp.diff(wrong_order, z) + J1) != 0 + # Crossing the conformal value 1/6 flips the curvature weight sign: the + # minimal coupling xi = 0 has 1/6 - xi > 0 while xi = 1 flips it. For + # the cutoff schemes I_2 > 0, so the value sign follows. + plus = exact_mass_inverse_newton_shift( + 1, 0, regulator=SHARP_PROPER_TIME_REGULATOR, cutoff=Lam + ) + minus = exact_mass_inverse_newton_shift( + 1, 1, regulator=SHARP_PROPER_TIME_REGULATOR, cutoff=Lam + ) + assert plus.curvature_weight_sign == 1 and minus.curvature_weight_sign == -1 + assert plus.value_sign == 1 and minus.value_sign == -1 + assert sp.simplify(plus.value + minus.value) != 0 + + +def test_zeta_value_sign_disagrees_with_the_curvature_weight_sign() -> None: + # Reviewer counterexample: N=1, xi=0, m^2 = mu^2 = 1 gives a negative + # total shift while the curvature weight is positive. + negative = exact_mass_inverse_newton_shift( + 1, + 0, + regulator=ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=1, + renormalization_scale=1, + ) + assert negative.curvature_weight_sign == 1 + assert negative.value.is_negative is True + assert negative.value_sign == -1 + assert sp.simplify(negative.value - (sp.EulerGamma - 1) / (12 * sp.pi)) == 0 + + # Massless power-subtracted branch: value is exactly zero, so the value + # sign is 0 even though the curvature weight stays positive. + zero = exact_mass_inverse_newton_shift( + 1, + 0, + regulator=ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=0, + renormalization_scale=1, + ) + assert zero.value == 0 + assert zero.curvature_weight_sign == 1 + assert zero.value_sign == 0 + + # Symbolic scale ratio: the value sign is undecidable, not guessed. + symbolic = exact_mass_inverse_newton_shift( + 1, + 0, + regulator=ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=m2, + renormalization_scale=mu, + ) + assert symbolic.value_sign is None + assert symbolic.curvature_weight_sign == 1 + + # Large-mass zeta branch flips the integral itself positive. + positive = exact_mass_inverse_newton_shift( + 1, + 0, + regulator=ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=sp.Integer(10), + renormalization_scale=1, + ) + assert positive.value_sign == 1 + + +def test_input_contracts_reject_ambiguous_schemes() -> None: + with pytest.raises(ValueError): + curvature_proper_time_integral("wrong_name", cutoff=Lam) + with pytest.raises(ValueError): + curvature_proper_time_integral(SMOOTH_PROPER_TIME_REGULATOR) + with pytest.raises(ValueError): + curvature_proper_time_integral( + SMOOTH_PROPER_TIME_REGULATOR, + cutoff=Lam, + renormalization_scale=mu, + ) + with pytest.raises(ValueError): + curvature_proper_time_integral(ZETA_POWER_SUBTRACTED_REGULATOR) + with pytest.raises(ValueError): + curvature_proper_time_integral( + ZETA_POWER_SUBTRACTED_REGULATOR, + cutoff=Lam, + renormalization_scale=mu, + ) + with pytest.raises(ValueError): + curvature_proper_time_integral( + SHARP_PROPER_TIME_REGULATOR, + cutoff=Lam, + mass_squared=sp.Symbol("neg", negative=True), + ) + with pytest.raises(ValueError): + exact_mass_inverse_newton_shift( + sp.Rational(3, 2), + 0, + regulator=SHARP_PROPER_TIME_REGULATOR, + cutoff=Lam, + ) + with pytest.raises(ValueError): + exact_mass_inverse_newton_shift( + 1, + sp.Symbol("xi", real=True), + regulator=SHARP_PROPER_TIME_REGULATOR, + cutoff=Lam, + ) + + +def test_module_is_target_blind() -> None: + import inspect + + from substrate_framework import scalar_one_loop_mass + + source = inspect.getsource(scalar_one_loop_mass) + for forbidden in ("6.674", "Planck", "M_pl", "6.708", "observed"): + assert forbidden not in source + + +def test_known_regulators_are_the_three_preregistered_schemes() -> None: + assert set(KNOWN_ONE_LOOP_REGULATORS) == { + SHARP_PROPER_TIME_REGULATOR, + SMOOTH_PROPER_TIME_REGULATOR, + ZETA_POWER_SUBTRACTED_REGULATOR, + } From 276521d4701571e0d43d1dd7298801806ca48984 Mon Sep 17 00:00:00 2001 From: axiom-marbell Date: Tue, 18 Aug 2026 12:11:48 +0200 Subject: [PATCH 2/2] promote: exact constant-mass one-loop regulator claims --- .../adjudication.yaml | 58 ++++ .../attempts/0001/result.yaml | 12 + .../attempts/0002/result.yaml | 15 + .../attempts/0003/result.yaml | 23 ++ .../evidence/candidate-comparison.yaml | 26 ++ .../evidence/consumer-audit.yaml | 37 +++ .../evidence/dependency-audit.yaml | 19 ++ .../evidence/formula-freeze.yaml | 50 ++++ .../evidence/literature-audit.yaml | 33 +++ .../evidence/nonduplication-audit.yaml | 16 ++ .../evidence/primary-provenance.yaml | 19 ++ .../proposal.yaml | 78 +++++ .../reviews/C-IGR-001-claim-review.md | 40 +++ .../reviews/C-IGR-002-claim-review.md | 40 +++ .../reviews/C-IGR-003-claim-review.md | 40 +++ .../reviews/impact_analysis.md | 7 + .../reviews/independent_exact_mass_review.py | 192 +++++++++++++ .../P230-exact-mass-regulator-rung/verify.py | 272 ++++++++++++++++++ docs/generated/claim-index.md | 27 ++ governance/claims.yaml | 138 +++++++++ governance/releases/current.yaml | 23 +- governance/releases/v0.161.0.yaml | 223 ++++++++++++++ .../efforts/P230-exact-mass-promotion.md | 77 +++++ memory/framework/claims/C-IGR-001.md | 45 +++ memory/framework/claims/C-IGR-002.md | 44 +++ memory/framework/claims/C-IGR-003.md | 45 +++ memory/framework/releases/v0.161.0.md | 233 +++++++++++++++ .../scalar_one_loop_mass.py | 106 +++---- tests/test_scalar_one_loop_mass.py | 72 ++++- 29 files changed, 1939 insertions(+), 71 deletions(-) create mode 100644 campaigns/P230-exact-mass-regulator-rung/adjudication.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/attempts/0001/result.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/attempts/0002/result.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/evidence/consumer-audit.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/evidence/literature-audit.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/evidence/nonduplication-audit.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/evidence/primary-provenance.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/proposal.yaml create mode 100644 campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-001-claim-review.md create mode 100644 campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-002-claim-review.md create mode 100644 campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-003-claim-review.md create mode 100644 campaigns/P230-exact-mass-regulator-rung/reviews/impact_analysis.md create mode 100644 campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py create mode 100644 campaigns/P230-exact-mass-regulator-rung/verify.py create mode 100644 governance/releases/v0.161.0.yaml create mode 100644 memory/codex/efforts/P230-exact-mass-promotion.md create mode 100644 memory/framework/claims/C-IGR-001.md create mode 100644 memory/framework/claims/C-IGR-002.md create mode 100644 memory/framework/claims/C-IGR-003.md create mode 100644 memory/framework/releases/v0.161.0.md diff --git a/campaigns/P230-exact-mass-regulator-rung/adjudication.yaml b/campaigns/P230-exact-mass-regulator-rung/adjudication.yaml new file mode 100644 index 00000000..d290130b --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/adjudication.yaml @@ -0,0 +1,58 @@ +campaign: P230 +title: Exact constant-mass one-loop regulator coefficient promotion +base_release: v0.160.0 +accepted_release: v0.161.0 +base_commit: 0beaac3 +adjudicated_at: '2026-08-18T11:49:56+02:00' +selected_candidates: + - {id: A, reason: Sharp_tail_integrals_are_exact_positive_and_keep_the_cutoff_explicit.} + - {id: B, reason: Smooth_weight_integrals_form_an_exact_convergent_Bessel_family.} + - {id: C, reason: Declared_power_subtraction_exposes_the_exact_scale_dependent_finite_part.} +selection: parallel_conditional_families_not_physical_regulator_choice +claims: + - {id: C-IGR-001, decision: accepted, verification: symbolic_verified, review: accepted, compatibility: compatible_extension, epistemic: active} + - {id: C-IGR-002, decision: accepted, verification: symbolic_verified, review: accepted, compatibility: compatible_extension, epistemic: active} + - {id: C-IGR-003, decision: accepted, verification: symbolic_verified, review: accepted, compatibility: compatible_extension, epistemic: active} +accepted_composition: + - {id: C-GRV-001, decision: unchanged_dimensional_and_independent_additive_baseline_ledger} +unpromoted_surfaces: + - {id: varying_mass_background, reason: Factorization_requires_spacetime_constant_m_squared.} + - {id: full_nonlocal_determinant, reason: P230_proves_only_the_displayed_local_coefficient_families.} + - {id: physical_regulator_or_renormalization_condition, reason: All_three_declared_schemes_remain_unselected.} + - {id: total_Newton_constant, reason: C-GRV-001_retains_an_independent_additive_baseline.} + - {id: attractive_gravity_or_sourced_geometry, reason: No_field_equation_solution_or_total_sign_is_derived.} + - {id: higher_curvature_sector, reason: Tau_minus_one_and_nonlocal_terms_are_outside_P230.} +evidence: + proposal: proposal.yaml + formula_freeze: evidence/formula-freeze.yaml + literature_audit: evidence/literature-audit.yaml + candidate_comparison: evidence/candidate-comparison.yaml + dependency_audit: evidence/dependency-audit.yaml + consumer_audit: evidence/consumer-audit.yaml + nonduplication_audit: evidence/nonduplication-audit.yaml + primary_provenance: evidence/primary-provenance.yaml + primary_verifier: verify.py + independent_rederivation: reviews/independent_exact_mass_review.py + claim_reviews: + - reviews/C-IGR-001-claim-review.md + - reviews/C-IGR-002-claim-review.md + - reviews/C-IGR-003-claim-review.md + impact_analysis: reviews/impact_analysis.md + final_attempt: attempts/0003/result.yaml +validation_commands: + - PYTHONPATH_src_python_campaign_verify + - PYTHONPATH_src_python_independent_exact_mass_review + - pytest_scalar_one_loop_mass_and_affected_consumers + - scripts_validate_sh_full_once_at_final_boundary + - git_diff_check_separately +integrated_gate: + status: passed + repository_workflow: ALL_REPOSITORY_WORKFLOW_CHECKS_PASS + full_pytest: 2263_passed_in_288_80_seconds + primary_verifier: ALL_27_CHECKS_PASS + independent_rederivation: ALL_21_CHECKS_PASS + targeted_affected_tests: 42_passed + final_record_validation: passed + git_diff_check: passed +debt: [] +status: accepted diff --git a/campaigns/P230-exact-mass-regulator-rung/attempts/0001/result.yaml b/campaigns/P230-exact-mass-regulator-rung/attempts/0001/result.yaml new file mode 100644 index 00000000..d8f3673f --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/attempts/0001/result.yaml @@ -0,0 +1,12 @@ +id: attempt-0001 +campaign: P230 +source_pr: 77 +reviewed_head: 04f26f2 +verdict: rejected_implementation_boundary +reusable_atoms: [sharp_integrals, smooth_integrals, power_subtracted_integrals, scheme_ledger] +failures: + - mass_resummed_I3_was_double_counted_with_unresummed_m2_I2 + - reported_sign_ignored_power_subtracted_integral_sign + - landed_unpromoted_dependencies_were_called_accepted +evidence: https://github.com/vantasnerdan/substrate-framework/pull/77#pullrequestreview-4957950666 +next_route: repair_formula_sign_and_authority_boundaries diff --git a/campaigns/P230-exact-mass-regulator-rung/attempts/0002/result.yaml b/campaigns/P230-exact-mass-regulator-rung/attempts/0002/result.yaml new file mode 100644 index 00000000..bf8fd95c --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/attempts/0002/result.yaml @@ -0,0 +1,15 @@ +id: attempt-0002 +campaign: P230 +source_prs: [77, 82] +source_heads: [a8a47f8, d415caf] +verdict: artifact_correct_claim_promotion_incomplete +repairs: + - mass_resummed_vacuum_uses_I3_only + - dI3_dm2_equals_minus_I2_added + - curvature_weight_sign_separated_from_full_value_sign + - focused_PR82_removed_false_accepted_language_from_code +remaining_gaps: + - proposal_still_lacked_exact_promotable_claim_statements + - arbitrary_background_language_exceeded_constant_mass_factorization + - no_independent_rederivation_campaign_reviews_registry_release_docs_or_accepted_memory +next_route: constant_mass_claim_promotion_transaction diff --git a/campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml b/campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml new file mode 100644 index 00000000..4ad66b81 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml @@ -0,0 +1,23 @@ +id: attempt-0003 +campaign: P230 +branch: research/p230-exact-mass-promotion +status: complete +verdict: three_exact_constant_mass_claims_ready_for_promotion +repairs: + - exact_claims_narrowed_to_constant_nonnegative_mass + - positive_self_adjoint_and_IR_reference_assumptions_declared + - local_coefficient_ceiling_separated_from_full_nonlocal_determinant + - inverse_Newton_factor_rederived_without_unpromoted_scientific_API + - direct_substitution_at_m2_zero_replaced_by_one_sided_limit_oracle +evidence: + primary_verifier: ALL_27_CHECKS_PASS + independent_rederivation: ALL_21_CHECKS_PASS + targeted_tests: 42_passed + package_import: pass + legacy_numpy_preflight: clean + gitnexus_impact: low_zero_affected_execution_flows + full_repository_validation: 2263_passed_in_288_80_seconds + final_record_validation: pass + git_diff_check: pass +integrated_gate: pass +debt: [] diff --git a/campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml b/campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml new file mode 100644 index 00000000..6e53ab31 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml @@ -0,0 +1,26 @@ +campaign: P230 +criteria_frozen_before_empirical_comparators: + - defining_integral_or_subtraction_correctness + - constant_mass_and_convention_consistency + - exact_dimensions_limits_and_derivative_identity + - independent_rederivation + - mutation_sensitivity + - assumption_and_parameter_economy + - importable_consumer_and_governance_closure +candidates: + A: + decision: retain_as_C-IGR-001 + strengths: [literal_positive_tail_integral, exact_E1_form, sharp_cutoff_compatibility] + premises: [positive_cutoff] + B: + decision: retain_as_C-IGR-002 + strengths: [convergent_smooth_weight, exact_Bessel_family, no_hard_step] + premises: [positive_cutoff, declared_smooth_weight] + C: + decision: retain_as_C-IGR-003 + strengths: [explicit_power_subtraction, exact_scale_dependence, no_power_term] + premises: [positive_subtraction_scale, declared_finite_part] +selection: parallel_exposure_not_physical_regulator_choice +exact_scheme_falsifier: + massless_I3_at_unit_cutoff: {sharp: 1/2, smooth: 1, power_subtracted: 0} +comparator_gate: closed_no_empirical_comparator_used diff --git a/campaigns/P230-exact-mass-regulator-rung/evidence/consumer-audit.yaml b/campaigns/P230-exact-mass-regulator-rung/evidence/consumer-audit.yaml new file mode 100644 index 00000000..a9c0a4e2 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/evidence/consumer-audit.yaml @@ -0,0 +1,37 @@ +campaign: P230 +direct_consumers: + - src/substrate_framework/__init__.py + - tests/test_scalar_one_loop_mass.py + - campaigns/P230-exact-mass-regulator-rung/verify.py +indirect_consumers: + - docs/generated/claim-index.md + - memory/framework/claims/C-IGR-001.md + - memory/framework/claims/C-IGR-002.md + - memory/framework/claims/C-IGR-003.md + - memory/framework/releases/v0.161.0.md +existing_runtime_callers_outside_module: [] +gitnexus_final_impact: + risk: low + affected_execution_flows: 0 + direct_internal_callers: + curvature_proper_time_integral: [exact_mass_vacuum_density_shift] + vacuum_proper_time_integral: [exact_mass_vacuum_density_shift] + upstream_runtime_callers: + exact_mass_inverse_newton_shift: [] + exact_mass_vacuum_density_shift: [] + regulator_scheme_ledger: [] + note: >- + detect-changes reported seven indexed files and one false-positive AGENTS + instruction symbol caused by GitNexus injection; source git diff confirmed + AGENTS unchanged after the injected block was removed. Governance files + exceed the graph size cap and are covered by repository validation. +replay: + targeted_tests: 42_passed + campaign_verifier: pass_27 + independent_rederivation: pass_21 + package_import: pass + governance_and_generated_consumers: pass + full_repository_validation: 2263_passed_in_288_80_seconds + final_record_validation: pass + git_diff_check: pass +debt: [] diff --git a/campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml b/campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml new file mode 100644 index 00000000..cf18f3c3 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml @@ -0,0 +1,19 @@ +campaign: P230 +accepted_dependencies: + C-GRV-001: + reused: conditional_additive_inverse_coupling_and_independent_baseline_ledger + not_reused: [field_spectrum, regulator, cutoff_ontology, coefficient, total_G] +approved_nonclaim_imports: + - positive_self_adjoint_boundaryless_D_E_with_constant_nonnegative_mass + - declared_real_scalar_determinant_heat_kernel_and_EH_matching_conventions + - exact_special_function_identities_and_cutoff_finite_part_definition +implementation_only_inputs: + scalar_induced_newton: + reused: SHARP_PROPER_TIME_REGULATOR_string_tag + scientific_outputs_imported: [] + covariant_sine_gordon_action: + reused: [] + optional_composition: constant_vacuum_V_double_prime_only +dependency_closure: pass +hidden_parameters: [] +debt: [] diff --git a/campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml b/campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml new file mode 100644 index 00000000..2052b45a --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml @@ -0,0 +1,50 @@ +campaign: P230 +scope: exact_constant_mass_local_one_loop_coefficient_families +domains: + cutoff: positive_exact + subtraction_scale: positive_exact + mass_squared: nonnegative_spacetime_constant_exact + field_count: positive_integer + non_minimal_coupling: exact_real_with_decidable_relation_to_one_sixth +operator_contract: + dimension: 4 + signature: Euclidean + type: positive_self_adjoint_boundaryless_real_scalar_Laplace_type + expression: D_E=-nabla_E^2+xi*R_E+m^2 + infrared_contract: adequate_convergence_or_declared_reference_subtraction + claim_ceiling: local_heat_kernel_coefficient_families_not_full_nonlocal_determinant +conventions: + effective_action: Gamma_E=(1/2)*ln(det(D_E)) + proper_time_sign: ln(lambda)=-integral_0^infinity exp(-tau*lambda)*d_tau/tau_up_to_input_independent_constant + heat_kernel_prefactor: (4*pi)^-2 + curvature_weight: 1/6-xi + determinant_weight: 1/2 + inverse_newton_matching_factor: 16*pi + per_field_inverse_newton_factor: (1-6*xi)/(12*pi) +families: + C-IGR-001: + regulator: sharp_lower_cutoff_tau0=Lambda^-2 + I2: Lambda^2*(exp(-z)-z*E1(z)) + I3: Lambda^4*exp(-z)/2-m^2*I2/2 + C-IGR-002: + regulator: smooth_weight_exp(-1/(Lambda^2*tau)) + I2: 2*Lambda^2*sqrt(z)*K1(2*sqrt(z)) + I3: 2*Lambda^4*z*K2(2*sqrt(z)) + C-IGR-003: + regulator: declared_power_subtracted_cutoff_finite_part_at_mu + I2: m^2*(log(m^2/mu^2)+EulerGamma-1) + I3: -(m^4/2)*(log(m^2/mu^2)+EulerGamma-3/2) +common_identities: + - dI3/d(m^2)=-I2 + - Delta(1/G)=N*((1-6*xi)/(12*pi))*I2 + - Delta(rho)=-(N/2)*(4*pi)^-2*I3 + - 1/G_total=B+Delta(1/G)_with_B_independent +exclusions: + - varying_V_double_prime_background_factorization + - full_determinant_or_nonlocal_action + - regulator_or_subtraction_condition_selection + - cutoff_to_substrate_scale_identification + - additive_baseline_selection + - total_Newton_constant_or_attractive_gravity + - tau_minus_one_higher_curvature_sector + - empirical_gravity_comparator diff --git a/campaigns/P230-exact-mass-regulator-rung/evidence/literature-audit.yaml b/campaigns/P230-exact-mass-regulator-rung/evidence/literature-audit.yaml new file mode 100644 index 00000000..ce20b778 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/evidence/literature-audit.yaml @@ -0,0 +1,33 @@ +campaign: P230 +primary_sources: + - citation: D.V. Vassilevich, Heat kernel expansion user's manual, arXiv:hep-th/0306138v3 + url: https://arxiv.org/abs/hep-th/0306138 + checked_equations: [1.16, 1.17, 1.18, 1.20, 2.2, 4.26, 4.27] + imported_scope: + - real_scalar_one_loop_determinant_weight + - proper_time_sign_and_cutoff_representation + - Laplace_type_operator_convention + - boundaryless_a0_and_a2_coefficients + nonimports: + - physical_scalar_identification + - regulator_selection + - full_determinant_from_a_finite_heat_kernel_truncation + - citation: M. Visser, Sakharov's induced gravity a modern perspective, arXiv:gr-qc/0204062v1 + url: https://arxiv.org/abs/gr-qc/0204062 + checked_equations: [7, 8, 9, 11, 13, 14, 15] + imported_scope: + - cutoff_effective_action_organization + - curvature_vacuum_and_higher_curvature_sector_separation + - independent_cosmological_and_higher_curvature_obligations + nonimports: + - zero_tree_level_gravity + - Planck_cutoff_identification + - physical_spectrum + - ignored_cosmological_or_higher_curvature_terms +source_consistency: + result: pass + note: >- + Vassilevich uses D=-(nabla^2+E), so D=-nabla^2+xi*R+m^2 + corresponds to E=-xi*R-m^2 and a2 density E+R/6=(1/6-xi)R-m^2. + P230 instead factors constant exp(-tau*m^2), leaving curvature weight + (1/6-xi)R and the exact derivative bridge to the unresummed mass term. diff --git a/campaigns/P230-exact-mass-regulator-rung/evidence/nonduplication-audit.yaml b/campaigns/P230-exact-mass-regulator-rung/evidence/nonduplication-audit.yaml new file mode 100644 index 00000000..ad9c737c --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/evidence/nonduplication-audit.yaml @@ -0,0 +1,16 @@ +campaign: P230 +identifier_search: + family: C-IGR-001_through_C-IGR-003 + prior_accepted_or_rejected_uses: 0 +nearest_claims: + C-GRV-001: dimensional_and_additive_baseline_ledger_only + C-VAC-002: conditional_Dirac_gauge_polarization_not_scalar_gravity_coefficients + C-VAC-003: conditional_complex_scalar_gauge_polarization_not_curved_scalar_heat_kernel +nearest_modules: + scalar_induced_newton: leading_massless_sharp_asymptotic_only_and_unpromoted + covariant_sine_gordon_action: action_and_fluctuation_slot_only_and_unpromoted +novel_surface: >- + P230 is the first accepted candidate to expose exact constant-mass I2/I3 + families for sharp, smooth, and declared power-subtracted prescriptions, + their mass-resummed vacuum composition, and their exact scheme ceiling. +result: nonduplicate_narrow_conditional_extension diff --git a/campaigns/P230-exact-mass-regulator-rung/evidence/primary-provenance.yaml b/campaigns/P230-exact-mass-regulator-rung/evidence/primary-provenance.yaml new file mode 100644 index 00000000..965f407d --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/evidence/primary-provenance.yaml @@ -0,0 +1,19 @@ +campaign: P230 +canonical_issue: 76 +source_prs: [77, 82] +promotion_branch: research/p230-exact-mass-promotion +base_commit: 0beaac3 +implementation_source_commit: d415caf +implementation_cherry_pick: a309008 +reviewed_failure_commit: 04f26f2 +repaired_source_commit: a8a47f8 +formula_freeze: evidence/formula-freeze.yaml +proposal: proposal.yaml +primary_verifier: verify.py +independent_rederivation: reviews/independent_exact_mass_review.py +claim_ids: [C-IGR-001, C-IGR-002, C-IGR-003] +authorship: + original_implementation: prime-agent + promotion_scope_and_oracles: root-agent + distinct_merger: pending_handoff +non_self_merge: enforced diff --git a/campaigns/P230-exact-mass-regulator-rung/proposal.yaml b/campaigns/P230-exact-mass-regulator-rung/proposal.yaml new file mode 100644 index 00000000..04e0f0c6 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/proposal.yaml @@ -0,0 +1,78 @@ +id: P230 +base_release: v0.160.0 +source_baseline: substrate-framework@0beaac3 +question: >- + For a declared positive self-adjoint boundaryless four-dimensional + Euclidean real-scalar Laplace-type operator + D_E=-nabla_E^2+xi*R_E+m^2 with constant + nonnegative m^2, what are the exact tau^-2 curvature-class and tau^-3 + vacuum-class one-loop proper-time coefficients under three preregistered + regulator prescriptions, and what conditional additive inverse-Newton and + vacuum shifts follow in the declared determinant and Einstein-Hilbert + conventions? +source_inventory: + - source: accepted C-GRV-001 + access: governance/claims.yaml and campaigns/P074-as3-induced-gravity-scaling + extracted: conditional additive inverse-coupling ledger, independent baseline, and the explicit ceiling that field content, coefficient, regulator, and cutoff identification remain premises + - source: landed conditional module src/substrate_framework/scalar_induced_newton.py + access: repository source at 0beaac3; PR #14 promoted no claims + extracted: reusable regulator tag and convention ledger only; all load-bearing coefficients are independently rederived for P230 + - source: landed conditional module src/substrate_framework/covariant_sine_gordon_action.py + access: repository source at 0beaac3; PR #25 promoted no claims + extracted: optional constant-vacuum composition V''(0)=m^2 only; no arbitrary-background mass factorization is imported + - source: Vassilevich arXiv:hep-th/0306138v3 equations 1.16-1.20, 2.2, and 4.27 + access: public primary source, independently checked + extracted: real-scalar determinant weight, proper-time sign, Laplace-type convention, and boundaryless a0/a2 heat-kernel coefficients + - source: Visser arXiv:gr-qc/0204062v1 equations 7-15 + access: public primary source, independently checked + extracted: cutoff effective-action organization, curvature and vacuum classes, and the need to retain cosmological and higher-curvature sectors +invariants: + - the determinant reading assumes a positive self-adjoint operator and the required infrared convergence or reference subtraction; only the displayed local coefficient families are claimed, not the full nonlocal determinant + - m^2 is a spacetime-constant nonnegative exact input; a varying V''(phi_bg(x)) is outside the factorized theorem + - the operator, real-scalar determinant weight 1/2, four-dimensional heat-kernel prefactor (4*pi)^-2, and Euclidean Einstein-Hilbert matching convention are declared conditional inputs + - mass is retained exactly in exp(-tau*m^2); the unresummed -m^2 coefficient is not added again + - regulator, cutoff, subtraction scale, field count, xi, additive baseline, and any physical scale identification remain explicit inputs + - no observed G, Planck scale, or empirical gravity comparator enters selection, formulas, tolerances, or tests + - accepted C-GRV-001 supplies only dimensional and additive-baseline closure; no landed-but-unpromoted API is treated as scientific authority + - accepted claims, releases, and generated docs change only in one reviewed promotion transaction +allowed_imports: + - C-GRV-001 + - a separately declared positive self-adjoint boundaryless four-dimensional Euclidean real-scalar Laplace-type operator with constant nonnegative mass and adequate infrared/reference treatment + - the one-real-scalar determinant, heat-kernel, and Einstein-Hilbert coefficient conventions stated in source_inventory + - SymPy exact algebra, expint E1, modified Bessel K, limits, and series + - mpmath high-precision adaptive quadrature as independent regression evidence + - substrate_framework.exact_symbolic input contracts + - the scalar_induced_newton regulator tag and API only as implementation reuse checked against an independent derivation +candidates: + - id: A + description: >- + Sharp proper-time lower cutoff tau_0=Lambda^-2 with exact mass, + I2=Lambda^2*(exp(-z)-z*E1(z)) and its linked I3 tail integral. + - id: B + description: >- + Smooth essential-singularity weight exp(-1/(Lambda^2*tau)) with exact + mass, giving the convergent modified-Bessel integral family. + - id: C + description: >- + Power-divergence-subtracted cutoff finite parts with declared scale mu, + preserving the logarithmic I2/I3 family and its scale dependence. +selection_criteria: + - exact agreement with the defining integrals or declared subtraction limits + - constant-mass and convention consistency without hidden background dependence + - dimensional consistency and exact massless, large-mass, and derivative limits + - independent rederivation without importing scalar_one_loop_mass + - sensitivity to prefactor, sign, special-function order or branch, double counting, and scale mutations + - explicit regulator and baseline provenance with no physical selection by numerical closeness + - importable API reuse and complete consumer/governance replay +claims_proposed: + - C-IGR-001 + - C-IGR-002 + - C-IGR-003 +comparators_blinded_until: >- + No empirical gravitational comparator belongs to P230. Existing PR #77 and + #82 text exposed internal exact values such as z=1 scheme ratios before this + promotion review; those are formula evaluations, not empirical comparators, + and they do not select a regulator or tolerance. Physical comparators remain + closed until a later renormalization-condition proposal freezes regulator, + scale, cutoff identification, baseline, and field content. +status: accepted diff --git a/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-001-claim-review.md b/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-001-claim-review.md new file mode 100644 index 00000000..e5979978 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-001-claim-review.md @@ -0,0 +1,40 @@ +# C-IGR-001 Claim Review + +## Claim Under Review +The claim is the sharp-cutoff constant-mass theorem. Conditional on the frozen P230 real-scalar determinant and heat-kernel conventions, for positive cutoff `Lambda`, constant `m2>=0`, and `z=m2/Lambda^2`, the exact local coefficient integrals are `I2=Lambda^2*(exp(-z)-z*E1(z))` and `I3=Lambda^4*exp(-z)/2-m2*I2/2`. They satisfy `dI3/dm2=-I2`, have massless limits `Lambda^2` and `Lambda^4/2`, and decay for large mass. The displayed inverse-Newton and vacuum shifts are additive conditional coefficients only. + +## Sourced Inputs +The review read accepted release v0.160.0, C-GRV-001, the corrected P230 proposal and formula freeze, Vassilevich equations 1.16-1.20, 2.2, 4.26-4.27, Visser equations 7-15, PR #77's failed and repaired heads, PR #82's focused implementation, the canonical module, and all P230 executable evidence. + +## Independence +The independent review script imports no `scalar_one_loop_mass` symbol. It solves the sharp tail problem from the boundary derivative, derives I3 by integration by parts, derives the determinant and Einstein-Hilbert factors separately, and only then compares the resulting formulas to the implementation through the primary verifier and tests. + +## Verification Status +The claim earns `symbolic_verified`. Exact differentiation proves the I2 tail equation and the I3 derivative bridge. The positive defining integral and the bound by `exp(-z)/z` prove the large-mass I2 limit without relying on SymPy's unevaluated E1 limit. Exact massless limits and dimensions are checked separately. Numerical values are regression evidence only. + +## Sensitivity and Counterexamples +Doubling the prefactor or replacing E1 by Ei breaks the tail derivative oracle. Adding `m2*I2` to the mass-resummed vacuum coefficient breaks the determinant-integrand oracle by a nonzero exact term. Flipping the proper-time determinant sign breaks the independent coefficient check. A varying `V''(phi_bg(x))` is a counterexample to factorization and is excluded by the constant-mass API and statement. + +## Framework Compatibility +The claim is a compatible conditional extension of C-GRV-001. It preserves the independent additive baseline and treats the field content, positive self-adjoint operator, infrared/reference treatment, regulator, cutoff, and physical interpretation as premises. It establishes neither a total Newton constant nor a full or nonlocal effective action. + +## Dependency and Consumer Replay +The accepted dependency is C-GRV-001 for the additive ledger only. The QFT conventions are explicit approved inputs, not hidden accepted claims. Direct consumers are the canonical module exports, tests, and P230 verifier; generated governance and memory consumers are replayed in the promotion transaction. No existing runtime caller depends on the API. + +## Competing Candidate Audit +Sharp, smooth, and power-subtracted prescriptions were registered together before empirical comparators. Candidate A is retained for its literal positive tail-integral definition, not because its value is close to a physical target. Candidates B and C remain separate accepted conditional claims. + +## Four-Axis Decision +Verification is `symbolic_verified`; review is `accepted`; compatibility is `compatible_extension`; epistemic status is `active` within the declared conditional model. The relationship is a new claim depending on C-GRV-001, with no challenge or supersession edge. + +## Promotion Transaction +The transaction materializes the P230 campaign, canonical API and tests, this review, the registry entry, release v0.161.0, generated docs and accepted memory, and the full replay. The source PRs remain provenance, not authority. + +## Continuation if Not Accepted +Not applicable after the exact constant-mass narrowing. Any future varying-mass, nonlocal, physical-regulator, total-coupling, or higher-curvature statement requires a separate proposal and evidence. + +## Done Gate +The claim is accepted only together with the final status-zero campaign verifiers, full repository validation, generated-state checks, and empty P230 debt ledger. + +## Cross-References +See P230 formula freeze, `verify.py`, `independent_exact_mass_review.py`, PRs #77/#82, C-GRV-001, `scalar_one_loop_mass.py`, and `tests/test_scalar_one_loop_mass.py`. diff --git a/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-002-claim-review.md b/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-002-claim-review.md new file mode 100644 index 00000000..f4d19a15 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-002-claim-review.md @@ -0,0 +1,40 @@ +# C-IGR-002 Claim Review + +## Claim Under Review +The claim is the smooth-weight constant-mass theorem. With P230's declared operator and positive cutoff, weight `exp(-1/(Lambda^2*tau))`, constant `m2>=0`, and `z=m2/Lambda^2`, the exact local coefficient integrals are `I2=2*Lambda^2*sqrt(z)*K1(2*sqrt(z))` and `I3=2*Lambda^4*z*K2(2*sqrt(z))`, continuously extended at zero. They satisfy `dI3/dm2=-I2`, have massless limits `Lambda^2` and `Lambda^4`, and decay for large mass. Their conditional additive shifts use the same independently typed determinant and matching factors. + +## Sourced Inputs +The review read v0.160.0, C-GRV-001, the P230 proposal and formula freeze, the primary heat-kernel sources, the reviewed PR implementations, the module, tests, and both campaign verifiers. + +## Independence +The independent script instantiates the standard modified-Bessel integral family directly, proves its differential recurrence, boundary data, and large-mass behavior, and corroborates both integrals by 60-digit adaptive quadrature at three preregistered positive z values. It imports no P230 scientific implementation. + +## Verification Status +The claim earns `symbolic_verified`. The standard exact K-integral identity is an explicit mathematical import; raw SymPy differentiation verifies the required K1/K2 recurrence and exact boundary and decay limits. High-precision quadrature is corroboration rather than the source of the verdict. + +## Sensitivity and Counterexamples +Replacing K1 by K2 in I2 breaks the exact differential recurrence. A wrong determinant sign breaks the independently typed vacuum coefficient. Removing the constant-mass premise would introduce derivative endomorphism terms and invalidate factorization; that stronger statement is excluded. + +## Framework Compatibility +The claim is a compatible conditional extension. It introduces an explicit smooth regulator premise rather than presenting the result as regulator independent. The exact massless I3 value differs from the sharp value by a factor of two, so no universal normalization is inferred. C-GRV-001's independent baseline remains untouched. + +## Dependency and Consumer Replay +The accepted dependency is C-GRV-001 only for the additive ledger. The smooth weight, determinant convention, positive self-adjoint operator, and infrared/reference treatment are declared inputs. Consumers are the canonical module, package exports, tests, verifier, generated docs, and accepted memory; no preexisting runtime process is affected. + +## Competing Candidate Audit +Candidate B survives on exact structural criteria. It is retained alongside A and C because P230 compares declared schemes; no numeric closeness or empirical gravity value selects it. + +## Four-Axis Decision +Verification is `symbolic_verified`; review is `accepted`; compatibility is `compatible_extension`; epistemic status is `active` within its conditional model. The relationship is a new claim depending on C-GRV-001, with no challenge or supersession. + +## Promotion Transaction +The implementation, tests, immutable campaign, this review, registry, release v0.161.0, generated docs, accepted memory, and full validation form one promotion boundary. + +## Continuation if Not Accepted +Not applicable. Extensions to other weights, varying masses, full determinants, or physical regulator choice require new claims. + +## Done Gate +Acceptance is conditional on the final integrated status-zero gate and empty debt ledger recorded in P230 attempt 0003. + +## Cross-References +See P230, C-IGR-001, C-IGR-003, the two campaign verifiers, `scalar_one_loop_mass.py`, and PRs #77/#82. diff --git a/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-003-claim-review.md b/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-003-claim-review.md new file mode 100644 index 00000000..606fb5b9 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-003-claim-review.md @@ -0,0 +1,40 @@ +# C-IGR-003 Claim Review + +## Claim Under Review +The claim is the declared power-subtracted constant-mass finite-part theorem and scheme ceiling. For positive scale `mu` and constant `m2>=0`, P230 defines `I2=m2*(log(m2/mu^2)+EulerGamma-1)` and `I3=-(m2^2/2)*(log(m2/mu^2)+EulerGamma-3/2)`, continuously extended by zero at `m2=0`. They are the exact finite parts after the frozen sharp-cutoff power and logarithmic subtractions, satisfy `dI3/dm2=-I2`, and retain exact scale derivatives. Together with the sharp and smooth families they prove that P230 supplies no scheme-independent physical normalization. + +## Sourced Inputs +The review read C-GRV-001, Vassilevich's zeta/proper-time relations and heat-kernel convention, Visser's cutoff sector organization, the P230 finite-part definition, PR #77's sign defect and repair, PR #82, the module, tests, and independent raw-limit script. + +## Independence +The independent review begins with the sharp tail expressions and takes the specified cutoff subtraction limits directly. It then inserts the declared scale, differentiates the resulting I2/I3 family, types the coefficient from the determinant and Einstein-Hilbert factors, and evaluates the sign counterexample without importing the implementation. + +## Verification Status +The claim earns `symbolic_verified`. SymPy evaluates the two exact cutoff finite-part limits and the derivative and scale identities. The zero-mass branch is proved by one-sided limits; direct substitution into `m2*log(m2)` is intentionally not used. + +## Sensitivity and Counterexamples +The original sign field is falsified at `m2=mu^2=1`, `xi=0`: the curvature weight is positive while the returned value is `(EulerGamma-1)/(12*pi)<0`. The massless continuous branch is zero. A shifted finite subtraction changes the frozen scheme and is not the same claim. Passing both cutoff and scale or defaulting the scale is rejected by the API. + +## Framework Compatibility +The claim explicitly exposes scale and subtraction dependence. It does not call the finite part regulator free, select a renormalization condition, erase a finite counterterm or C-GRV-001 baseline, or identify a physical cutoff. At unit cutoff and zero mass the vacuum-class sharp, smooth, and power-subtracted values are exactly `1/2`, `1`, and `0`, establishing the scheme ceiling without a comparator. + +## Dependency and Consumer Replay +The accepted dependency is C-GRV-001 for the independent baseline ledger. The finite-part prescription and scale are approved declared inputs. Consumers are the canonical API, tests, P230 verifier, docs, registry, release, and accepted memory; the final replay checks every one. + +## Competing Candidate Audit +Candidate C was preregistered alongside A and B and is retained because it exposes a distinct exact subtraction family. It is not selected as the physical regulator; all three remain available with their assumptions and contrasts. + +## Four-Axis Decision +Verification is `symbolic_verified`; review is `accepted`; compatibility is `compatible_extension`; epistemic status is `active` conditionally. The relationship is a new claim depending on C-GRV-001, with no challenge or supersession. + +## Promotion Transaction +The accepted boundary includes the constant-mass code correction, raw-limit reviewer, this claim review, immutable P230 campaign, registry, release v0.161.0, generated docs and memory, and full validation. + +## Continuation if Not Accepted +Not applicable. A selected physical scheme, matching condition, total coupling, or comparator requires a later proposal. + +## Done Gate +Acceptance requires the recorded integrated gate and empty debt ledger; a passing finite-part identity alone is insufficient. + +## Cross-References +See P230 formula freeze and literature audit, C-IGR-001/002, PRs #77/#82, both campaign verifiers, and C-GRV-001. diff --git a/campaigns/P230-exact-mass-regulator-rung/reviews/impact_analysis.md b/campaigns/P230-exact-mass-regulator-rung/reviews/impact_analysis.md new file mode 100644 index 00000000..08b7e3a6 --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/reviews/impact_analysis.md @@ -0,0 +1,7 @@ +# P230 Impact Analysis + +The canonical change adds one exact conditional module and three accepted claims. GitNexus on the reviewed PR #82 implementation reported LOW runtime risk, zero affected execution flows, and no upstream runtime consumers of the composed APIs. P230 removes the module's scientific call to the unpromoted leading-coefficient API and rederives the factor locally, so the only retained cross-module runtime dependency is the sharp-regulator string tag. + +Direct consumers are the package export table, `tests/test_scalar_one_loop_mass.py`, and the P230 primary verifier. Indirect promotion consumers are the registry, v0.161.0 release, generated claim index, and generated accepted memory. No existing public contract changes; the new APIs remain additive. Because claim governance, release state, generated consumers, and a new accepted scientific module all change together, the final validation boundary is full despite the LOW runtime graph risk. + +The final indexed compare reports LOW risk and zero affected processes. Upstream impact finds one internal direct caller for each integral (`exact_mass_vacuum_density_shift`) and no callers for either composed shift or the scheme ledger. The graph also reports an AGENTS instruction symbol because its analyzer temporarily injected a block; source `git diff` confirms that block was removed and AGENTS is unchanged. Governance files exceed GitNexus's default size cap, so registry, release, docs, and memory consumers are covered by the repository validator and full suite rather than omitted from the dependency replay. diff --git a/campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py b/campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py new file mode 100644 index 00000000..167d024d --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py @@ -0,0 +1,192 @@ +#!/usr/bin/env python3 +"""Independent raw-algebra review of C-IGR-001 through C-IGR-003. + +This script imports no scalar_one_loop_mass implementation symbol. It builds +the coefficient families, determinant factors, mutations, and quadrature +comparisons directly from the frozen proposal conventions. +""" + +from __future__ import annotations + +import mpmath as mp +import sympy as sp + +from substrate_framework.verification import CheckLedger + + +def run() -> int: + checks = CheckLedger("P230/independent-C-IGR-review") + tau0, mass2, cutoff, scale, z = sp.symbols( + "tau0 m2 Lambda mu z", positive=True + ) + + # C-IGR-001: solve the sharp tail problem independently. + sharp2_tau = ( + sp.exp(-mass2 * tau0) + - mass2 * tau0 * sp.expint(1, mass2 * tau0) + ) / tau0 + checks.check( + "C-IGR-001 raw tail derivative", + sp.simplify( + sp.diff(sharp2_tau, tau0) + + tau0**-2 * sp.exp(-mass2 * tau0) + ) + == 0, + ) + sharp2 = sp.simplify(sharp2_tau.subs(tau0, cutoff**-2)) + sharp3 = sp.simplify( + sp.exp(-mass2 / cutoff**2) * cutoff**4 / 2 + - mass2 * sharp2 / 2 + ) + checks.check( + "C-IGR-001 raw I3 derivative and massless limits", + sp.simplify(sp.diff(sharp3, mass2) + sharp2) == 0 + and sp.limit(sharp2, mass2, 0, "+") == cutoff**2 + and sp.limit(sharp3, mass2, 0, "+") == cutoff**4 / 2, + ) + wrong_branch = ( + sp.exp(-mass2 * tau0) + - mass2 * tau0 * sp.Ei(mass2 * tau0) + ) / tau0 + checks.mutation_sensitive( + "C-IGR-001 E1 branch is load-bearing", + lambda candidate: sp.simplify( + sp.diff(candidate, tau0) + + tau0**-2 * sp.exp(-mass2 * tau0) + ) + == 0, + sharp2_tau, + (wrong_branch,), + ) + + # C-IGR-002: independently instantiate the standard K integral family. + smooth1 = 2 * sp.besselk(0, 2 * sp.sqrt(z)) + smooth2_unit = 2 * sp.sqrt(z) * sp.besselk(1, 2 * sp.sqrt(z)) + smooth3_unit = 2 * z * sp.besselk(2, 2 * sp.sqrt(z)) + checks.check( + "C-IGR-002 raw Bessel recurrence", + sp.simplify(sp.diff(smooth2_unit, z) + smooth1) == 0 + and sp.simplify(sp.diff(smooth3_unit, z) + smooth2_unit) == 0, + ) + checks.check( + "C-IGR-002 raw boundary and decay data", + sp.limit(smooth2_unit, z, 0, "+") == 1 + and sp.limit(smooth3_unit, z, 0, "+") == 1 + and sp.limit(smooth2_unit, z, sp.oo) == 0 + and sp.limit(smooth3_unit, z, sp.oo) == 0, + ) + wrong_order = 2 * sp.sqrt(z) * sp.besselk(2, 2 * sp.sqrt(z)) + checks.mutation_sensitive( + "C-IGR-002 Bessel order is load-bearing", + lambda candidate: sp.simplify(sp.diff(candidate, z) + smooth1) == 0, + smooth2_unit, + (wrong_order,), + ) + + mp.mp.dps = 60 + for z_text in ("0.1", "1.0", "2.5"): + z_value = mp.mpf(z_text) + integral2 = mp.quad( + lambda t: mp.exp(-z_value * t - 1 / t) / t**2, + [0, 1, mp.inf], + ) + integral3 = mp.quad( + lambda t: mp.exp(-z_value * t - 1 / t) / t**3, + [0, 1, mp.inf], + ) + closed2 = 2 * mp.sqrt(z_value) * mp.besselk(1, 2 * mp.sqrt(z_value)) + closed3 = 2 * z_value * mp.besselk(2, 2 * mp.sqrt(z_value)) + checks.check( + f"C-IGR-002 independent 60-digit quadrature z={z_text}", + abs(integral2 - closed2) < mp.mpf("1e-48") * abs(closed2) + and abs(integral3 - closed3) < mp.mpf("1e-48") * abs(closed3), + ) + + # C-IGR-003: independently take the declared sharp-cutoff finite parts. + finite2_unscaled = sp.limit( + sharp2_tau - 1 / tau0 - mass2 * sp.log(tau0), + tau0, + 0, + "+", + ) + sharp3_tau = sp.exp(-mass2 * tau0) / (2 * tau0**2) - mass2 * sharp2_tau / 2 + finite3_unscaled = sp.limit( + sharp3_tau + - 1 / (2 * tau0**2) + + mass2 / tau0 + + mass2**2 * sp.log(tau0) / 2, + tau0, + 0, + "+", + ) + zeta2 = mass2 * ( + sp.log(mass2 / scale**2) + sp.EulerGamma - 1 + ) + zeta3 = -mass2**2 * ( + sp.log(mass2 / scale**2) + + sp.EulerGamma + - sp.Rational(3, 2) + ) / 2 + checks.check( + "C-IGR-003 raw cutoff finite parts", + sp.simplify( + finite2_unscaled + - mass2 * (sp.log(mass2) + sp.EulerGamma - 1) + ) + == 0 + and sp.simplify( + finite3_unscaled + + mass2**2 + * (sp.log(mass2) + sp.EulerGamma - sp.Rational(3, 2)) + / 2 + ) + == 0, + ) + checks.check( + "C-IGR-003 raw derivative and scale identities", + sp.simplify(sp.diff(zeta3, mass2) + zeta2) == 0 + and sp.simplify(scale * sp.diff(zeta2, scale) + 2 * mass2) == 0 + and sp.simplify(scale * sp.diff(zeta3, scale) - mass2**2) == 0, + ) + + # Independent coefficient typing and scheme ceiling. + xi = sp.Symbol("xi", real=True) + per_field = sp.simplify( + (16 * sp.pi) + * sp.Rational(1, 2) + * (4 * sp.pi) ** -2 + * (sp.Rational(1, 6) - xi) + ) + checks.check( + "conditional inverse-Newton coefficient is rederived", + sp.simplify(per_field - (1 - 6 * xi) / (12 * sp.pi)) == 0, + ) + vacuum_prefactor = -sp.Rational(1, 2) * (4 * sp.pi) ** -2 + checks.mutation_sensitive( + "proper-time determinant sign is load-bearing", + lambda candidate: sp.simplify(candidate - vacuum_prefactor) == 0, + vacuum_prefactor, + (-vacuum_prefactor,), + ) + checks.check( + "exact massless scheme spread forbids a universal regulator value", + sp.Rational(1, 2) != 1 + and sp.limit(zeta2, mass2, 0, "+") == 0 + and sp.limit(zeta3, mass2, 0, "+") == 0, + ) + negative_value = sp.simplify( + per_field.subs(xi, 0) * zeta2.subs({mass2: 1, scale: 1}) + ) + checks.check( + "power-subtracted value sign depends on the declared scale ratio", + negative_value.is_negative is True + and sp.simplify( + negative_value - (sp.EulerGamma - 1) / (12 * sp.pi) + ) + == 0, + ) + return checks.finish() + + +if __name__ == "__main__": + raise SystemExit(run()) diff --git a/campaigns/P230-exact-mass-regulator-rung/verify.py b/campaigns/P230-exact-mass-regulator-rung/verify.py new file mode 100644 index 00000000..eece684f --- /dev/null +++ b/campaigns/P230-exact-mass-regulator-rung/verify.py @@ -0,0 +1,272 @@ +#!/usr/bin/env python3 +"""Primary exact verifier for C-IGR-001 through C-IGR-003.""" + +from __future__ import annotations + +import inspect + +import sympy as sp + +from substrate_framework import scalar_one_loop_mass as implementation +from substrate_framework.scalar_one_loop_mass import ( + SMOOTH_PROPER_TIME_REGULATOR, + ZETA_POWER_SUBTRACTED_REGULATOR, + curvature_proper_time_integral, + exact_mass_inverse_newton_shift, + exact_mass_vacuum_density_shift, + regulator_scheme_ledger, + vacuum_proper_time_integral, +) +from substrate_framework.scalar_induced_newton import SHARP_PROPER_TIME_REGULATOR +from substrate_framework.verification import CheckLedger + + +def run() -> int: + checks = CheckLedger("P230/C-IGR") + cutoff, scale, mass2 = sp.symbols("Lambda mu m2", positive=True) + tau0, z = sp.symbols("tau0 z", positive=True) + + sharp2 = curvature_proper_time_integral( + SHARP_PROPER_TIME_REGULATOR, + cutoff=cutoff, + mass_squared=mass2, + ) + sharp2_expected = cutoff**2 * ( + sp.exp(-mass2 / cutoff**2) + - mass2 * sp.expint(1, mass2 / cutoff**2) / cutoff**2 + ) + checks.check( + "C-IGR-001 sharp I2 closed form", + sp.simplify(sharp2 - sharp2_expected) == 0, + ) + sharp_tail = ( + sp.exp(-mass2 * tau0) + - mass2 * tau0 * sp.expint(1, mass2 * tau0) + ) / tau0 + checks.check( + "C-IGR-001 sharp tail derivative fixes the integral", + sp.simplify( + sp.diff(sharp_tail, tau0) + + tau0**-2 * sp.exp(-mass2 * tau0) + ) + == 0, + ) + u = sp.Symbol("u", nonnegative=True) + checks.check( + "C-IGR-001 large-mass tail is squeezed to zero", + sp.factor(1 - 1 / (1 + u) ** 2).is_nonnegative is True + and sp.limit(sp.exp(-z) / z, z, sp.oo) == 0, + ) + + sharp3 = vacuum_proper_time_integral( + SHARP_PROPER_TIME_REGULATOR, + cutoff=cutoff, + mass_squared=mass2, + ) + checks.check( + "C-IGR-001 sharp I3 integration-by-parts form", + sp.simplify( + sharp3 + - sp.exp(-mass2 / cutoff**2) * cutoff**4 / 2 + + mass2 * sharp2 / 2 + ) + == 0, + ) + + smooth2 = curvature_proper_time_integral( + SMOOTH_PROPER_TIME_REGULATOR, + cutoff=cutoff, + mass_squared=mass2, + ) + smooth3 = vacuum_proper_time_integral( + SMOOTH_PROPER_TIME_REGULATOR, + cutoff=cutoff, + mass_squared=mass2, + ) + dimensionless2 = 2 * sp.sqrt(z) * sp.besselk(1, 2 * sp.sqrt(z)) + dimensionless3 = 2 * z * sp.besselk(2, 2 * sp.sqrt(z)) + checks.check( + "C-IGR-002 smooth Bessel closed forms", + sp.simplify(smooth2 - cutoff**2 * dimensionless2.subs(z, mass2 / cutoff**2)) + == 0 + and sp.simplify( + smooth3 - cutoff**4 * dimensionless3.subs(z, mass2 / cutoff**2) + ) + == 0, + ) + checks.check( + "C-IGR-002 smooth recurrence and massless boundaries", + sp.simplify( + sp.diff(dimensionless3, z) + dimensionless2 + ) + == 0 + and sp.limit(dimensionless2, z, 0, "+") == 1 + and sp.limit(dimensionless3, z, 0, "+") == 1, + ) + checks.check( + "C-IGR-002 smooth large-mass limits", + sp.limit(dimensionless2, z, sp.oo) == 0 + and sp.limit(dimensionless3, z, sp.oo) == 0, + ) + + zeta2 = curvature_proper_time_integral( + ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=mass2, + renormalization_scale=scale, + ) + zeta3 = vacuum_proper_time_integral( + ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=mass2, + renormalization_scale=scale, + ) + checks.check( + "C-IGR-003 declared power-subtracted finite parts", + sp.simplify( + zeta2 + - mass2 + * (sp.log(mass2 / scale**2) + sp.EulerGamma - 1) + ) + == 0 + and sp.simplify( + zeta3 + + mass2**2 + * ( + sp.log(mass2 / scale**2) + + sp.EulerGamma + - sp.Rational(3, 2) + ) + / 2 + ) + == 0, + ) + checks.check( + "C-IGR-003 scale derivatives remain explicit", + sp.simplify(scale * sp.diff(zeta2, scale) + 2 * mass2) == 0 + and sp.simplify(scale * sp.diff(zeta3, scale) - mass2**2) == 0, + ) + + for regulator, kwargs in ( + ( + SHARP_PROPER_TIME_REGULATOR, + {"cutoff": cutoff, "mass_squared": mass2}, + ), + ( + SMOOTH_PROPER_TIME_REGULATOR, + {"cutoff": cutoff, "mass_squared": mass2}, + ), + ( + ZETA_POWER_SUBTRACTED_REGULATOR, + {"mass_squared": mass2, "renormalization_scale": scale}, + ), + ): + i2 = curvature_proper_time_integral(regulator, **kwargs) + i3 = vacuum_proper_time_integral(regulator, **kwargs) + checks.check( + f"{regulator} exact dI3/dm2 bridge", + sp.simplify(sp.diff(i3, mass2) + i2) == 0, + ) + + field_count, xi = sp.Integer(3), sp.Rational(1, 12) + shift = exact_mass_inverse_newton_shift( + field_count, + xi, + regulator=SHARP_PROPER_TIME_REGULATOR, + cutoff=cutoff, + mass_squared=0, + ) + independent_coefficient = sp.simplify( + field_count + * (16 * sp.pi) + * sp.Rational(1, 2) + * (4 * sp.pi) ** -2 + * (sp.Rational(1, 6) - xi) + ) + checks.check( + "conditional inverse-Newton factor is independently typed", + sp.simplify(shift.value - independent_coefficient * cutoff**2) == 0, + ) + + vacuum = exact_mass_vacuum_density_shift( + 2, + regulator=SHARP_PROPER_TIME_REGULATOR, + cutoff=cutoff, + mass_squared=mass2, + ) + expected_vacuum = -sp.Rational(2, 2) * (4 * sp.pi) ** -2 * sharp3 + checks.check( + "mass-resummed vacuum composition uses I3 once", + sp.simplify(vacuum.value - expected_vacuum) == 0, + ) + spurious = -sp.Rational(2, 2) * (4 * sp.pi) ** -2 * mass2 * sharp2 + checks.mutation_sensitive( + "vacuum mass double count is rejected", + lambda candidate: sp.simplify(candidate - expected_vacuum) == 0, + vacuum.value, + (vacuum.value + spurious,), + ) + + wrong_sharp = 2 * sharp_tail + checks.mutation_sensitive( + "sharp prefactor is load-bearing", + lambda candidate: sp.simplify( + sp.diff(candidate, tau0) + tau0**-2 * sp.exp(-mass2 * tau0) + ) + == 0, + sharp_tail, + (wrong_sharp,), + ) + wrong_order = 2 * sp.sqrt(z) * sp.besselk(2, 2 * sp.sqrt(z)) + smooth1 = 2 * sp.besselk(0, 2 * sp.sqrt(z)) + checks.mutation_sensitive( + "smooth Bessel order is load-bearing", + lambda candidate: sp.simplify(sp.diff(candidate, z) + smooth1) == 0, + dimensionless2, + (wrong_order,), + ) + + negative_zeta = exact_mass_inverse_newton_shift( + 1, + 0, + regulator=ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=1, + renormalization_scale=1, + ) + checks.check( + "zeta full-value sign is not inferred from curvature weight", + negative_zeta.curvature_weight_sign == 1 + and negative_zeta.value_sign == -1, + ) + ledger = regulator_scheme_ledger(1, 0, 1) + checks.check( + "massless scheme spread is exact and regulator selection stays open", + ledger.sharp_value == 1 + and ledger.smooth_value == 1 + and ledger.zeta_value == 0 + and vacuum_proper_time_integral( + SHARP_PROPER_TIME_REGULATOR, cutoff=1, mass_squared=0 + ) + == sp.Rational(1, 2) + and vacuum_proper_time_integral( + SMOOTH_PROPER_TIME_REGULATOR, cutoff=1, mass_squared=0 + ) + == 1, + ) + + source = inspect.getsource(implementation) + signature = inspect.signature(curvature_proper_time_integral) + checks.check( + "constant-mass scope and implementation independence are explicit", + "spacetime-constant nonnegative" in source + and "varying effective mass" in source + and "leading_scalar_newton_shift_coefficient" not in source + and "background" not in signature.parameters, + ) + checks.check( + "no empirical comparator or legacy trapezoid access enters", + all(token not in source for token in ("6.674", "M_pl", "np.trapz", 'getattr(np, "trapz")')), + ) + return checks.finish() + + +if __name__ == "__main__": + raise SystemExit(run()) diff --git a/docs/generated/claim-index.md b/docs/generated/claim-index.md index 5d41e1a5..4573c2a0 100644 --- a/docs/generated/claim-index.md +++ b/docs/generated/claim-index.md @@ -534,6 +534,33 @@ Let G_r=G/G_ref, L_r=L/a_ref, C, and b0 be separately supplied positive exact di - Compatibility: `compatible_extension` - Dependencies: C-GRV-001, C-RGE-003, C-IDN-001 +## C-IGR-001 + +Conditional on C-GRV-001's independent additive inverse-coupling baseline ledger, separately declare a positive self-adjoint boundaryless four-dimensional Euclidean real-scalar Laplace-type operator D_E=-nabla_E^2+xi*R_E+m2 with spacetime-constant m2>=0, adequate infrared convergence or reference subtraction, Gamma_E=(1/2)*ln(det(D_E)), heat-kernel prefactor (4*pi)^-2, and Euclidean Einstein-Hilbert matching factor 16*pi. For a positive sharp proper-time cutoff Lambda with tau0=Lambda^-2 and z=m2/Lambda^2, the exact local curvature and vacuum coefficient integrals are I2=Lambda^2*(exp(-z)-z*E1(z)) and I3=Lambda^4*exp(-z)/2-m2*I2/2, continuously extended at m2=0. They obey dI3/dm2=-I2, have massless values Lambda^2 and Lambda^4/2, and decay in the large-mass limit. For positive integer N and exact xi, the declared conditional additive coefficients are Delta(1/G)=N*(1-6*xi)*I2/(12*pi) and Delta(rho)=-(N/2)*(4*pi)^-2*I3. These are exact local coefficient families, not a full or nonlocal determinant, varying-mass theorem, selected regulator, cutoff ontology, total Newton constant, attractive- gravity result, sourced geometry, or empirical prediction. + +- Accepted in: `v0.161.0` +- Verification: `symbolic_verified` +- Compatibility: `compatible_extension` +- Dependencies: C-GRV-001 + +## C-IGR-002 + +Under C-IGR-001's declared operator, determinant, local-coefficient, matching, constant-mass, and baseline assumptions, replace the sharp cutoff by the explicitly declared smooth proper-time weight exp(-1/(Lambda^2*tau)). For positive Lambda, m2>=0, and z=m2/Lambda^2, the exact coefficient integrals are I2=2*Lambda^2*sqrt(z)*K1(2*sqrt(z)) and I3=2*Lambda^4*z*K2(2*sqrt(z)), continuously extended at m2=0. They obey dI3/dm2=-I2, have massless values Lambda^2 and Lambda^4, and decay for large mass. The conditional additive coefficients are the same typed compositions Delta(1/G)=N*(1-6*xi)*I2/(12*pi) and Delta(rho)=-(N/2)*(4*pi)^-2*I3. The exact massless I3 value is twice C-IGR-001's sharp value at the same Lambda, so this claim exposes rather than removes regulator dependence and establishes no physical regulator selection or total gravitational normalization. + +- Accepted in: `v0.161.0` +- Verification: `symbolic_verified` +- Compatibility: `compatible_extension` +- Dependencies: C-GRV-001, C-IGR-001 + +## C-IGR-003 + +Under C-IGR-001's declared operator, determinant, local-coefficient, matching, constant-mass, and baseline assumptions, declare instead the power-subtracted sharp-cutoff finite-part prescription at a positive scale mu. For constant m2>=0 its exact continuously extended coefficient family is I2=m2*(log(m2/mu^2)+EulerGamma-1) and I3=-(m2^2/2)*(log(m2/mu^2)+EulerGamma-3/2). These are the finite limits after the P230-frozen power and logarithmic subtractions, obey dI3/dm2=-I2, mu*dI2/dmu=-2*m2, and mu*dI3/dmu=m2^2, and vanish continuously at m2=0. The conditional additive shifts use the same typed factors as C-IGR-001. At unit scale and cutoff with m2=0, the sharp, smooth, and power-subtracted I3 values are exactly 1/2, 1, and 0; moreover the finite-part I2 can change sign with m2/mu^2. Therefore P230 derives exact scheme and scale dependence, not a regulator-free coefficient, selected subtraction, finite counterterm, total Newton constant, or physical normalization. + +- Accepted in: `v0.161.0` +- Verification: `symbolic_verified` +- Compatibility: `compatible_extension` +- Dependencies: C-GRV-001, C-IGR-001, C-IGR-002 + ## C-IRR-001 In the standard fundamental SU(3) convention of C-LIE-001 with Y=2*T_8/sqrt(3), let p and q be arbitrary nonnegative integer Dynkin labels and use the U(3) Gelfand-Tsetlin top row (p+q,q,0). The exact Weyl dimension is (p+1)(q+1)(p+q+2)/2, the quadratic Casimir is (p^2+p*q+q^2+3*p+3*q)/3, and the C-LIE-002 center triality is p+2*q modulo three. Every interlacing pattern p+q>=m12>=q>=m22>=0 and m12>=m11>=m22 gives one basis state with I=(m12-m22)/2, I3=m11-(m12+m22)/2, and Y=m12+m22-2*(p+2*q)/3; these patterns are complete, their count equals the Weyl dimension, and aggregation gives the exact weight multiplicities and multiplicity-free SU(2)xU(1) branching rows. Conditional on an exact rational target hypercharge and explicitly supplied finite nonnegative bounds max_p and max_q, enumerating all labels in the rectangular domain returns every irrep containing that hypercharge and preserves all minimum-dimension ties. At target Y=1 the unique global minimum is (1,1) of dimension eight with I=1/2; the next distinct dimension is ten and contains both (0,3) with I=1/2 and (3,0) with I=3/2. This is a mathematical representation and kinematic-filter theorem only. It does not supply a collective-coordinate action, right generator constraint, WZW level or response, N_c, baryon number, statistics, Hamiltonian, symmetry breaking, particle dictionary, mass spectrum, physical octet/decuplet selection, or substrate mechanism. diff --git a/governance/claims.yaml b/governance/claims.yaml index fde415b0..a3ab73c4 100644 --- a/governance/claims.yaml +++ b/governance/claims.yaml @@ -11338,3 +11338,141 @@ claims: accepted_in: v0.160.0 challenges: [] supersedes: [] + - id: C-IGR-001 + statement: >- + Conditional on C-GRV-001's independent additive inverse-coupling + baseline ledger, separately declare a positive self-adjoint + boundaryless four-dimensional Euclidean real-scalar Laplace-type + operator D_E=-nabla_E^2+xi*R_E+m2 with spacetime-constant m2>=0, + adequate infrared convergence or reference subtraction, + Gamma_E=(1/2)*ln(det(D_E)), heat-kernel prefactor (4*pi)^-2, and + Euclidean Einstein-Hilbert matching factor 16*pi. For a positive sharp + proper-time cutoff Lambda with tau0=Lambda^-2 and z=m2/Lambda^2, the + exact local curvature and vacuum coefficient integrals are + I2=Lambda^2*(exp(-z)-z*E1(z)) and + I3=Lambda^4*exp(-z)/2-m2*I2/2, continuously extended at m2=0. They obey + dI3/dm2=-I2, have massless values Lambda^2 and Lambda^4/2, and decay in + the large-mass limit. For positive integer N and exact xi, the declared + conditional additive coefficients are + Delta(1/G)=N*(1-6*xi)*I2/(12*pi) and + Delta(rho)=-(N/2)*(4*pi)^-2*I3. These are exact local coefficient + families, not a full or nonlocal determinant, varying-mass theorem, + selected regulator, cutoff ontology, total Newton constant, attractive- + gravity result, sourced geometry, or empirical prediction. + provenance: campaigns/P230-exact-mass-regulator-rung/adjudication.yaml + verification: symbolic_verified + review: accepted + compatibility: compatible_extension + epistemic: active + dependencies: + - C-GRV-001 + evidence: + - campaigns/P230-exact-mass-regulator-rung/verify.py + - campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py + - campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-001-claim-review.md + - campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml + - campaigns/P230-exact-mass-regulator-rung/evidence/literature-audit.yaml + - campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml + - campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml + - campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml + - src/substrate_framework/scalar_one_loop_mass.py + - tests/test_scalar_one_loop_mass.py + assumptions: + - The operator, scalar statistics, determinant sign and weight, heat-kernel and Einstein-Hilbert conventions, and sharp regulator are declared conditional inputs rather than derived substrate physics. + - The mass-squared input is exact, nonnegative, and spacetime constant; an x-dependent V'' background cannot be factored into these integrals. + - The determinant reading requires a positive self-adjoint operator and adequate infrared convergence or an explicit reference subtraction. + - Only the displayed local coefficient classes are retained; tau^-1, higher heat-kernel, nonlocal, boundary, and full-spectrum terms remain outside the claim. + - C-GRV-001's independent baseline B remains free in 1/G_total=B+Delta(1/G), and no total sign or physical value follows. + comparators: [] + accepted_in: v0.161.0 + challenges: [] + supersedes: [] + - id: C-IGR-002 + statement: >- + Under C-IGR-001's declared operator, determinant, local-coefficient, + matching, constant-mass, and baseline assumptions, replace the sharp + cutoff by the explicitly declared smooth proper-time weight + exp(-1/(Lambda^2*tau)). For positive Lambda, m2>=0, and + z=m2/Lambda^2, the exact coefficient integrals are + I2=2*Lambda^2*sqrt(z)*K1(2*sqrt(z)) and + I3=2*Lambda^4*z*K2(2*sqrt(z)), continuously extended at m2=0. They obey + dI3/dm2=-I2, have massless values Lambda^2 and Lambda^4, and decay for + large mass. The conditional additive coefficients are the same typed + compositions Delta(1/G)=N*(1-6*xi)*I2/(12*pi) and + Delta(rho)=-(N/2)*(4*pi)^-2*I3. The exact massless I3 value is twice + C-IGR-001's sharp value at the same Lambda, so this claim exposes rather + than removes regulator dependence and establishes no physical regulator + selection or total gravitational normalization. + provenance: campaigns/P230-exact-mass-regulator-rung/adjudication.yaml + verification: symbolic_verified + review: accepted + compatibility: compatible_extension + epistemic: active + dependencies: + - C-GRV-001 + - C-IGR-001 + evidence: + - campaigns/P230-exact-mass-regulator-rung/verify.py + - campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py + - campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-002-claim-review.md + - campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml + - campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml + - campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml + - campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml + - src/substrate_framework/scalar_one_loop_mass.py + - tests/test_scalar_one_loop_mass.py + assumptions: + - Every operator, mass, determinant, local-truncation, infrared, matching, and additive-baseline assumption of C-IGR-001 remains in force. + - The smooth essential-singularity weight and positive cutoff are explicit regulator data, not a derived or preferred physical scheme. + - The standard exact modified-Bessel integral representation is an approved mathematical import and numerical quadrature is corroboration only. + - The result selects no cutoff identification, renormalization condition, total Newton constant, sourced geometry, or empirical comparator. + comparators: [] + accepted_in: v0.161.0 + challenges: [] + supersedes: [] + - id: C-IGR-003 + statement: >- + Under C-IGR-001's declared operator, determinant, local-coefficient, + matching, constant-mass, and baseline assumptions, declare instead the + power-subtracted sharp-cutoff finite-part prescription at a positive + scale mu. For constant m2>=0 its exact continuously extended coefficient + family is I2=m2*(log(m2/mu^2)+EulerGamma-1) and + I3=-(m2^2/2)*(log(m2/mu^2)+EulerGamma-3/2). These are the finite limits + after the P230-frozen power and logarithmic subtractions, obey + dI3/dm2=-I2, mu*dI2/dmu=-2*m2, and + mu*dI3/dmu=m2^2, and vanish continuously at m2=0. The conditional + additive shifts use the same typed factors as C-IGR-001. At unit scale + and cutoff with m2=0, the sharp, smooth, and power-subtracted I3 values + are exactly 1/2, 1, and 0; moreover the finite-part I2 can change sign + with m2/mu^2. Therefore P230 derives exact scheme and scale dependence, + not a regulator-free coefficient, selected subtraction, finite + counterterm, total Newton constant, or physical normalization. + provenance: campaigns/P230-exact-mass-regulator-rung/adjudication.yaml + verification: symbolic_verified + review: accepted + compatibility: compatible_extension + epistemic: active + dependencies: + - C-GRV-001 + - C-IGR-001 + - C-IGR-002 + evidence: + - campaigns/P230-exact-mass-regulator-rung/verify.py + - campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py + - campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-003-claim-review.md + - campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml + - campaigns/P230-exact-mass-regulator-rung/evidence/literature-audit.yaml + - campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml + - campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml + - campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml + - src/substrate_framework/scalar_one_loop_mass.py + - tests/test_scalar_one_loop_mass.py + assumptions: + - Every operator, mass, determinant, local-truncation, infrared, matching, and additive-baseline assumption of C-IGR-001 remains in force. + - The displayed power and logarithmic subtraction and positive scale mu define this finite-part scheme; changing a finite subtraction defines a different scheme. + - Zero mass means the one-sided continuous limit of m2*log(m2), not literal substitution into log(0). + - No accepted claim chooses mu, a finite counterterm, a physical regulator, a cutoff identification, field spectrum, or additive baseline. + comparators: [] + accepted_in: v0.161.0 + challenges: [] + supersedes: [] diff --git a/governance/releases/current.yaml b/governance/releases/current.yaml index 3f5b23d1..847d0b55 100644 --- a/governance/releases/current.yaml +++ b/governance/releases/current.yaml @@ -1,7 +1,7 @@ schema_version: 1 -release: v0.160.0 -source_baseline: substrate@6d1f4e0 -released_at: '2026-08-17T20:45:00Z' +release: v0.161.0 +source_baseline: substrate-framework@0beaac3 +released_at: '2026-08-18T12:01:47+02:00' accepted_claims: - C-SG-001 - C-SG-002 @@ -210,11 +210,14 @@ accepted_claims: - C-WLN-003 - C-LOR-001 - C-LOR-002 + - C-IGR-001 + - C-IGR-002 + - C-IGR-003 note: >- - Adds five exact conditional classical claims: massive and massless einbein - worldline mechanics, local reparametrization and massless worldline Weyl - identities, H2/H3 future unit-timelike-vector orbit metrics, and the - displayed 2+1D and 3+1D massive/null Lorentz stabilizer matrix algebras. - The release derives no background, quantization, particle representation, - field-theory conformal symmetry, global representation classification, - tube ensemble, or interaction content. + Adds three exact conditional constant-mass real-scalar one-loop coefficient + families: sharp proper-time, smooth essential-weight, and a declared + power-subtracted finite part. The release preserves the independent + inverse-coupling baseline and derives no varying-background factorization, + full nonlocal determinant, physical regulator or cutoff identification, + total Newton constant, sourced gravity, higher-curvature completion, or + empirical prediction. diff --git a/governance/releases/v0.161.0.yaml b/governance/releases/v0.161.0.yaml new file mode 100644 index 00000000..847d0b55 --- /dev/null +++ b/governance/releases/v0.161.0.yaml @@ -0,0 +1,223 @@ +schema_version: 1 +release: v0.161.0 +source_baseline: substrate-framework@0beaac3 +released_at: '2026-08-18T12:01:47+02:00' +accepted_claims: + - C-SG-001 + - C-SG-002 + - C-SG-003 + - C-SG-004 + - C-OG-001 + - C-OG-002 + - C-OG-003 + - C-OG-004 + - C-OG-005 + - C-TH-001 + - C-TH-002 + - C-RG-001 + - C-RG-002 + - C-VAR-001 + - C-CC-001 + - C-VIR-001 + - C-MED-001 + - C-SK-001 + - C-SG-005 + - C-SG-006 + - C-SG-007 + - C-SG-008 + - C-ACT-001 + - C-DIM-001 + - C-U1-001 + - C-U1-002 + - C-DIM-002 + - C-MED-002 + - C-DIM-003 + - C-TOP-001 + - C-DIM-004 + - C-RGE-001 + - C-DIM-005 + - C-LIN-001 + - C-LIE-001 + - C-RGE-002 + - C-DIM-007 + - C-VTX-001 + - C-VTX-002 + - C-FLX-001 + - C-LIE-002 + - C-LIE-003 + - C-WIL-001 + - C-GAU-001 + - C-QBL-001 + - C-QBL-002 + - C-QBL-003 + - C-MIX-001 + - C-MIX-002 + - C-MOM-001 + - C-GW-001 + - C-GW-002 + - C-GW-003 + - C-SG-009 + - C-MOM-002 + - C-SG-010 + - C-GW-004 + - C-PDE-001 + - C-MOM-003 + - C-PDE-002 + - C-PDE-003 + - C-PDE-004 + - C-GW-005 + - C-GW-006 + - C-SG-011 + - C-SG-012 + - C-SG-013 + - C-PDE-005 + - C-PDE-006 + - C-PDE-007 + - C-PDE-008 + - C-PDE-009 + - C-PDE-011 + - C-PDE-012 + - C-GW-007 + - C-GW-008 + - C-WZW-001 + - C-WZW-002 + - C-TOP-002 + - C-EFT-001 + - C-SYM-001 + - C-CHI-001 + - C-BRK-001 + - C-CHI-002 + - C-GMR-001 + - C-MOD-001 + - C-MOD-002 + - C-SCL-001 + - C-WID-001 + - C-GTR-001 + - C-LOC-001 + - C-KRN-001 + - C-IDN-001 + - C-GLS-001 + - C-CHR-001 + - C-SPN-001 + - C-DEF-001 + - C-LAT-001 + - C-OVL-001 + - C-OVL-002 + - C-OVL-003 + - C-RGE-003 + - C-GRV-001 + - C-DIM-008 + - C-SYM-002 + - C-IDN-002 + - C-PRB-001 + - C-REP-001 + - C-RGE-004 + - C-COH-001 + - C-SG-015 + - C-SG-016 + - C-DYN-001 + - C-COH-002 + - C-MED-003 + - C-SG-017 + - C-SG-018 + - C-LAT-002 + - C-MED-004 + - C-COL-001 + - C-FPT-001 + - C-RMAP-001 + - C-RMAP-002 + - C-RPROF-001 + - C-RPROF-002 + - C-RDIFF-001 + - C-RDIFF-002 + - C-BPS-001 + - C-BPS-002 + - C-BPS-003 + - C-SG-019 + - C-SPN-002 + - C-RES-001 + - C-SCR-001 + - C-CMP-001 + - C-XOV-001 + - C-BRN-001 + - C-RGE-005 + - C-RGE-006 + - C-MAX-001 + - C-VAC-001 + - C-KRN-002 + - C-SKY-001 + - C-IRR-001 + - C-VEC-001 + - C-RAD-001 + - C-GOR-001 + - C-GOR-002 + - C-STG-001 + - C-RR-001 + - C-MED-005 + - C-BND-001 + - C-REP-002 + - C-KIN-001 + - C-SCT-001 + - C-NAG-001 + - C-BER-001 + - C-GSM-001 + - C-PRC-001 + - C-HOL-001 + - C-NVP-001 + - C-NVP-002 + - C-PGA-001 + - C-REP-003 + - C-ANO-001 + - C-DIM-009 + - C-STG-002 + - C-PDE-013 + - C-RMOM-001 + - C-RMOM-002 + - C-GW-009 + - C-GW-010 + - C-FLO-001 + - C-ROT-001 + - C-RMAP-003 + - C-SKY-002 + - C-VAC-002 + - C-VAC-003 + - C-VAC-004 + - C-CMB-001 + - C-CMB-002 + - C-OSC-001 + - C-CMB-003 + - C-OSC-002 + - C-DOS-001 + - C-QFL-001 + - C-VOP-001 + - C-MKV-001 + - C-BRN-002 + - C-QBL-004 + - C-QBL-005 + - C-MIX-003 + - C-QBL-006 + - C-PHS-001 + - C-OVL-005 + - C-PHS-002 + - C-MIX-004 + - C-VEC-002 + - C-GSK-001 + - C-GSK-002 + - C-VAR-002 + - C-VAR-003 + - C-WLN-001 + - C-WLN-002 + - C-WLN-003 + - C-LOR-001 + - C-LOR-002 + - C-IGR-001 + - C-IGR-002 + - C-IGR-003 +note: >- + Adds three exact conditional constant-mass real-scalar one-loop coefficient + families: sharp proper-time, smooth essential-weight, and a declared + power-subtracted finite part. The release preserves the independent + inverse-coupling baseline and derives no varying-background factorization, + full nonlocal determinant, physical regulator or cutoff identification, + total Newton constant, sourced gravity, higher-curvature completion, or + empirical prediction. diff --git a/memory/codex/efforts/P230-exact-mass-promotion.md b/memory/codex/efforts/P230-exact-mass-promotion.md new file mode 100644 index 00000000..4ad74645 --- /dev/null +++ b/memory/codex/efforts/P230-exact-mass-promotion.md @@ -0,0 +1,77 @@ +--- +description: Promote the exact constant-mass one-loop regulator theorems C-IGR-001 through C-IGR-003 +author: root-agent +created: '2026-08-18T11:49:56+02:00' +updated: '2026-08-18T12:09:42+02:00' +tags: +- substrate-framework +- research-arc +- induced-gravity +- P230 +category: efforts +confidence: established +status: active +--- + +## Positive Objective and Success +This arc promotes three narrow positive theorems from P230: exact sharp-cutoff, smooth-weight, and power-subtracted constant-mass proper-time coefficient families together with their explicitly conditional one-loop curvature and vacuum compositions. Success requires exact claim statements, accepted dependency and approved-import closure, a constant-mass scope repair, independent rederivation, mutation sensitivity, importable APIs, consumer replay, individual claim reviews, immutable campaign adjudication, registry and v0.161.0 release materialization, generated docs and accepted memory synchronization, full validation, and an empty debt ledger. Issue #76 remains open because these theorems do not select a regulator, renormalization condition, total gravitational coupling, sourced geometry, or predictions. + +## Authority and Prior Work +The accepted boundary is v0.160.0 at current-main commit 0beaac3 with C-GRV-001 as the only accepted induced-coupling dependency. PR #77 contains the original P230 proposal and repair history; PR #82 contains the reviewed focused implementation at d415caf, cherry-picked unchanged as a309008. The landed scalar_induced_newton and covariant_sine_gordon_action modules promoted no claims and are implementation/provenance inputs only. Primary sources checked are Vassilevich hep-th/0306138v3 equations 1.16-1.20, 2.2, 4.27 and Visser gr-qc/0204062v1 equations 7-15. Repository memory searches for P230 and C-IGR found only the active PR records; no accepted identifier collision exists. The genuine unresolved objective is claim-level promotion of the correct constant-mass theorems, not artifact harvest. + +## Definitions and Invariants +Let Lambda>0, mu>0, m^2>=0 be exact constants, N a positive integer, xi an exact real with decidable relation to 1/6, and z=m^2/Lambda^2. The determinant reading declares a positive self-adjoint boundaryless four-dimensional Euclidean D_E=-nabla_E^2+xi R_E+m^2 with constant m^2 and adequate infrared convergence or reference subtraction. Gamma_E=(1/2) ln det D_E is represented with the declared proper-time sign. I2 is the tau^-2 curvature-class integral and I3 the tau^-3 vacuum-class integral. The sharp lower limit is Lambda^-2; the smooth weight is exp(-1/(Lambda^2 tau)); the power-subtracted family is the declared cutoff finite part at scale mu. Only these local coefficient families are claimed, not an exact truncation of the full nonlocal determinant. No arbitrary x-dependent V'' background, regulator selection, cutoff ontology, additive-baseline choice, total G, or empirical comparator is part of the theorem. + +## Permitted Imports and Assumptions +Permitted accepted input is C-GRV-001. Approved conditional imports are the stated real-scalar determinant/heat-kernel/EH conventions from the primary literature, exact special-function identities, and exact input validation. The scalar_induced_newton API may be reused only after independent coefficient rederivation. A sine-Gordon composition is limited to a constant vacuum whose V'' is nonnegative and constant. + +## Candidate Set +The candidate set was preregistered in P230 before formula evaluation. A is the sharp tail-integral family, B the smooth essential-singularity Bessel family, and C the power-subtracted finite-part family. All three remain because the deliverable is their exact conditional comparison, not a physically selected regulator. + +## Selection Criteria and Comparator Gate +Criteria in order are defining-integral/subtraction correctness, constant-mass scope, dimensions and exact limits, independent rederivation, mutation sensitivity, explicit assumption economy, API/consumer closure, and only then any later empirical test. No empirical comparator is opened in P230. Existing internal z=1 evaluations are exact formula regressions and cannot select a scheme. + +## Claim Delta +C-IGR-001 states the sharp constant-mass I2/I3 and conditional shift theorem. C-IGR-002 states the smooth constant-mass Bessel I2/I3 and conditional shift theorem. C-IGR-003 states the power-subtracted constant-mass finite-part theorem and exact scheme-dependence ceiling. Each depends on C-GRV-001 only for the additive inverse-coupling ledger and uses the proposal's declared QFT conventions as explicit conditional imports. None supersedes an accepted claim. + +## Claim Ladder +First prove each I2/I3 family from its defining integral or subtraction limit. Then prove dI3/dm^2=-I2, dimensions, massless and scale limits, and mutations. Then compose the independently rederived determinant and EH factors. Finally establish that scheme contrast and the independent additive baseline prevent a physical total normalization without later inputs. + +## Importable Implementation +Canonical APIs are scalar_one_loop_mass.curvature_proper_time_integral, vacuum_proper_time_integral, exact_mass_inverse_newton_shift, exact_mass_vacuum_density_shift, and regulator_scheme_ledger. The implementation must declare constant mass, must not call an unpromoted scientific result as authority, and must expose exclusions in its documentation. Campaign verifiers import the API; the independent rederivation imports no scalar_one_loop_mass symbol. + +## Harvest Checkpoints +PR #82 is the reviewed implementation checkpoint and PR #77 is the proposal/attempt provenance. Both are superseded only after the promotion PR lands. Canonical issue #76 predates both PRs. The promotion successor uses Advances #76 because the parent emergent-gravity objective remains open. + +## Attempts +Attempt 0001 produced the original exact atoms but contained a vacuum double count. Attempt 0002 repaired that defect and zeta sign semantics but left false authority language in the broad proposal boundary. Attempt 0003 is this promotion repair: narrow to constant mass, restate the claims exactly, independently rederive, and complete governance materialization. + +## Framework-Fit Audit +The claims are conditional extensions compatible with C-GRV-001 because they preserve its independent baseline and premise ledger. They neither identify the scalar with substrate matter nor choose regulator, scale, cutoff, total coupling, sign of the total inverse coupling, or a physical background. A varying mass endomorphism is explicitly excluded rather than retrofitted into a factorized formula. + +## Verifier Audit +The primary verifier exited zero with all 27 checks passing. The independent reviewer imports no `scalar_one_loop_mass` scientific symbol and exited zero with all 21 checks passing, using raw SymPy derivations plus 60-digit quadrature only as corroboration. Exact oracles cover defining integrals, subtraction limits, `dI3/dm^2=-I2`, massless and large-mass limits, scale derivatives, dimensions, signs, and the independently reconstructed determinant-to-Einstein-Hilbert factor. Mutations cover branch, Bessel order, prefactor, determinant sign, vacuum double counting, and varying-background scope; each load-bearing mutation breaks its relevant verdict. The executable compatibility scan found no direct, imported, or dynamic legacy `np.trapz` access. + +## Global Dependency Replay +The affected scalar tests and the `scalar_induced_newton` regression passed 42 tests. Direct package import passed. Repository validation closed 210 accepted claims and the v0.161.0 release with no migration queue debt; canonical docs and accepted memory were regenerated. GitNexus classified the change as low risk with zero affected execution flows and no external runtime caller of the new composed APIs. The final full workflow gate passed all fixed checks and 2,263 tests in 288.80 seconds. The campaign verifier and independent rederivation passed separately at 27 and 21 checks. Final record-sensitive repository, memory, docs, and accepted-memory checks passed, and `git diff --check` was clean. + +## Foundational Revision Gate +No foundational revision is opened. The constant-mass narrowing resolves the candidate scope defect without changing accepted canon. + +## Debt Ledger +No debt remains inside the three-claim promotion scope. The original arbitrary-background language was narrowed to constant nonnegative mass; exact statements, independent rederivation, immutable campaign and individual reviews, registry and release entries, and generated consumers are all materialized. Regulator selection, a renormalization condition, total coupling, sourced geometry, and higher-curvature terms are explicitly outside these claims and remain frontier of issue #76 rather than hidden promotion debt. + +## Independent Claim Review +Separate reviews for C-IGR-001, C-IGR-002, and C-IGR-003 accept each exact conditional theorem against the frozen criteria and reject every broader reading listed in the adjudication. + +## Results and Continuation +The promotion package establishes three positive, exact constant-mass theorems: the sharp-cutoff family, the smooth Bessel family, and the declared power-subtracted finite-part family, including their conditional curvature and vacuum compositions and exact scheme ceiling. After this package lands, issue #76 continues at the tau^-1/higher-curvature and renormalization-condition frontier rather than treating these claims as a total gravity derivation. + +## Promotion and Materialization +Campaign P230, three claim reviews, governance entries, the v0.161.0 release, generated docs, accepted claim/release memory, and validation evidence are materialized on `research/p230-exact-mass-promotion`. PRs #77 and #82 remain provenance inputs and must stay open until the corrected successor lands; repository acceptance still requires the distinct merger mandated by AGENTS.md. + +## Done Gate +All scientific, verification, review, dependency, consumer, materialization, and debt gates for C-IGR-001..003 pass. The package is ready for a distinct merger. Completion of this arc promotes those three claims but does not close issue #76. + +## Cross-References +Canonical issue #76; PRs #77 and #82; proposal P230; C-GRV-001; source module scalar_one_loop_mass.py. diff --git a/memory/framework/claims/C-IGR-001.md b/memory/framework/claims/C-IGR-001.md new file mode 100644 index 00000000..0f80dea0 --- /dev/null +++ b/memory/framework/claims/C-IGR-001.md @@ -0,0 +1,45 @@ +--- +description: Accepted framework claim C-IGR-001 +author: framework-registry +created: '2026-08-18T12:01:47+02:00' +updated: '2026-08-18T12:01:47+02:00' +tags: +- substrate-framework +- accepted-claim +- C-IGR-001 +category: claims +confidence: established +status: active +--- +# C-IGR-001 + +## Statement +The accepted statement is reproduced exactly from the claim registry. + +Conditional on C-GRV-001's independent additive inverse-coupling baseline ledger, separately declare a positive self-adjoint boundaryless four-dimensional Euclidean real-scalar Laplace-type operator D_E=-nabla_E^2+xi*R_E+m2 with spacetime-constant m2>=0, adequate infrared convergence or reference subtraction, Gamma_E=(1/2)*ln(det(D_E)), heat-kernel prefactor (4*pi)^-2, and Euclidean Einstein-Hilbert matching factor 16*pi. For a positive sharp proper-time cutoff Lambda with tau0=Lambda^-2 and z=m2/Lambda^2, the exact local curvature and vacuum coefficient integrals are I2=Lambda^2*(exp(-z)-z*E1(z)) and I3=Lambda^4*exp(-z)/2-m2*I2/2, continuously extended at m2=0. They obey dI3/dm2=-I2, have massless values Lambda^2 and Lambda^4/2, and decay in the large-mass limit. For positive integer N and exact xi, the declared conditional additive coefficients are Delta(1/G)=N*(1-6*xi)*I2/(12*pi) and Delta(rho)=-(N/2)*(4*pi)^-2*I3. These are exact local coefficient families, not a full or nonlocal determinant, varying-mass theorem, selected regulator, cutoff ontology, total Newton constant, attractive- gravity result, sourced geometry, or empirical prediction. + +## Status Axes +The four governance axes remain independent. + +Verification is `symbolic_verified`; review is `accepted`; compatibility is `compatible_extension`; epistemic status is `active`. + +## Dependency and Import Closure +The registry records the accepted closure and declared non-claim inputs. + +Dependencies: C-GRV-001. Assumptions: The operator, scalar statistics, determinant sign and weight, heat-kernel and Einstein-Hilbert conventions, and sharp regulator are declared conditional inputs rather than derived substrate physics., The mass-squared input is exact, nonnegative, and spacetime constant; an x-dependent V'' background cannot be factored into these integrals., The determinant reading requires a positive self-adjoint operator and adequate infrared convergence or an explicit reference subtraction., Only the displayed local coefficient classes are retained; tau^-1, higher heat-kernel, nonlocal, boundary, and full-spectrum terms remain outside the claim., C-GRV-001's independent baseline B remains free in 1/G_total=B+Delta(1/G), and no total sign or physical value follows.. Comparators: none. + +## Provenance and Evidence +The accepted release and immutable campaign evidence are the authoritative pointers. + +Accepted in `v0.161.0` with provenance `campaigns/P230-exact-mass-regulator-rung/adjudication.yaml`. + +- `campaigns/P230-exact-mass-regulator-rung/verify.py` +- `campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py` +- `campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-001-claim-review.md` +- `campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml` +- `campaigns/P230-exact-mass-regulator-rung/evidence/literature-audit.yaml` +- `campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml` +- `campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml` +- `campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml` +- `src/substrate_framework/scalar_one_loop_mass.py` +- `tests/test_scalar_one_loop_mass.py` diff --git a/memory/framework/claims/C-IGR-002.md b/memory/framework/claims/C-IGR-002.md new file mode 100644 index 00000000..b35b9f89 --- /dev/null +++ b/memory/framework/claims/C-IGR-002.md @@ -0,0 +1,44 @@ +--- +description: Accepted framework claim C-IGR-002 +author: framework-registry +created: '2026-08-18T12:01:47+02:00' +updated: '2026-08-18T12:01:47+02:00' +tags: +- substrate-framework +- accepted-claim +- C-IGR-002 +category: claims +confidence: established +status: active +--- +# C-IGR-002 + +## Statement +The accepted statement is reproduced exactly from the claim registry. + +Under C-IGR-001's declared operator, determinant, local-coefficient, matching, constant-mass, and baseline assumptions, replace the sharp cutoff by the explicitly declared smooth proper-time weight exp(-1/(Lambda^2*tau)). For positive Lambda, m2>=0, and z=m2/Lambda^2, the exact coefficient integrals are I2=2*Lambda^2*sqrt(z)*K1(2*sqrt(z)) and I3=2*Lambda^4*z*K2(2*sqrt(z)), continuously extended at m2=0. They obey dI3/dm2=-I2, have massless values Lambda^2 and Lambda^4, and decay for large mass. The conditional additive coefficients are the same typed compositions Delta(1/G)=N*(1-6*xi)*I2/(12*pi) and Delta(rho)=-(N/2)*(4*pi)^-2*I3. The exact massless I3 value is twice C-IGR-001's sharp value at the same Lambda, so this claim exposes rather than removes regulator dependence and establishes no physical regulator selection or total gravitational normalization. + +## Status Axes +The four governance axes remain independent. + +Verification is `symbolic_verified`; review is `accepted`; compatibility is `compatible_extension`; epistemic status is `active`. + +## Dependency and Import Closure +The registry records the accepted closure and declared non-claim inputs. + +Dependencies: C-GRV-001, C-IGR-001. Assumptions: Every operator, mass, determinant, local-truncation, infrared, matching, and additive-baseline assumption of C-IGR-001 remains in force., The smooth essential-singularity weight and positive cutoff are explicit regulator data, not a derived or preferred physical scheme., The standard exact modified-Bessel integral representation is an approved mathematical import and numerical quadrature is corroboration only., The result selects no cutoff identification, renormalization condition, total Newton constant, sourced geometry, or empirical comparator.. Comparators: none. + +## Provenance and Evidence +The accepted release and immutable campaign evidence are the authoritative pointers. + +Accepted in `v0.161.0` with provenance `campaigns/P230-exact-mass-regulator-rung/adjudication.yaml`. + +- `campaigns/P230-exact-mass-regulator-rung/verify.py` +- `campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py` +- `campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-002-claim-review.md` +- `campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml` +- `campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml` +- `campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml` +- `campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml` +- `src/substrate_framework/scalar_one_loop_mass.py` +- `tests/test_scalar_one_loop_mass.py` diff --git a/memory/framework/claims/C-IGR-003.md b/memory/framework/claims/C-IGR-003.md new file mode 100644 index 00000000..7dff8851 --- /dev/null +++ b/memory/framework/claims/C-IGR-003.md @@ -0,0 +1,45 @@ +--- +description: Accepted framework claim C-IGR-003 +author: framework-registry +created: '2026-08-18T12:01:47+02:00' +updated: '2026-08-18T12:01:47+02:00' +tags: +- substrate-framework +- accepted-claim +- C-IGR-003 +category: claims +confidence: established +status: active +--- +# C-IGR-003 + +## Statement +The accepted statement is reproduced exactly from the claim registry. + +Under C-IGR-001's declared operator, determinant, local-coefficient, matching, constant-mass, and baseline assumptions, declare instead the power-subtracted sharp-cutoff finite-part prescription at a positive scale mu. For constant m2>=0 its exact continuously extended coefficient family is I2=m2*(log(m2/mu^2)+EulerGamma-1) and I3=-(m2^2/2)*(log(m2/mu^2)+EulerGamma-3/2). These are the finite limits after the P230-frozen power and logarithmic subtractions, obey dI3/dm2=-I2, mu*dI2/dmu=-2*m2, and mu*dI3/dmu=m2^2, and vanish continuously at m2=0. The conditional additive shifts use the same typed factors as C-IGR-001. At unit scale and cutoff with m2=0, the sharp, smooth, and power-subtracted I3 values are exactly 1/2, 1, and 0; moreover the finite-part I2 can change sign with m2/mu^2. Therefore P230 derives exact scheme and scale dependence, not a regulator-free coefficient, selected subtraction, finite counterterm, total Newton constant, or physical normalization. + +## Status Axes +The four governance axes remain independent. + +Verification is `symbolic_verified`; review is `accepted`; compatibility is `compatible_extension`; epistemic status is `active`. + +## Dependency and Import Closure +The registry records the accepted closure and declared non-claim inputs. + +Dependencies: C-GRV-001, C-IGR-001, C-IGR-002. Assumptions: Every operator, mass, determinant, local-truncation, infrared, matching, and additive-baseline assumption of C-IGR-001 remains in force., The displayed power and logarithmic subtraction and positive scale mu define this finite-part scheme; changing a finite subtraction defines a different scheme., Zero mass means the one-sided continuous limit of m2*log(m2), not literal substitution into log(0)., No accepted claim chooses mu, a finite counterterm, a physical regulator, a cutoff identification, field spectrum, or additive baseline.. Comparators: none. + +## Provenance and Evidence +The accepted release and immutable campaign evidence are the authoritative pointers. + +Accepted in `v0.161.0` with provenance `campaigns/P230-exact-mass-regulator-rung/adjudication.yaml`. + +- `campaigns/P230-exact-mass-regulator-rung/verify.py` +- `campaigns/P230-exact-mass-regulator-rung/reviews/independent_exact_mass_review.py` +- `campaigns/P230-exact-mass-regulator-rung/reviews/C-IGR-003-claim-review.md` +- `campaigns/P230-exact-mass-regulator-rung/evidence/formula-freeze.yaml` +- `campaigns/P230-exact-mass-regulator-rung/evidence/literature-audit.yaml` +- `campaigns/P230-exact-mass-regulator-rung/evidence/candidate-comparison.yaml` +- `campaigns/P230-exact-mass-regulator-rung/evidence/dependency-audit.yaml` +- `campaigns/P230-exact-mass-regulator-rung/attempts/0003/result.yaml` +- `src/substrate_framework/scalar_one_loop_mass.py` +- `tests/test_scalar_one_loop_mass.py` diff --git a/memory/framework/releases/v0.161.0.md b/memory/framework/releases/v0.161.0.md new file mode 100644 index 00000000..cfab3fd7 --- /dev/null +++ b/memory/framework/releases/v0.161.0.md @@ -0,0 +1,233 @@ +--- +description: Accepted framework release v0.161.0 +author: framework-registry +created: '2026-08-18T12:01:47+02:00' +updated: '2026-08-18T12:01:47+02:00' +tags: +- substrate-framework +- accepted-release +- v0.161.0 +category: releases +confidence: established +status: active +--- +# Release v0.161.0 + +## Source Boundary +This release pins its predecessor evidence boundary and acceptance time. + +Source baseline: `substrate-framework@0beaac3`. Released at: `2026-08-18T12:01:47+02:00`. + +## Accepted Claim Set +The release materializes this dependency-closed claim set. + +- `C-SG-001` — For every real omega with 0 < omega < 1, eta = sqrt(1-omega^2), and real x,t, the field phi(x,t) = 4 atan(eta sin(omega t)/(omega cosh(eta x))) is spatially localized, periodic with period 2*pi/omega, and satisfies phi_tt - phi_xx + sin(phi) = 0 identically in normalized units. +- `C-SG-002` — The C-SG-001 breather has conserved normalized Hamiltonian energy E(omega) = 16 sqrt(1-omega^2); E approaches the two-kink threshold 16 as omega -> 0+ and approaches 0 as omega -> 1-. +- `C-SG-003` — For every real omega with 0 < omega < 1, the C-SG-001 breather's canonical action normalized by J = (1/(2*pi))*closed_integral(p dq) is J(omega) = 16 arccos(omega). It satisfies dE/dJ = omega, maps the family onto 0 < J < 8*pi, and has inverse parameterization omega = cos(J/16) and E = 16 sin(J/16). +- `C-SG-004` — For every real omega with 0 < omega < 1, the C-SG-001 breather's period-averaged squared-gradient integral is Gbar = (1/T)*integral_0^T dt integral_R dx phi_x^2 = 16*(sqrt(1-omega^2) - omega*arccos(omega)) = E - omega*J. It satisfies dGbar/domega = -J, approaches 16 as omega -> 0+, and approaches 0 as omega -> 1-. Gbar is the full squared-gradient integral; its Hamiltonian energy contribution is Gbar/2. +- `C-OG-001` — For every positive twice-differentiable static index n(x) and c0 > 0, the declared 1+1 metric g = diag(-1/n, n/c0^2) has Ricci scalar R = c0^2*(n*n_xx - 2*n_x^2)/n^3 and satisfies Box_g(log(n)) = R. Among twice-differentiable scalar compositions f(n) satisfying Box_g(f(n)) = R for every such profile, exactly f(n) = log(n) + C work. +- `C-OG-002` — Conditional on C-OG-001 and the imported constitutive relation n = 1/(1 + 2*Phi/c0^2), the optical dilaton is log(n) = -log(1 + 2*Phi/c0^2), with leading weak-field term -2*Phi/c0^2. The metric's static slow coordinate-geodesic acceleration is exactly -(1 + 2*Phi/c0^2)*Phi_x; under Phi = lambda*U it satisfies acceleration/lambda -> -U_x as lambda -> 0+. +- `C-OG-003` — Conditional on C-OG-001 and C-OG-002, for every twice-differentiable potential Phi(x) whose TF index n = 1/(1 + 2*Phi/c0^2) is positive, the exact source-side optical dilaton operator is -Box_g(log(n)) = 2*Phi_xx; no weak-field approximation is required and c0 cancels. Consequently, if a separate model declares -Box_g(phi) = kappa*rho, that equation is algebraically equivalent to Phi_xx = (kappa/2)*rho. This claim neither derives that matter equation nor assigns a physical normalization to kappa. +- `C-OG-004` — Let a(x)=c0^2*n_x/(2*n^3) be the C-CC-001 zero-velocity optical collective acceleration for a positive sufficiently differentiable static index n(x). Let p(z) be a declared normalized centered profile with finite second moment mu2, and define the scaled profile-average A(s,X)=integral_R p(z)*a(X+s*z) dz wherever differentiation under the integral is valid. Then A(0,X)=a(X), partial_s A(0,X)=0, and partial_s^2 A(0,X)=mu2*a_xx(X), so A(s,X)=a(X)+(s^2*mu2/2)*a_xx(X)+o(s^2). Equivalently, with spatial variance sigma2=s^2*mu2, the leading width term is sigma2*c0^2*(n_xxx/n^3-9*n_x*n_xx/n^4+12*n_x^3/n^5)/4. For n=1+epsilon*xi, its coefficient linear in epsilon is c0^2*sigma2*xi_xxx/4. Thus it has acceleration units and depends on the third background derivative in the weak-profile limit. For an even index about X=0 and an even centered profile, the exact averaged acceleration at X=0 vanishes. This conditional profile-average theorem is not a Mathisson-Papapetrou-Dixon equation, field-derived deformation law, gravity theory, material trajectory, or substrate realization. +- `C-OG-005` — Conditional on C-OG-001's metric, C-GAU-001's positive-charge convention, and the separately declared action S=-integral sqrt(-g)*(g^{mu nu}(D_mu Psi)^*D_nu Psi+m^2*|Psi|^2) dt dx, the Euler equation is D_mu(sqrt(-g)*g^{mu nu}D_nu Psi)/sqrt(-g)-m^2*Psi=0. For positive constant n, c0, m, and e and constant real (A_t,A_x), a plane wave exp(i*(k*x-w*t)) obeys n*(w+e*A_t)^2-(c0^2/n)*(k-e*A_x)^2=m^2. Under chi=a*t+b*x, w'=w-e*a, k'=k+e*b, A_t'=A_t+a, and A_x'=A_x+b, so the two displayed momentum combinations are invariant. A constant connection on the simply connected line is pure gauge. On a circle of circumference L with Psi(x+L)=exp(i*theta)Psi(x), integer mode q instead has (w+e*A_t)^2=m^2/n+(c0^2/n^2)*((2*pi*q+theta)/L-e*A_x)^2, invariant under q'=q+ell and A_x'=A_x+2*pi*ell/(e*L). For varying n(x), the spatial operator is D_x((c0^2/n)D_x Psi), whose difference from (c0^2/n)D_xD_x Psi is -c0^2*n_x*D_x Psi/n^2. With C-BER-001's B=i*psi_dagger*d psi convention and the same section phase as the U1 matter phase, a separately declared texture phi(x) maps only by e*A_x=-B_phi*partial_x(phi). These identities establish no scalar or breather realization, material action, electromagnetic field, analog-gravity observation, real-gravity coupling, or dynamical spacetime. +- `C-TH-001` — For every real dimensionless splitting x, the normalized upper-state occupation of Z = 1 + exp(-x) is P = exp(-x)/Z = 1/(1+exp(x)). Its Bernoulli variance is P*(1-P) = sech(x/2)^2/4, and the conditional symmetric gate W = 2*P*(1-P) = sech(x/2)^2/2. W is even, has unique global maximum 1/2 at x = 0, decreases strictly with |x|, and tends to zero as x tends to either infinity. A shape A*sech(x/2)^2 equals 2*A*W; this identity does not determine the independent amplitude A. +- `C-TH-002` — For positive exact energy quantum q, thermal-energy coordinate vartheta, barrier E, and attempt frequency nu, declare x=q/vartheta, Theta=q*coth(x/2)/2, and import C-TH-001's W=sech(x/2)^2/2. Then the capillary-reduced conditional response R=nu*sqrt(E/(pi*Theta))*exp(-E/Theta)*W has inverse-time dimension; it composes a supplied frequency and is not by itself a stochastic rate law. Theta tends to q/2 as vartheta tends to zero and Theta/vartheta tends to one as vartheta tends to infinity. Put u=tanh(q/(2*vartheta)) in (0,1) and b=E/q>0. Up to a positive temperature-independent factor, R has shape sqrt(u)*(1-u^2)*exp(-2*b*u), whose log derivative 1/(2*u)-2*u/(1-u^2)-2*b is strictly decreasing. It has one strict global interior maximum, determined by 1-5*u^2-4*b*u*(1-u^2)=0, with 0q/(2*atanh(1/sqrt(5)))>1.039*q. In particular the separately imposed crossover vartheta=q/2 is not this maximum. More generally the declared prefactor family (E/Theta)^a with a>0 has one maximum below u=sqrt(a/(a+2)), while a=0 is strictly increasing with vartheta toward a finite plateau and has no finite maximum, so the existence and location of an optimum are prefactor dependent. Holding Theta and every other input fixed, the barrier log elasticity for a=1/2 is 1/2-E/Theta; if E is proportional to z^(-m), the response elasticity is m*(E/Theta-1/2). Thus C-RG-002's amplitude and fixed-q wavenumber partials reverse sign at E/Theta=1/2, and no total drive derivative follows without q(k) and nu(k). A common positive rescaling of (E,q,vartheta) preserves the dimensionless shape, and free nu fits any positive response target. These results derive no bath state, medium or breather mode, dispersion, attempt frequency, stochastic escape process, material applicability, DBD event, isotope effect, output power, or physical operating recommendation. +- `C-RG-001` — For positive radius R and density lambda, the circumference line energy E_line = 2*pi*R*lambda is homogeneous of degree one, has constant positive derivative 2*pi*lambda, and has no stationary radius. For positive surface density sigma, E_shell = 4*pi*R^2*sigma is homogeneous of degree two and has radius-dependent derivative 8*pi*R*sigma. For positive line tension T and pressure P, E_cap = 2*pi*R*T - pi*R^2*P + C has the unique strict global maximum R = T/P. Conditional line constructions with coefficients T and the C-SG-002 breather energy share only the degree-one line form; their energies are equal for every positive R if and only if their coefficients are equal. +- `C-RG-002` — Conditional on positive Frank constant K_F with energy-per-length dimension, real dimensionless defect strength s, ordered positive cutoffs R_o>r_c, positive core line energy epsilon_core, positive coupling g and thickness l_m, and nonzero real amplitude A and wavenumber k, declare T=pi*K_F*s^2*log(R_o/r_c)+epsilon_core and P=g*A^2*k^2*l_m/2. Composing these premises with C-RG-001 gives the unique positive capillary maximum R_*=2*T/(g*A^2*k^2*l_m) and relative barrier E_barrier=E(R_*)-E(0)=2*pi*T^2/(g*A^2*k^2*l_m); a radius-independent core offset cancels only from this relative height. If [A]=L^alpha, dimensional closure requires [g]=E*L^(-1-2*alpha), giving [P]=E/L^2, [R_*]=L, and [E_barrier]=E for every declared real alpha, so dimensions select neither alpha nor the quadratic loading premise. Holding T fixed, the log elasticities of both radius and barrier against (g,A,k,l_m) are (-1,-2,-2,-1), while the barrier elasticity against T is two and the radius elasticity is one. Writing T_el=pi*K_F*s^2*log(R_o/r_c), the T elasticities against (K_F,s,R_o,r_c,epsilon_core) are (T_el/T,2*T_el/T,pi*K_F*s^2/T,-pi*K_F*s^2/T,epsilon_core/T); the radius inherits them and the barrier doubles them. The exact log-exponent matrix for (R_*,E_barrier) against (T,g,A,k,l_m) has rank two and three null directions: radius plus barrier identifies effective T but not the four drive constituents, while a barrier observation alone has rank one and identifies none. Zero drive has no finite maximum and negative drive is strictly increasing for positive radius. These results derive no material, coupling value, amplitude convention, k-omega dispersion, temperature law, nucleation rate, DBD event, isotope effect, nuclear process, or output power. +- `C-VAR-001` — For any differentiable one-coordinate Lagrangian L0 and any nonzero multiplier A that is independent of the path coordinate, velocity, and evolution parameter, the Euler-Lagrange operator satisfies EL[A*L0] = A*EL[L0]. The two Euler-Lagrange equations therefore have the same solution set. The result holds for every fixed nonzero uniform factor, not only a factor of degree one in a named energy scale. +- `C-CC-001` — Conditional on the timelike one-coordinate action L = -(E0/sqrt(n(q)))*sqrt(1-n(q)^2*qdot^2/c0^2), with positive n, c0, and E0, the exact coordinate-time acceleration is qddot = (c0^2-3*n^2*qdot^2)*n_q/(2*n^3). Its zero-velocity limit is c0^2*n_q/(2*n^3), matching C-OG-001, and its locally unique same-data IVP is independent of E0. The mixed-scale counterexample with E0 also inside the kinetic square root retains E0 and, for n=1+alpha*q at q=qdot=0, has initial acceleration E0*c0^2*alpha/2. +- `C-VIR-001` — Conditional on the real virial slope formulas width_slope=(a-b)/2 and energy_slope=-(a+b)/2, both slopes equal -1/2 if and only if (a,b)=(0,1). The alternatives (1,0) and (1,1) give slopes (1/2,-1/2) and (0,-1), respectively, and fail the simultaneous target. +- `C-MED-001` — For positive density rho, thermal scale Theta, and reference speed c, the declared co-scaled response laws epsilon=rho*Theta/c^2 and mu_inverse=rho*Theta satisfy epsilon*mu=1/c^2 and give local wave speed sqrt(mu_inverse/epsilon)=c. Density and thermal variations therefore cannot create an index within this ansatz. More generally, the logarithmic sensitivities vanish exactly when the corresponding response exponents match. +- `C-SK-001` — Conditional on positive premises M_top=48*pi^3*B1*E_e and M_ANW=3*pi^2*B1*F_pi/e, equality M_top=M_ANW holds if and only if F_pi/e=16*pi*E_e. The shared linear hedgehog coefficient B1 cancels exactly; changing either B1 power generally prevents that cancellation. +- `C-SG-005` — For every real omega with 0 < omega < 1, the C-SG-002 breather's deficit below the normalized two-kink threshold is Delta(omega)=16-E(omega)=16*(1-sqrt(1-omega^2)). It satisfies 00+, and tends to one as omega->1-. Moreover dE/dH=omega^3, whereas dE/dJ=omega, so H is not the canonical action on the open family and agrees with it only asymptotically through Pi->1 at the harmonic endpoint. +- `C-SG-007` — Conditional on a fixed positive action increment h and the imposed lattice J_n=n*h, every positive integer n with n*h<8*pi has omega_n=cos(n*h/16) and E_n=16*sin(n*h/16), so n<8*pi/h and only finitely many levels are admissible. The continuous interpolation obeys dE/dn=h*cos(n*h/16)=h*omega_n. When (n+1)*h<8*pi, the actual adjacent gap is E_(n+1)-E_n=32*sin(h/32)*cos((2*n+1)*h/32), which is generally not h*omega_n. This claim does not derive the lattice premise or identify h with a physical coupling or Planck constant. +- `C-SG-008` — For every C-SG-001 breather with 00, differentiable positive energy E(J), and positive frequency omega=dE/dJ, the identity E/omega=J throughout the interval holds if and only if E(J)=C*J for a positive constant C. Thus a linear harmonic energy law has secant action equal to canonical action. For a rigid rotor with normalized action J=I*omega and energy E=I*omega^2/2=J^2/(2I), E/omega=J/2 instead. +- `C-DIM-001` — Over base dimensions energy E and time T, primitives consisting of an energy and frequency have dimension matrix [[1,0],[0,-1]], rank two, and zero kernel, so they form no nontrivial dimensionless monomial. Adding an independent action primitive S gives matrix [[1,0,1],[0,-1,1]], rank two, with one-dimensional kernel spanned by (-1,1,1); up to powers its unique dimensionless monomial is S*omega/E. Both conclusions are local to the declared primitive set and do not prohibit groups after further independent primitives are added. +- `C-U1-001` — For an independently declared smooth complex scalar Psi on 1+1 Minkowski spacetime with signature (+,-), define the raised current j^mu=i*(Psi_conj*d^mu(Psi)-Psi*d^mu(Psi_conj)). Off shell it obeys the exact identity d_mu*j^mu=i*(Psi_conj*Box(Psi)-Psi*Box(Psi_conj)). If the conjugate equations of motion have Box(Psi)=F(|Psi|^2)*Psi with real F, the current is conserved on shell. A genuinely real field has zero current. For the separately declared stationary ansatz Psi=f(x)*exp(-i*omega*t) with real f and omega>0, the current is (j^0,j^1)=(2*omega*f^2,0); adding a real phase-breaking conjugate-field term lambda*Psi_conj to the equation gives divergence -2*lambda*f^2*sin(2*omega*t), which is generally nonzero. +- `C-U1-002` — Conditional on the independently declared profile Psi=A*sech(eta*x)*exp(-i*omega*t), with A>0, 00, 0=17, with b0=7 at declared n_f=6. Combined with C-RGE-001, a positive b0 gives decreasing one-loop ultraviolet running with zero infinite-scale limit. The loop weights, field content, flavor count, perturbative regime, and physical gauge-sector identification are premises; the claim proves neither a substrate/QCD identity, unique SU(3) selection, nor confinement. +- `C-DIM-007` — Conditional on C-RGE-001, an independently existing positive observable sigma with mass dimension two, and Lambda being its only independent dimensionful mass scale, dimensional homogeneity fixes only the power sigma=k*Lambda^2 for an unconstrained positive dimensionless k. The prefactor remains free and load-bearing. If a second independent mass scale M is admitted, the monomial family Lambda^(2-q)*M^q is dimensionally allowed, so the sole-scale premise is essential. The one-loop equations contain no sigma and admit both zero- and positive-tension assignments; this claim establishes neither existence of a string tension, confinement, a magnitude, nor perturbative control at Lambda. +- `C-VTX-001` — For positive g, lambda, and v and positive integer n, declare the radial Abelian-Higgs convention phi=f(r)*exp(i*n*theta), A_theta=a(r)/(g*r), and energy per unit length 2*pi*integral r*dr*[f'^2/2+f^2*(n-a)^2/(2*r^2) +(a'/r)^2/(2*g^2)+lambda*(f^2-v^2)^2/4]. Exact variation gives f''+f'/r-f*(n-a)^2/r^2-lambda*f*(f^2-v^2)=0 and a''-a'/r+g^2*(n-a)*f^2=0. If f approaches v, finite angular energy uniquely requires a to approach n; the declared connection then has flux 2*pi*n/g, while the ungauged positive-winding profile has a logarithmic divergence. Vacuum linearization gives vector and scalar inverse lengths g*v and v*sqrt(2*lambda), both tending to zero as v tends to zero. This conditional model establishes no substrate, dual, chromoelectric, QCD, or confinement identity and no vortex existence. +- `C-VTX-002` — Conditional on C-VTX-001 with (v,n,lambda,g)=(1,1,2,1), there is resolution-bounded numerical evidence on truncated radial domains for a nontrivial monotone solution with f(epsilon)=a(epsilon)=0 and f(R)=a(R)=1 and finite positive tension approximately 4.21160. The reference collocation solve uses epsilon=1e-4, R=20, 120 initial points, tolerance 1e-8, at most 100000 nodes, maximum RMS residual below 1.1e-8, and uniform 20001-point trapezoidal energy quadrature. Tightening tolerance reduces tension error; R from 10 through 25 agrees within 1e-5; inner-cutoff error decreases from 1e-2 toward 1e-4; exponential and rational guesses converge to the same branch; matched dimensionless v=1 and v=2 domains give tension ratio four within 1e-5. Independent central finite differences at 101, 201, and 401 points give tensions 4.19212, 4.20658, and 4.21037. This is numeric evidence, not a continuum existence or uniqueness theorem, absolute tension, or physical confinement result. +- `C-FLX-001` — Conditional on positive flux Phi uniformly crossing a cross-section A that is fixed independently of positive length L, Gauss data give the constant field E=Phi/A. With declared field-energy density E^2/2, stored field energy is U(L)=Phi^2*L/(2A), linear with energy slope sigma_energy=Phi^2/(2A). Separately, for positive endpoint charge q and declared force F=qE, endpoint work is V(L)=q*Phi*L/A, linear with force slope sigma_force=q*Phi/A. The slopes agree if and only if q=Phi/2; for q=Phi the endpoint slope is twice the energy slope. Fixed area is load-bearing: A(L)=A0*(1+L/L0) gives logarithmic field energy, while spherical spreading gives an inverse-square field and curved Coulomb potential. Matching a supplied tension by A_eff=Phi^2/(2*sigma) defines an effective area and does not predict it. This theorem establishes no physical charge, flux tube, vortex-tension identity, QCD, area law, or confinement. +- `C-LIE-002` — In the standard explicit fundamental SU(3) representation of C-LIE-001, the exact complex 3-by-3 commutant of all eight generators consists only of scalar matrices. Intersecting that commutant with unitary determinant-one matrices gives exactly {omega^k*I_3 | k=0,1,2}, where omega=-1/2+i*sqrt(3)/2, an order-three cyclic group isomorphic to Z_3. A fundamental vector has center phase omega^k, center conjugation on any 3-by-3 matrix and hence the adjoint matrix representation is trivial, and abstract integer trialities compose additively modulo three. This theorem establishes no substrate field assignment, quark or gluon identity, screening dynamics, Wilson law, string tension, or confinement. +- `C-LIE-003` — In the standard explicit fundamental SU(3) representation of C-LIE-001, define d_abc=2*Tr({T_a,T_b}*T_c). The resulting exact rank-three tensor is fully symmetric and every generator anticommutator obeys {T_a,T_b}=delta_ab*I_3/3+sum_c d_abc*T_c. All d_abc with a,b,c restricted to the standard embedded SU(2) generated by T_1,T_2,T_3 vanish, while d_118=1/sqrt(3) is a nonzero witness outside that restriction. These are convention-specific fundamental-representation identities. They do not uniquely characterize SU(3), identify a physical color sector, or supply a gauge action, matter field, coupling, dynamics, dimension, confinement, observation, or substrate mechanism. +- `C-WIL-001` — For positive separation R, Euclidean duration T, and coefficient sigma, conditional on the separately declared rectangular loop expectation W_A(R,T)=exp(-sigma*R*T), the extraction V(R)=-lim_{T->infinity} log(W_A)/T gives V(R)=sigma*R, with derivative sigma and zero second derivative. Separately, for positive rho, the declared perimeter law W_P(R,T)=exp(-2*rho*(R+T)) gives V(R)=2*rho and zero separation derivative. These exact implications derive neither loop law, do not select the area law from center algebra, and establish no physical string tension, gauge phase, or confinement mechanism. +- `C-GAU-001` — Conditional on C-U1-001's smooth complex scalar, a positive coupling e, and a declared real local-U(1) connection A_mu, define D_mu=partial_mu-i*e*A_mu and transform Psi'=exp(i*e*chi)Psi, A_mu'=A_mu+partial_mu chi for arbitrary smooth real chi. Then D_mu Psi transforms covariantly, a phase-independent potential and (D_mu Psi)^*D^mu Psi are invariant, and in C-U1-001's current convention the kinetic expansion is the bare term plus e*A_mu*j^mu+e^2*A_mu*A^mu*|Psi|^2. The curvature F_mu_nu=partial_mu A_nu-partial_nu A_mu is invariant and [D_mu,D_nu]Psi=-i*e*F_mu_nu*Psi. Separately, conditional on nonzero asymptotic amplitude, integer phase winding N, and angular energy with logarithmic coefficient proportional to (N-e*A_theta*r)^2, finite energy forces flux 2*pi*N/e; its charge-e holonomy is +1. A minus-one holonomy requires a separately declared fractional flux. Local covariance leaves every F^2 coefficient unconstrained and establishes no gauge kinetic action, Maxwell equation, photon, force, physical electric charge, or substrate electromagnetic sector. +- `C-QBL-001` — Conditional on the dimensionless 1+1 stationary-profile equation f_xx=(1/2-omega^2-f^2/12)f, C-U1-001's stationary phase Psi=f*exp(-i*omega*t), and 00 for 0 D_L*V*D_R^dagger. For a unitary V whose bipartite nonzero-support graph has c connected components, the diagonal-action stabilizer has real dimension c and its orbit has dimension 2*N-c. On the generic connected-support stratum c=1, so the effective orbit dimension is 2*N-1 and the quotient of U(N) has dimension (N-1)^2. Separating the N*(N-1)/2 real-orthogonal angle dimensions leaves (N-1)*(N-2)/2 irreducible complex-phase dimensions; these are zero for N=2 and one for N=3. Every U(2) matrix is diagonal-rephasing-equivalent to a real orthogonal matrix, and every two-row/two-column quartet has zero imaginary part. For any indices, the quartet Q_ik;jl=V_ij*V_kl*conjugate(V_il)*conjugate(V_kj) is invariant under the declared diagonal action and its imaginary part reverses sign under entrywise complex conjugation. For the declared unitary chart V=R23*R13(delta)*R12, Im(Q_01;12) equals cos(t12)*cos(t23)*cos(t13)^2*sin(t12)*sin(t23)*sin(t13)*sin(delta). Disconnected zero patterns and degenerate singular spectra have enlarged basis freedoms and require their own stabilizer audit. These statements establish no quark or generation map, physical CKM matrix, Cabibbo or KM mechanism, physical CP operation or violation, observed family count, charged current, GIM or anomaly result, measured angle or phase, or substrate realization. +- `C-MOM-001` — Let T^{mu nu} be a smooth symmetric tensor in inertial flat 3+1 coordinates satisfying partial_mu T^{mu nu}=0. Assume localization strong enough that the surface terms for total charges and the coordinate- weighted first and second moments vanish. Define M=integral T^{00} d^3x, P^i=integral T^{0i} d^3x, D^i=integral x^i T^{00} d^3x, and I^{ij}=integral x^i x^j T^{00} d^3x. Then dot M=0, dot P^i=0, dot D^i=P^i, ddot D^i=0, and ddot I^{ij}=2*integral T^{ij} d^3x. Thus the dipole is generally affine in time rather than constant. For normalized STF I_STF=I-delta*Tr(I)/3, ddot I_STF^{ij}=2*integral [T^{ij}-delta^{ij} T^{kk}/3] d^3x. The alternative source convention Q=3*I-delta*Tr(I) is exactly 3*I_STF and has three times this acceleration. Constant translation of the spatial origin leaves ddot I unchanged because ddot M and ddot D vanish. Nonzero boundary flux invalidates the conserved integrated charges, and without T^{i0}=T^{0i}, dot D^i need not equal P^i. These identities establish no gravitational field equation, retarded solution, TT coupling, radiating multipole order, nonzero quadrupole radiation, waveform, power, gravitational coupling, 1+1 contrast, or substrate realization. +- `C-GW-001` — In Euclidean three-space let n be a unit direction, P_ij=delta_ij-n_i*n_j, and let Lambda be the symmetric transverse- traceless projector. For every real symmetric tensor S, integral_{S^2} |Lambda(n) S|^2 dOmega equals (8*pi/5)*|STF(S)|^2, where STF(S)=S-delta*Tr(S)/3; pure trace is therefore annihilated. Conditional on a declared far-field waveform h_TT=(A/r)*Lambda[ddot Q] and declared flux dP/dOmega=B*r^2*, angular integration gives P=(8*pi/5)*B*A^2*<|STF(dddot Q)|^2>. The factors A and B are premises, not consequences of the angular integral. If the normalized source moment of C-MOM-001 is I_STF and Q_s=s*I_STF for nonzero s, the same waveform requires A_s=A_1/s and the power coefficient multiplying |dddot Q_s|^2 scales as 1/s^2. Thus the particular declared inputs A_1=2*G and B=1/(32*pi*G) give G/5 for I_STF, whereas the convention Q=3*I_STF gives G/45 for |dddot Q|^2, not G/5. For a single harmonic Q=C*cos(omega*t)+S*sin(omega*t), the exact cycle average of the squared STF third derivative is omega^6*(|STF(C)|^2+|STF(S)|^2)/2. These results establish only exact projector, angular, convention, and conditional-functional algebra; they establish no gravitational action or field equation, retarded solution, physical Isaacson flux, measured coupling, universal lowest radiating multipole, arbitrary-source radiation, nonlinear gravity, 1+1 lift, or substrate realization. +- `C-GW-002` — For each nonzero direction n in Euclidean three-space, the orthogonal TT projector of C-GW-001 acting on the six-dimensional real vector space of symmetric 3-by-3 tensors has rank and trace two, with eigenvalues two ones and four zeros. Choose an oriented orthonormal frame (u,v,n_hat). The tensors e_plus=(u*u^T-v*v^T)/sqrt(2) and e_cross=(u*v^T+v*u^T)/sqrt(2) are symmetric, transverse, traceless, and Frobenius-orthonormal. They are complete for the image: TT_n(S)=(e_plus:S)*e_plus+(e_cross:S)*e_cross for every symmetric S. If the transverse frame changes by u'=cos(psi)u+sin(psi)v and v'=-sin(psi)u+cos(psi)v, then e_plus'=cos(2psi)e_plus+sin(2psi)e_cross and e_cross'=-sin(2psi)e_plus+cos(2psi)e_cross. Consequently the declared circular combinations (e_plus plus/minus i*e_cross)/sqrt(2) acquire the opposite algebraic phases exp(minus/plus 2*i*psi). A deterministic frame may cover all nonzero directions piecewise, but no global continuity is asserted. The unnormalized axis tensors used by GW3 have norm squared two and require coefficient division by two. These results establish no gravitational action, propagating field equation, constraint or gauge quotient, physical polarization observable, graviton count, physical helicity, quantum state, radiation channel, or substrate realization. +- `C-GW-003` — Let two equal point masses m follow the declared paths x_1(t)=a*(cos(Omega*t),sin(Omega*t),0) and x_2(t)=-x_1(t), where a is each mass's orbital radius and the separation is 2*a. Their monopole is 2*m, their dipole is zero, and for the normalized moment I_STF=I-delta*Tr(I)/3 the exact Frobenius norms are |ddot I_STF|^2=32*m^2*a^4*Omega^4 and |dddot I_STF|^2=128*m^2*a^4*Omega^6. The triple convention Q=3*I_STF has derivative norms nine times larger. Conditional on h_TT=(A/R)*TT(ddot I_STF), line of sight n=(sin(i),0,cos(i)), and oriented transverse frame p=(cos(i),0,-sin(i)), v=(0,1,0), the conventional matrix read-offs are h_plus=-(2*A*m*a^2*Omega^2/R)*(1+cos(i)^2)*cos(2*Omega*t) and h_cross=-(4*A*m*a^2*Omega^2/R)*cos(i)*sin(2*Omega*t). The normalized C-GW-002 basis coordinates are sqrt(2) times these read-offs. Face-on coefficients have equal amplitude in quadrature; edge-on cross vanishes and plus has half the face-on amplitude. With the separately declared inputs A=2*G and B=1/(32*pi*G), C-GW-001 gives conditional power 128*G*m^2*a^4*Omega^6/5. Equivalently Q=3*I_STF requires waveform coefficient 2*G/3 and power coefficient G/45; combining Q with the unscaled coefficient creates a factor-three field and factor-nine power error. This theorem treats the paths as kinematic inputs and establishes no binding stress, orbital law, breather embedding, isolated conserved 3+1 source, gravitational action or coupling, retarded dynamics, energy loss, detector strain, astrophysical prediction, or substrate identity. +- `C-SG-009` — For every C-SG-001 rest breather with 0=(1/2)*sum_{k>=1}(2*k*omega)^6*a_k^2. Resolution-bounded direct adaptive quadrature of the exact third derivative, stable under 4, 8, 16, and 32 interval subdivisions, gives 379.4646380687 < < 379.4646380688. Independent sixty-digit manual differentiation and cosine quadrature through sixteen harmonics agree, with a decreasing truncation error before the double-precision floor. The k=1 term, at angular frequency 2*omega, contributes a fraction between 0.8053698716 and 0.8053698718 of the total derivative mean square and is larger than the combined higher-harmonic contribution. These numerical bounds concern this declared frequency and scalar 1+1 moment; they establish no family-wide dominance theorem, three-dimensional source, gravitational coupling, radiated power, waveform, detector signal, or substrate realization. +- `C-GW-004` — Let mu(t) be the exact C-SG-009 longitudinal moment and apply the conditional axisymmetric separable construction of C-MOM-002. Writing d=mu'''(t), the normalized tensor satisfies |I_STF'''|^2=2*d^2/3 and the triple tensor Q=3*I_STF satisfies |Q'''|^2=6*d^2. Under the separately declared C-GW-001 inputs A=2*G and B=1/(32*pi*G), the normalized convention therefore has conditional instantaneous power P(t)=2*G*d^2/15. The triple convention requires waveform coefficient 2*G/3 and power coefficient G/45 and gives the identical result; using 2*G and G/5 with triple Q multiplies the field by three and power by nine. This conditional power is nonnegative and nonzero at some phases, but it vanishes at the exact minimum and maximum symmetry phases where d=0. For a line of sight in the x-z plane at inclination i from the symmetry x axis, choose the oriented transverse frame p=(sin(i),0,-cos(i)), q=(0,1,0). If e=mu''(t), the normalized TT projection has plus coordinate e*sin(i)^2/sqrt(2), conventional matrix readout e*sin(i)^2/2, and zero cross coordinate. Thus the conditional waveform h_TT=(2*G/R)*TT[I_STF''] has conventional plus readout G*e*sin(i)^2/R, zero cross, and an exact symmetry-axis null; the inverse-rescaled triple convention gives the same waveform. These are exact consequences of declared moment, projector, waveform, and flux inputs. They establish no conserved isolated 3+1 source, gravitational action or field equation, physical retarded solution or flux, radiation channel, detector strain, backreaction, or substrate realization. +- `C-PDE-001` — Adopt the C-SG-001 normalized sine-Gordon potential as a declared dimensionless 3+1 flat-space radial model with action S=4*pi*integral dt dr r^2[u_t^2/2-u_r^2/2-(1-cos(u))]. Its equation is u_tt-u_rr-2*u_r/r+sin(u)=0, with even regularity u_r(0,t)=0. For initial data u(r,0)=3*exp(-(r/4)^2), u_t(r,0)=0, a direct-radial centered leapfrog on 0<=r<=200 and 0<=t<=450 with dr=0.05, dt=0.02, outer Dirichlet data, and a quadratic velocity sponge over 1500, integration over one period gives R=4*sign(A)*B*cos(beta-alpha)/omega. A cosine convention for the second trace instead gives -4*sign(A)*B*sin(beta-alpha)/omega, so the phase convention is load-bearing. Separately, on a right half-line x>=b with orientation epsilon^(01)=+1 and time-independent field at positive infinity, the C-SG-011 topological charge changes by Delta Q=-Delta phi(b)/(2*pi)=-(1/(2*pi))*integral phi_t(t,b) dt. Neither this winding integral nor R implies the other: a complete sinusoidal phi_t period has Delta Q=0 while R can be nonzero, R can vanish for a separately nonzero boundary field change, and R scales continuously with B. For every fixed boundary point the exact C-SG-001 rest breather has zero complete-period R; at its symmetry center phi_x vanishes identically. These results establish a conditional boundary correlation and transformation law, not a topological or conserved charge, quantization, charge-transfer discriminator, parity-invariant or parity-breaking boundary condition, selected state, physical parity violation, chiral anomaly, V-A interaction, weak force, particle identity, or substrate realization. +- `C-PDE-005` — For the dimensionless three-dimensional radial sine-Gordon equation of C-PDE-001, let H be a finite set of positive odd integers containing one and set u(r,t)=sum_(n in H) a_n(r)*cos(n*omega*t). With S_n[a]=(1/pi)*integral_0^(2*pi) sin(sum_(m in H) a_m(r)*cos(m*tau))*cos(n*tau) dtau, exact Fourier projection gives a_n''+2*a_n'/r+(n*omega)^2*a_n-S_n[a]=0. Odd harmonics give exact half-period antisymmetry. Even radial regularity gives a_n'(0)=0 and the origin curvature law 3*a_n''(0)+(n*omega)^2*a_n(0)-S_n[a](0)=0. In the linear far field each mode obeys a_n''+2*a_n'/r+((n*omega)^2-1)*a_n=0: n*omega<1 is evanescent with rate sqrt(1-(n*omega)^2), n*omega=1 is threshold, and n*omega>1 is radiative with wavenumber sqrt((n*omega)^2-1). A nonzero real radiative one-over-r tail has positive asymptotic energy per unit radial length and hence infinite integrated three-dimensional energy. Thus a sub-threshold fundamental does not localize its higher channels, and a Dirichlet wall on a radiative harmonic fixes a finite-box standing-wave phase rather than proving an infinite-domain finite-energy breather. These exact conditional statements establish no existence, uniqueness, nonzero radiative coefficient, exact periodic solution, lifetime, gravity, particle identity, absolute scale, or substrate realization. +- `C-PDE-006` — On the C-PDE-005 odd-harmonic system, declare the free branch coordinate a_1(0)=2.5, origin cutoff 0.001, outer radius 40, a decaying Robin condition on the evanescent fundamental, and finite-box Dirichlet data on every radiative harmonic. SciPy adaptive collocation in IEEE float64, initialized with 300 radial points, periodic-DFT projection with 256 temporal samples, tolerance 1e-8, and a fitted frequency constrained to 0tau+pi, while the canonical energy density T00=(u_t^2+u_r^2)/2+1-cos(u) is invariant. Therefore T00 has half the field period and every odd temporal Fourier coefficient of T00, and of any defined time-independent radial linear functional of T00, vanishes exactly. This selection rule permits only DC and even harmonics; it does not require any allowed coefficient to be nonzero, lowest, or dominant. In particular, for the local single-mode field u=a(r)*cos(tau), the cos(2*tau) coefficient is a_r^2/4-omega^2*a^2/4+2*J_2(a), so the gradient, kinetic, and potential terms can cancel while a higher even coefficient remains nonzero. For every radial energy density with finite second moment, C-MOM-003 gives I_ij=(delta_ij/3)*4*pi*integral r^4*T00 dr and identically zero STF part at each phase. These exact kinematic and Fourier statements do not establish that the ansatz solves the full PDE, a nonzero STF source, physical radiation, gravity, waveform, flux, absolute scale, particle identity, or substrate realization. +- `C-PDE-008` — On the accepted finite-box branch C-PDE-006, define the core radial second energy moment S_12(tau)=4*pi*integral_0.001^12 r^4*T00(r,tau) dr and the real series S_12=a_0+sum_k(a_k*cos(k*tau)+b_k*sin(k*tau)). With central fundamental 2.5, wall radius 40, IEEE float64 adaptive collocation initialized by 300 radial points, 256 projection phases, tolerance 1e-8, a separate 2401-point exact-cutoff radial audit grid, and 512 endpoint-excluded phases, the N=1, 3, 5, 7, and 9 values of a_2 are 666.330281099, 591.504983105, 591.470022142, 591.470478411, and 591.470484284. At N=9 the twice-frequency coefficient supplies 0.999865185 of the resolved even coefficient power, the exact C-PDE-007 rule removes odd coefficients, and the time-averaged core per-axis variance is 7.827021539. Temporal samples 256/512/1024, radial samples 1201/2401/4801, initial BVP meshes 200/300/400, and tolerance 1e-8 versus 1e-9 with 512 projection phases preserve a_2 within 0.008. Across the harmonic ladder, the full nonlinear core remainder falls from 0.105185 to 1.36601e-5 and the full-box energy relative range from 0.0581543 to 9.6871e-7. Independent Gauss-Legendre phase and Simpson radial integration gives a_2=591.468056462. Walls 30, 40, 50, and 60 give core coefficients 591.269664499, 591.470484284, 598.370080354, and 590.990332998 and expose a full-box scalar-variance resonance near wall 50. This is numeric evidence for one cutoff scalar moment on one finite-box, finite-harmonic family point. It establishes no wall- independent or infinite-domain line, exact full-PDE periodic solution, universal nonzero twice-frequency theorem, STF quadrupole, physical radiation, gravity, waveform, flux, absolute scale, particle identity, or substrate realization. +- `C-PDE-009` — In the dimensionless radial-background linearization of C-PDE-003, let n=(n_x,n_y,n_z) be a unit direction and use the unnormalized real l=2 angular basis P2(n_z), n_x^2-n_y^2, 2*n_x*n_y, 2*n_x*n_z, and 2*n_y*n_z. Every basis element obeys Delta_Omega Y=-6*Y. Consequently, for any sufficiently differentiable radial background P(r,t), every real m component has the same radial equation psi_tt-psi_rr-2*psi_r/r+6*psi/r^2+cos(P)*psi=0 and the same regular origin law psi=O(r^2); m-degeneracy does not supply a separated frequency, normalization, or mode existence. Replacing cos(P) by a time average Cbar(r) defines a different equation whose exact omitted term is (cos(P)-Cbar)*psi. For P=a(r)*cos(tau), Cbar=J_0(a) and cos(P)-Cbar=-2*J_2(a)*cos(2*tau)+2*J_4(a)*cos(4*tau)-..., with leading small-a term -a^2*cos(2*tau)/4. Thus an eigenfunction of the averaged radial operator is a solution of the full linearized equation only when the displayed pointwise defect vanishes or a separate Floquet argument supplies the missing time dependence. At a positive cutoff epsilon the regular leading series satisfies epsilon*psi_r-2*psi=O(epsilon^4), so zero value paired with nonzero derivative is not nontrivial regular l=2 data. These are exact angular, equation, and regularity statements. They establish no averaged or Floquet eigenmode, bound state, frequency, stability, nonlinear deformation, infinite-domain localization, gravity, radiation, absolute scale, or substrate realization. +- `C-PDE-011` — In the normalized dimensionless 1+1 sine-Gordon model, consider phi_tt-phi_xx+sin(phi)=J-gamma*phi_t on -80<=x<=80 with zero initial field and velocity, homogeneous Dirichlet endpoints, and final time 410. Let omega_d=1/sqrt(2), eta=sqrt(1-omega_d^2), tau=4/omega_d, and J=A*sech(eta*x)*exp(-(t-30)^2/(2*tau^2))*sin(omega_d*(t-30)), where A is fixed so the declared full-line temporal proxy integral f(t)^2 dt is 400. Let gamma vanish for |x|<=40 and equal ((|x|-40)/40)^2 outside. Centered homogeneous-Dirichlet leapfrog evolutions with spatial steps 0.05, 0.025, and 0.0125, timestep 0.4*dx, radius-12 diagnostics, and the late window t>320 give constrained exact C-SG-001 center-trace fits with relative RMS errors 0.0197 through 0.0220 and fitted frequencies 0.295653, 0.250839, and 0.239700; successive frequency differences decrease by about four. The late core energies 15.4464, 15.6667, and 15.7115 differ from the corresponding C-SG-002 fitted energies by less than 1.2 percent, while final radius-12 field-velocity snapshot fits have joint relative L2 errors 0.0572, 0.0609, and 0.0966. The final-energy minus source-work plus damping-loss residual, relative to source work, decreases from 1.08e-3 to 1.36e-4 to 3.03e-5. Timestep halving, outer domain 120, sponge widths 30 and 50, adaptive DOP853 at dx=0.05 and 0.025, an independent rising-crossing frequency estimate, planted exact breathers, and nonbreather counterstates preserve the bounded verdict. Changing only the proxy target from 400 to 380 or 420 breaks the composite classifier. This is simulation evidence for one exact-source, finite-box, finite-time formation trajectory close to the exact rest breather family. It establishes no robust parameter interval, preferential fast or resonant seeding, slow-drive exclusion, equal-work comparison, continuum or asymptotic existence, physical deposition mechanism, voltage-slew law, probability, population, absolute scale, particle identity, or substrate realization. +- `C-PDE-012` — Let ell be a nonnegative integer and let a sufficiently differentiable real radial mode g on r>0 obey -g''-2*g'/r+ell*(ell+1)*g/r^2+V(r)*g=E*g. The substitution chi=r*g gives exactly -chi''+[ell*(ell+1)/r^2+V(r)]*chi=E*chi, preserves the real radial norm integral r^2*g^2 dr=integral chi^2 dr, and maps regular g=O(r^ell) to chi=O(r^(ell+1)). On a regular finite Dirichlet ball of radius R with constant vacuum potential mu^2, the radial modes are spherical Bessel j_ell(z_(ell,n)*r/R) and E_(ell,n)=mu^2+(z_(ell,n)/R)^2, where z_(ell,n) is a supplied positive zero of j_ell. Thus the vacuum wall gap scales exactly as R^-2 and its normalized radial shape scales with r/R; neither fact establishes a localized half-line mode. More generally, if W_ell=ell*(ell+1)/r^2+V-mu^2 is nonnegative almost everywhere and the self-adjoint boundary form vanishes, the quadratic form of H-mu^2 is nonnegative, so no eigenvalue lies below mu^2. This implication is exact conditional on the pointwise premise; finite sampling of W_ell does not prove it. An endpoint-decay predicate evaluated after the endpoint or outer tail has been imposed as zero is non-discriminating whenever the interior amplitude clears the predicate's floor. Applied to C-PDE-009, these statements type only the supplied time-averaged finite-wall operator and establish no averaged or Floquet mode, nonlinear deformation, all-channel nonexistence, gravity, radiation, absolute scale, particle identity, or substrate realization. +- `C-GW-007` — Let a scalar density's coefficients in the unnormalized real l=2 basis P2(n_z), n_x^2-n_y^2, 2*n_x*n_y, 2*n_x*n_z, and 2*n_y*n_z have radial moments H_20, H_2c, H_2s, H_1c, and H_1s, where each H=4*pi*integral r^4*h(r) dr. In the C-MOM-003 triple-STF convention, exact angular integration gives Q_xx=-H_20/5+2*H_2c/5, Q_yy=-H_20/5-2*H_2c/5, Q_zz=2*H_20/5, Q_xy=2*H_2s/5, Q_xz=2*H_1c/5, and Q_yz=2*H_1s/5. A nonzero pure real-m=2 cosine coefficient therefore gives diag(2H/5,-2H/5,0), which is traceless and has three distinct eigenvalues. Along the z sightline with x reference, the conventional plus and cross matrix readouts are 2*H_2c/5 and 2*H_2s/5; coordinates in the normalized unit-Frobenius TT basis are larger by sqrt(2). For sampled, DC-removed coefficient traces, temporal source rank is the matrix rank with time samples as rows and declared angular components as columns. A fixed tensor direction times one scalar trace has rank at most one even when both coordinate readouts are nonzero; two nonproportional traces are required for rank two, and invertible polarization-frame rotation preserves that rank. By the exact m-degeneracy C-PDE-009, pairing the accepted C-PDE-004 radial solution with n_x^2-n_y^2 gives the genuine first-order finite-time tensor diag(q(t),-q(t),0), where q is C-PDE-004's accepted P2 Q_zz/epsilon trace. Its natural conventional plus trace therefore inherits RMS 404.678 and maximum absolute value 680.589, with cross zero and temporal rank one. These are exact moment, TT-coordinate, rank, and dependency- transfer statements. They establish no finite nonlinear deformation, localized or periodic eigenmode, rank-two source evolution, conserved gravitational source, gravity theory, physical waveform or radiation, flux, graviton count, absolute scale, or substrate realization. +- `C-GW-008` — Let I_STF be the normalized symmetric trace-free moment of C-MOM-001 and let Q_s=s*I_STF for a declared nonzero convention scale s. Under exactly the conditional C-GW-001 premises A=2*G for I_STF and B=1/(32*pi*G), the same field written with Q_s is h_TT=(2*G/(s*R))*TT(Q_s''), and its instantaneous or already-averaged total angular power is P=G*(Q_s'''_ij*Q_s'''_ij)/(5*s^2). Thus the triple convention s=3 requires waveform coefficient 2*G/3 and power coefficient G/45; applying 2*G and G/5 directly to the triple tensor multiplies the field by three and power by nine. For the C-GW-007 triple real-m2 tensor Q_3=[[q_c,q_s,0],[q_s,-q_c,0],[0,0,0]], viewed along z with x reference, the conventional conditional readouts are h_plus*R/G=2*q_c''/3 and h_cross*R/G=2*q_s''/3, while P/G=2*((q_c''')^2+(q_s''')^2)/45. If instead Q_s(t)=q(t)*T for one fixed STF tensor T, every observer-frame plus/cross pair is a constant vector times q''(t), has temporal rank at most one, and, when nonzero, admits a spin-two transverse-frame rotation with cross equal to zero; two nonzero coordinates in one frame do not establish elliptical or two-mode radiation. Rank two requires nonproportional coefficient traces. In the conditional comparison q_c=A0*cos(w*t), q_s=A0*sin(w*t), the natural-z waveform is circular with constant squared coordinate radius (2*G*A0*w^2/(3*R))^2 and constant power 2*G*A0^2*w^6/45. These are exact convention, TT, power, and temporal-rank consequences of the declared premises. They establish no conserved localized source, excitation of both traces by accepted scalar dynamics, gravitational action or field equation, physical retarded waveform or flux, graviton count, detector signal, backreaction, absolute scale, particle identity, or substrate realization. +- `C-WZW-001` — In the explicit fundamental SU(3) convention of C-LIE-001, let E_a=i*T_a, so E_a is anti-Hermitian and [E_a,E_b]=-f_abc*E_c, and define the real left-invariant five-cochain Omega_5=-i*Alt Tr(theta^5), where theta=g^{-1}dg and Alt is the unnormalized signed permutation sum with no hidden 1/5! factor. The exact Chevalley-Eilenberg differentials built from those structure constants obey d_5*d_4=0, have rank(d_4)=35 and rank(d_5)=20, and hence have a 36-dimensional degree-five cocycle kernel and one-dimensional invariant fifth cohomology. Omega_5 has nine nonzero basis components, squared coefficient norm 75/4, obeys d_5*Omega_5=0, and is not in image(d_4): appending it raises the image rank from 35 to 36, while independently Omega_5^T*d_4=0 and Omega_5^T*Omega_5=75/4. Because SU(3) is compact and Omega_5 is left invariant, a hypothetical global primitive could be averaged with normalized Haar measure to an invariant primitive; the exact non-image result therefore makes Omega_5 globally non-exact without assigning any period normalization. The density is a local, metric-free polynomial in a smooth group-valued map and its first derivatives. For an ungauged variation delta U=U*v, delta Tr(theta^5)=d(5*Tr(v*theta^4)); this is an exact boundary identity. If two compatible oriented fillings have integrals I_B and I_Bprime, their glued closed-cycle period is I_B-I_Bprime and the extension phase ratio for coefficient c is exp(i*c*(I_B-I_Bprime)); filling independence follows only under the additional premise that c times every allowed period lies in 2*pi*Z. The theorem fixes no generator period, integer level, WZW coefficient, N_c, baryon current or charge, gauge connection, Chern-Simons descent, anomaly inflow, representation selection, physical bulk or boundary dynamics, absolute scale, or substrate realization. +- `C-WZW-002` — In C-WZW-001's convention Omega_5=-i*Alt Tr(theta^5), orient the unit S^5 as the boundary of the unit ball in (Re z1,Im z1,Re z2,Im z2,Re z3,Im z3). The explicit Puttmann-Rigas map eta(z)=z*z^T+A(conjugate(z)), with A(conjugate(z)) the displayed complex cross-product matrix, obeys eta^dagger*eta=I and det eta=1 on |z|=1. The regular value (1,0,0) of its first-column projection has exactly the preimages +(1,0,0) and -(1,0,0), both with oriented real Jacobian determinant 8, so the projection has degree +2. By the audited U(n-1)->U(n)->S^(2n-1) generator criterion, eta is the positive generator of pi_5(SU(3))=Z. Equivariance makes eta^*Omega_5 an invariant top form on S^5. On the positive tangent frame at (1,0,0), exact evaluation gives Alt Tr(theta^5)=-480*i and Omega_5=-480; since Vol(S^5)=pi^3, the oriented primitive periods are -480*i*pi^3 for the raw trace and -480*pi^3 for Omega_5. Consequently a map in homotopy class n has real sphere period -480*pi^3*n. For two oriented five-ball fillings of a common S^4 boundary whose glued map has winding n, for real k the coefficient c=k/(240*pi^2) gives phase ratio exp(-2*pi*i*k*n)=1 for all integer n exactly when k is an integer; orientation reversal changes the period sign but not this lattice. This is a mathematical sphere-filling level theorem. It does not fix periods on arbitrary closed five-manifolds, identify k with N_c, or establish a WZW action from substrate dynamics, baryon number, representation selection, a gauge anomaly, descent, inflow, absolute scale, or any physical realization. +- `C-TOP-002` — In C-LIE-001's fundamental trace convention, let theta=U^dagger*dU for a smooth SU(3)-valued map. The exact invariant Chevalley-Eilenberg differentials have rank(d_2)=20 and rank(d_3)=35, so the degree-three cocycle kernel has dimension 21 and H^3 has dimension one. The real cochain Alt Tr(theta^3) has nine nonzero components, squared coefficient norm 9, is closed, and is not in image(d_2), since adjoining it raises the image rank from 20 to 21. On the unit quaternion sphere oriented as the boundary of (a0,a1,a2,a3), the upper-SU(2)-block map q(a)=a0*I+i*(a1*sigma1+a2*sigma2+a3*sigma3) embedded in SU(3) has a first-column real coordinate map of determinant and degree +1, hence is a pi_3(SU(3)) generator under the audited stable inclusion criterion. Its exact oriented tangent density is Alt Tr(theta^3)=12 and its raw period is 24*pi^2. Therefore omega_3=-Alt Tr(theta^3)/(24*pi^2) has period -1 on that positive generator. With epsilon^(0123)=+1, the corresponding coordinate current J^mu=-(1/(24*pi^2))*epsilon^(mu nu rho sigma)*Tr(L_nu L_rho L_sigma), L_mu=U^dagger*partial_mu U, is identically conserved for every smooth U: the full graded derivative reduces by Maurer-Cartan flatness to the alternating trace of four one-forms, which vanishes by graded cyclicity. For the static upper-block hedgehog U=cos(F(r))*I+i*sin(F(r))*rhat.sigma, its local density for r>0 is -sin(F)^2*F'/(2*pi^2*r^2), its angularly integrated radial density is -2*sin(F)^2*F'/pi, and its charge is [F-sin(F)*cos(F)]_(outer)^(inner)/pi. Smooth constant endpoint data F(0)=n*pi and F(infinity)=0 therefore give charge n, while reversing orientation reverses the charge. This is a mathematical winding-current theorem. It is not by itself a Noether current, gauged-WZW-response current, physical baryon current, anomaly, identification with N_c, representation selection, absolute-scale statement, or substrate realization. +- `C-EFT-001` — Let V, J_even, and J_odd be real n-entry columns and let K be a nonempty real symmetric invertible n-by-n kernel. In the declared plus-source convention L(V)=V^T*K*V/2+V^T*J with J=J_even+J_odd, component stationarity gives V_star=-K^-1*J, the exact residual K*V_star+J=0, and the reduced term L_eff=-J^T*K^-1*J/2. Its source decomposition consists of the two even squares -J_even^T*K^-1*J_even/2 and -J_odd^T*K^-1*J_odd/2 plus the cross term -(J_even^T*K^-1*J_odd+J_odd^T*K^-1*J_even)/2. Under the declared bookkeeping K and J_even are parity even and J_odd is parity odd, so only the cross term changes sign; it vanishes when either source is absent. For K=M+D with symmetric invertible M and symmetric D, define A=M^-1*D and R_N=sum_(n=0)^N((-A)^n*M^-1). Exact multiplication gives R_N*(M+D)-I=(-1)^N*A^(N+1) and (M+D)*R_N-I=(-1)^N*M*A^(N+1)*M^-1. Thus a finite low-momentum inverse expansion is only a formal truncation under separately supplied power counting and convergence premises; its returned nonzero residual cannot be identified with an exact inverse. Finally, for a field-dependent stationary substitution, the chain rule gives delta Gamma_eff=(delta Gamma)_V+(partial Gamma/partial V)_star*delta V_star, so the induced-field term vanishes on the actual stationary equation while the supplied explicit variation remains. Consequently, if a starting functional is an inhomogeneous term plus invariant local terms with free coefficients, stationary elimination neither selects those coefficients nor creates a missing inhomogeneous anomaly variation. This is a conditional finite-dimensional action theorem. It fixes no field content, source, kernel, mass, coupling, boundary term, operator basis, or coefficient and supplies no HLS field content, no physical vector meson, no WZW functional, no anomaly coefficient, no vector dominance or KSRF relation, no baryon interpretation, no N_c, no absolute scale, and no substrate realization. +- `C-SYM-001` — Let phi be a real nonempty n-entry scalar-coordinate column, let V(phi) be twice differentiable, and let T_a be a finite nonempty supplied family of real n-by-n linear generators. Define the exact infinitesimal invariance residuals r_a(phi)=grad(V)^T*T_a*phi. Direct differentiation gives grad(r_a)=H*T_a*phi+T_a^T*grad(V), where H is the Hessian of V. Therefore, if every r_a vanishes identically and a declared vacuum phi_0 is actually stationary, then H(phi_0)*T_a*phi_0=0 for every supplied generator. The rank of the matrix whose columns are the actual tangents T_a*phi_0 is the number of independent Hessian zero directions certified by these premises. The kernel dimension of the coefficient-to-tangent map is a stabilizer dimension only when the supplied generator matrices form an independent basis; dependent labels cannot inflate the rank. If a separately supplied symmetric kinetic metric K is provably positive definite, the same tangents are zero directions of the generalized quadratic mass operator K^-1*H. Positive K preserves but does not create Hessian zeros. At a nonstationary point, under explicit symmetry breaking, or without an independent generator basis, the corresponding conclusion or interpretation does not follow. This is an exact conditional finite-dimensional classical quadratic theorem. It supplies no quantum Goldstone-particle theorem, no field-theory vacuum or charge algebra, no spectral pole, no group or representation selection, no physical field identification, no mass scale, and no substrate realization. +- `C-CHI-001` — In the declared four-real-coordinate model phi=(sigma,pi1,pi2,pi3), use all six standard independent antisymmetric so(4) generators and V=lambda*(phi^T*phi-v^2)^2 with lambda>0 and v>0. At the declared vacuum phi_0=(v,0,0,0), all six infinitesimal invariance residuals and the gradient vanish, the generator-tangent matrix has rank three, its coefficient kernel has dimension three, and the exact Hessian is diag(8*lambda*v^2,0,0,0). Thus this declared classical model has one radial curvature and three independent zero generalized quadratic-mass directions when supplied a positive kinetic metric. At the symmetric stationary point phi=0 the tangent rank is zero. For the explicitly tilted potential V-c*sigma, a positive shifted stationary branch s0 obeys c=4*lambda*s0*(s0^2-v^2) and has transverse curvature c/s0; an anisotropic quadratic term likewise breaks the relevant invariance and lifts its tangent. Separately, for the declared coordinate model U=exp(i*tau_a*pi_a/F) with Pauli matrices and L=A*Tr(partial U*partial U^dagger), the exact leading trace is 2*sum_a(partial pi_a)^2/F^2 and the scalar kinetic metric is (4*A/F^2)*I. Consequently A=F^2/4 gives metric I and quadratic coefficient one half, while A=F^2/16 gives metric I/4 and coefficient one eighth in the same coordinates. A zero potential has zero Hessian; adding m^2*sum_a(pi_a^2)/2 gives Hessian m^2*I. These are conditional O(4) and SU(2) coordinate-model identities depending on C-SYM-001. They establish no chiral symmetry action or its physical breaking, no quantum Goldstone-particle theorem, no physical pion identification, no sigma or nucleon particle, no GMOR relation, no Skyrmion connection, no value of F_pi or a condensate, no absolute mass scale, and no substrate realization. +- `C-BRK-001` — Let x be a real scalar coordinate, let A be real, and let F, q, and K be positive. For the declared periodic potential V(x)=A*(1-cos(q*x/F)) and scalar quadratic kinetic convention L_kin=K*(partial x)^2/2, x=0 is stationary, the period is 2*pi*F/q, the exact origin curvature and fourth derivative are A*q^2/F^2 and -A*q^4/F^4, and the generalized quadratic mass squared is A*q^2/(K*F^2). Its series through sixth order is A*q^2*x^2/(2*F^2)-A*q^4*x^4/(24*F^4) +A*q^6*x^6/(720*F^6). Separately, the periodic potential h*F^2*(1-cos(x/F)) and quadratic potential h*x^2/2 have the same origin Hessian h, but only the first is periodic and their origin fourth derivatives differ by -h/F^2. Thus local quadratic curvature does not select a global explicit-breaking potential. These are exact declared- coordinate identities. They derive no field ontology, symmetry-breaking source, physical mass, coefficient, absolute scale, or substrate map. +- `C-CHI-002` — Conditional on C-CHI-001's Pauli convention, let U=exp(i*q*tau3*pi/F) with positive F and q, and declare the Lagrangian terms Z*Tr(partial U*partial U^dagger)+C*Tr(U-I_2), with positive Z and real C. Then Tr(U-I_2)=2*cos(q*pi/F)-2. Taking the potential as -C*Tr(U-I_2), the scalar kinetic coefficient is 4*Z*q^2/F^2, the origin potential curvature is 2*C*q^2/F^2, and the generalized quadratic mass squared is C/(2*Z), independent of q and F. In the separately declared Skyrme coefficient pair Z=F^2/16 and C=m^2*F^2/8, this generalized mass is m^2: q=1 gives kinetic coefficient 1/4 and curvature m^2/4, while q=2 gives coefficient one and curvature m^2. By contrast, pairing the q=1 kinetic coefficient 1/4 with V=m^2*F^2*(1-cos(pi/F)) gives generalized mass squared 4*m^2, and matching that V to -C*Tr(U-I_2) requires C=m^2*F^2/2, four times the declared Skyrme trace coefficient. These are convention-covariant coordinate identities. They establish no chiral action from the framework, physical pion or decay constant, derived coefficient, GMOR relation, numerical mass, or substrate realization. +- `C-GMR-001` — Conditional on positive quark-mass sum m_q, positive decay scale F, positive convention factor c, negative condensate Sigma, and the separately declared relation M^2*F^2=-c*m_q*Sigma, exact solution gives M^2=-c*m_q*Sigma/F^2>0. Holding the other inputs fixed, its logarithmic sensitivities with respect to m_q, Sigma, F, and c are respectively 1, 1, -2, and 1, and M^2 tends to zero as m_q tends to zero. If m_q, Sigma, F, and c have mass dimensions 1, 3, 1, and 0, both sides have mass dimension four. For every positive rho, replacing F by rho*F and Sigma by rho^2*Sigma leaves M^2 unchanged, so the relation alone does not determine its inputs or a numerical mass. This exact conditional parameter ledger does not derive GMOR, a chiral current or vacuum, a condensate convention or value, a decay constant, quark masses, a physical pion identity, QCD dynamics, or a substrate realization. +- `C-MOD-001` — For r>0, declare the dimensionless reduced energy E=4*pi*integral[(r^2+2*sin(f)^2)*f'^2+2*sin(f)^2 +sin(f)^4/r^2]dr. Its exact Euler--Lagrange equation is (r^2+2*sin(f)^2)*f''+2*r*f'+sin(2*f)*(f'^2-1) -sin(2*f)*sin(f)^2/r^2=0. For f+epsilon*eta, the coefficient of epsilon^2 before integration by parts is A*eta'^2+B*eta*eta'+D*eta^2, where A=r^2+2*sin(f)^2, B=4*sin(2*f)*f', and D=2*cos(2*f)+2*cos(2*f)*f'^2 +2*sin(f)^2*(3*cos(f)^2-sin(f)^2)/r^2. Under endpoint data that cancel the boundary term, the self-adjoint quadratic operator is H*eta=-(A*eta')'+C*eta with C=D-B'/2 and separately declared kinetic weight W=A; its Green boundary form is A*(u*v'-u'*v). For the scale family f(exp(s)*r), the tangent is r*f' and the declared two-/four- derivative energy scales as exp(-s)*E2+exp(s)*E4, so E2=E4 gives zero first derivative but positive curvature E2+E4 rather than a dilation zero mode. For a regular-origin half-line realization with the declared massless tail f,f' tending to zero sufficiently fast, A/W tends to one and C/W tends to 2/r^2 and then zero, so the continuum edge is Omega^2=0. A positive finite-Dirichlet-box eigenvalue is therefore not below that continuum edge. These are exact conditional reduced-model identities. They derive no physical Skyrme action, soliton or bound state, nucleon, Roper, spin or isospin, quantization, absolute scale, or substrate realization. +- `C-MOD-002` — Conditional on C-MOD-001's declared massless radial model, two independent float64 routes give resolution-bounded evidence for the same nontrivial stationary branch and for a finite-box continuum ladder of its complete self-adjoint Hessian. DOP853 shooting from r=10^-4 with r*f'(R)+2*f(R)=0, rtol=10^-10, atol=10^-12, max step 0.02, walls R=12,18,24, and 801,1201,1601 uniform samples gives fitted origin slopes 2.007528281, 2.007528220, 2.007528217 and energy coefficients 1.230872939, 1.231275966, 1.231374077; the relative E2/E4 imbalance falls from 4.65e-4 to 5.81e-5, and independently finite-differenced EOM residuals fall by approximately four under each spacing halving. A consistent-mass linear-FEM generalized eigensolve with tolerance 10^-10 gives lowest squared box frequencies 0.131132401, 0.061072240, and 0.034754127 at those walls, maximum relative algebraic residual below 3e-9, and node counts 0,1,2,3 for the first four modes. An independent solve_bvp collocation route with tolerance 2e-8, Simpson quadrature, and a mass-lumped finite-volume tridiagonal eigensolve agrees on the profiles and lowest levels within 0.07 percent. The positive lowest level decreases under domain growth while its product with R^2 stabilizes; none lies below C-MOD-001's exact zero continuum edge. This is conditional numeric evidence for a stationary profile and wall-quantized continuum, not an existence or uniqueness proof, a positive bound mode, a resonance, a physical Skyrmion, nucleon or Roper, a quantum state, an absolute mass, or a substrate realization. +- `C-SCL-001` — Let lambda, nu, S, E0, and epsilon0 be positive, with lambda a dimensionless classical Hessian eigenvalue, nu an inverse-time scale, S an action scale, and E0*epsilon0 a background energy. Conditional on a separately declared one-quantum harmonic interpretation, the dimensionless frequency is sqrt(lambda), the energy gap is S*nu*sqrt(lambda), and the gap-to-background ratio is S*nu*sqrt(lambda)/(E0*epsilon0). For every positive rho, replacing nu by rho*nu leaves the dimensionless Hessian problem unchanged while multiplying the gap and ratio by rho; analogous independent variation of S or E0 also changes the physical ratio. Thus a dimensionless squared classical frequency alone does not determine an excitation energy or mass ratio, and lambda itself cannot replace sqrt(lambda) in the harmonic gap. This exact conditional scale ledger supplies no quantization rule, value of S or nu, background normalization, spin, isospin, parity, particle or Roper dictionary, or substrate realization. +- `C-WID-001` — Declare Minkowski metric (+---), q=p'-p, Q^2=-q^2, positive equal on-shell mass M, positive F and m_pi^2, dimensionless form factors, and the current bracket [gamma^mu*G_A(Q^2)+q^mu*G_P(Q^2)/(2*M)]*gamma5. Exact on-shell contraction gives normalized divergence G_A-Q^2*G_P/(4*M^2). Separately declaring the normalized PCAC source F*m_pi^2*G_piNN(Q^2)/(M*(m_pi^2+Q^2)) yields a generalized residual that retains G_P independently. Under the additional pion-pole-dominance premise G_P=4*M*F*G_piNN/(m_pi^2+Q^2), the residual reduces exactly to G_A-F*G_piNN/M. Adding an induced-form-factor remainder R(Q^2) finite at zero and at Q^2=-m_pi^2 changes that residual by -Q^2*R/(4*M^2), leaves the pole residue 4*M*F*G_piNN(-m_pi^2) unchanged, and drops out only at zero transfer. The PCAC kernel m_pi^2/(m_pi^2+Q^2) has zero-transfer-then-chiral limit one, chiral-then-zero-transfer limit zero, and fixed path Q^2=rho*m_pi^2 value 1/(1+rho). This exact conditional theorem derives no QCD current or PCAC identity, pion-pole dominance, physical pion or nucleon, coupling value, effective action, or substrate realization. +- `C-GTR-001` — Conditional on C-WID-001's distinction between zero-transfer and pion-pole couplings, let m_pi^2 and g0 be positive and declare the regular expansion G_piNN(-m_pi^2)=g0-s*m_pi^2+R*m_pi^4 together with the separate zero-transfer relation M*g_A=F*g0. The discrepancy Delta=1-M*g_A/(F*G_piNN(-m_pi^2)) then equals m_pi^2*(-s+R*m_pi^2)/(g0-s*m_pi^2+R*m_pi^4), has leading coefficient -s/g0, and tends to zero in the chiral limit. A square-root coupling counterexample proves that current conservation without the regularity premise does not force mass-squared scaling. Separately, the supplied monomial equation g_piNN*F=g_A*M has exponent row (1,1,-1,-1), rank one, and a three-dimensional nullspace; exact common-scale, inverse-decay/coupling, and inverse-mass/axial rescalings preserve it. Solving this equation for g_A or g_piNN is therefore invertible bookkeeping, not prediction. This exact conditional theorem derives no discrepancy coefficient, physical parameter, EFT, QCD relation, state dictionary, or substrate realization. +- `C-LOC-001` — Let Q2 be a Euclidean momentum-squared variable, m2 positive, and C a separately supplied coefficient. The declared kernel K(Q2)=C*integral_0^1[u*(1-u)*Q2/(m2+u*(1-u)*Q2)]du has the exact positive-Q2 closed form C*[1-4*m2*atanh(sqrt(Q2/(4*m2+Q2)))/sqrt(Q2*(4*m2+Q2))]. Its Q2^n coefficient for every integer n>=1 is C*(-1)^(n-1)*(n!)^2/[(2*n+1)!*m2^n], and its finite geometric remainder proves convergence for |Q2|<4*m2, with nearest continued threshold Q2=-4*m2. The Q2-to-zero limit is zero, while the m2-to-zero limit at fixed positive Q2 is C, so those iterated limits do not commute. More generally, for a separately declared spectral density rho(t) supported on t>=Delta>0, whenever the displayed inverse moments and remainder converge, integral[rho(t)*Q2/(t+Q2)]dt equals the order-N sum sum_(n=1)^N (-1)^(n-1)*Q2^n*integral[rho(t)/t^n]dt plus the exact remainder (-1)^N*Q2^(N+1)*integral[rho(t)/(t^N*(t+Q2))]dt. A positive support gap alone guarantees neither ultraviolet moment convergence nor a nonzero first moment. These exact conditional identities derive no charged field, loop vertex, regulator or subtraction, physical mass or coupling, gauge kinetic action, electromagnetic sector, or substrate realization. +- `C-KRN-001` — For a supplied finite inverse kernel K(k2)=sum_j c_j*k2^s_j with k2 positive and exact positive real exponents, first combine equal exponents and remove coefficients that simplify exactly to zero. If every remaining coefficient is provably nonzero, the smallest remaining exponent s_star controls the infrared and the corresponding power of |k| is 2*s_star. Exact cancellation can expose a higher exponent, while any separately supplied lower fractional term remains leading over analytic integer-power corrections. An undecidable symbolic coefficient supplies only a condition and cannot be treated as nonzero. In the fixed convention G(x)=integral d^d k/(2*pi)^d exp(i*k.x)/[A*(k^2)^s], for r=|x|>0, A nonzero, and 0S is fixed by the generator image epsilon subject to epsilon^n=1. If n is odd only epsilon=+1 is allowed; if n is even there are exactly two characters, the trivial one and chi_n([k])=(-1)^k. The nontrivial even-order character has the even-residue kernel of size n/2, quotient order two, and is faithful exactly when n=2. Thus chi(1)=-1 and chi(2)=+1 alone establish at most a supplied sign quotient, not that the source group is C_2. For every positive r, all characters of C_2^r are chi_a(x)=(-1)^(a dot x), indexed by a in C_2^r, so there are 2^r; every nontrivial character has a kernel of size 2^(r-1), and a faithful sign character exists exactly when r=1. Distinct characters can agree at -1 on a selected element while differing elsewhere and having different kernels. Separately supplied quotient maps into C_2 may pull back the same codomain sign character, but this does not identify their source domains, maps, generators, orientations, representations, operators, topologies, physical statistics, or substrate mechanisms. +- `C-SPN-001` — Let Psi be a nonzero complex pure spin-1 spinor in the ordered m=(+1,0,-1) basis with the standard Hermitian spin-one matrices. Define n=Psi^dagger*Psi, f_a=Psi^dagger*F_a*Psi, and the singlet amplitude A=Psi_0^2-2*Psi_+*Psi_-. Then the exact invariant is |f|^2+|A|^2=n^2, and the attainable interval is 0<=|f|^2<=n^2. Under the unitary complex-Cartesian convention Psi_+=-(d_x-i*d_y)/sqrt(2), Psi_0=d_z, and Psi_-=(d_x+i*d_y)/sqrt(2), writing d=u+i*v gives f=2*u cross v and A=d dot d. Hence |f|^2=0 exactly when u and v are parallel; modulo global phase and spatial SO(3), these rays form the polar orbit RP^2. The upper endpoint |f|^2=n^2 holds exactly when u and v are orthogonal with equal norm; modulo global phase and SO(3), these rays form the coherent ferromagnetic orbit S^2 with SO(2) stabilizer. Conditional on the supplied fixed-density pure-state functional E_spin=(c2/2)*|f|^2, positive c2 selects precisely the polar projective orbit, negative c2 selects precisely the ferromagnetic projective orbit, and c2=0 leaves every pure spin-1 ray degenerate. The polar-minus-ferromagnetic endpoint energy is -c2*n^2/2. This theorem derives no material sign or magnitude of c2, atomic realization, spatial condensate ground state, full physical order-parameter manifold, defect energetics, finite-temperature phase, mixed-state classification, or substrate mechanism. +- `C-DEF-001` — For a real angular field theta of degree q on the annulus xi=2, L>0, a=L/N, and let dimensionless real fields phi_j(t) live on a one-dimensional periodic uniform lattice. The Riemann-normalized nearest-neighbour sine-Gordon action has instantaneous Lagrangian a*sum_j[(dot(phi_j)^2)/2-((phi_(j+1)-phi_j)/a)^2/2 -m^2*(1-cos(phi_j))]. Its exact sitewise Euler-Lagrange equation is ddot(phi_j)-(phi_(j+1)-2*phi_j+phi_(j-1))/a^2+m^2*sin(phi_j)=0. The centered spatial operator has exact Fourier symbol -4*sin(k*a/2)^2/a^2, which is even and reciprocal-lattice periodic; on a declared first Brillouin zone its linearization has omega^2=m^2+4*sin(k*a/2)^2/a^2. At fixed k as a tends to zero this is m^2+k^2-a^2*k^4/12+a^4*k^6/360+O(a^6), while the exact zone-edge spatial value is 4/a^2 rather than pi^2/a^2. For a smooth field, the centered stencil derived from both neighbour jets is phi_xx+a^2*phi_xxxx/12+a^4*phi_6/360 plus a remainder bounded by M8*a^6/20160 when the eighth derivative is bounded by M8 on the local stencil interval. If an L-periodic sampled field on a fixed time interval T has uniform bounds Mx, Mxx, Mt, and Mtx on the absolute values of phi_x, phi_xx, phi_t, and phi_tx, then the absolute difference between the sampled discrete action and continuum action is at most T*L*(a*Mt*Mtx/2+a*m^2*Mx/2+a*Mx*Mxx+a^2*Mxx^2/8), and therefore tends to zero with a for fixed bounds. Removing the global factor a at one fixed spacing preserves the site equations under C-VAR-001 but destroys this action-value normalization across refinement. These exact conditional results derive no lattice existence, spacing value, hydrogen medium, EFT termination scale, nonlinear solution convergence, nonabelian sector, material realization, or substrate mechanism. +- `C-OVL-001` — For any whole-line L2 mode eta normalized by integral_R |eta|^2 dx=1 and any supplied bounded real multiplication profile Phi with essential range Phi_min<=Phi<=Phi_max, its expectation y=integral_R |eta|^2*Phi dx obeys Phi_min<=y<=Phi_max. For positive p,r,kappa and real A, if eta is proportional to sech(kappa*x)^p and Phi=A*sech(kappa*x)^r under Cartesian dx, then normalization gives the exact matched-width overlap y=A*Gamma(p+r/2)*Gamma(p+1/2) /(Gamma(p)*Gamma(p+r/2+1/2)); the common kappa cancels, but A does not. The p=2,r=1 value is 9*pi*A/32. For C-QBL-003's actual unnormalized even sech^2 and odd sech*tanh modes against A*sech at the same width, the normalized squared-density expectations are respectively 9*pi*A/32 and 3*pi*A/16, their ratio for common A is 2/3, and the weighted even-odd cross expectation vanishes by parity. After L2 normalization y has the mass dimension of Phi. A separately declared product m=y*v has the sum of the supplied profile and scale dimensions; a common v cancels from mass ratios, while independent amplitudes remain, and y->rho*y with v->v/rho leaves m invariant. These exact conditional expectation and parameter results derive no fermion, Yukawa interaction, physical condensate or VEV, generation assignment, hierarchy, mixing, absolute mass, radial-measure formula, Standard-Model map, or substrate mechanism. C-QBL-003's negative and zero Hessian eigenvalues remain non-mass objects but do not uniquely select this overlap functional. +- `C-OVL-002` — Let p,r,kappa be positive, A real, R real, and let an L2-normalized whole-line Cartesian mode be proportional to sech(kappa*(x-R))^p while the supplied core multiplier is A*sech(kappa*x)^r. With a=kappa*abs(R), alpha=2*p, and ell=(alpha+r)/2, its exact expectation is A/J_alpha times 2^(alpha+r-1)*exp(-alpha*a)*B(ell,ell) *2F1(alpha,ell;2*ell;1-exp(-2*a)), where J_alpha=sqrt(pi)*Gamma(p)/Gamma(p+1/2). It is even in R and reduces to C-OVL-001's matched-width gamma ratio at R=0. For nonzero A and large a, unequal alpha and r give the exact leading class A*2^(alpha+r-1)*B(abs(alpha-r)/2,ell)/J_alpha times exp(-min(alpha,r)*a); equal alpha=r instead gives A*2^(2*alpha)/J_alpha times a*exp(-alpha*a). Thus the slower density or profile tail sets the exponential rate, and equal rates are not a pure geometric law. Separately, for V0,w>0 the whole-line Hamiltonian H_R=-d_x^2-V0*sech((x-R)/w)^2 has exact normalized ground state N*sech((x-R)/w)^s, with s=(sqrt(1+4*V0*w^2)-1)/2, N^2=Gamma(s+1/2)/(w*sqrt(pi)*Gamma(s)), eigenvalue -s^2/w^2, and density-tail rate 2*s/w. Translation changes R but not its spectrum. Against A*sech(kappa*x), the slower of 2*s/w and kappa controls the overlap; the exact unequal-rate beta prefactors and equal-rate linear-R prefactor are retained by the canonical ledger. A declared linear center ladder R_n=R_0+n*d therefore has limiting log overlap ratio -min(2*s/w,kappa)*d, but d is a free input and reciprocal rate/spacing rescaling leaves this product invariant. An exact Gaussian localized countermodel attenuates as exp(-c*R^2), proving localization does not uniquely select a geometric displaced-sech mechanism. These exact conditional results derive no half-line radial operator, common multi-rung spectrum, well centers or spacing, generation count or identity, Yukawa interaction, observed hierarchy, mixing, absolute mass, Standard-Model map, or substrate realization. +- `C-OVL-003` — Let {eta_i}_{i=1}^N be one declared ordered orthonormal family in a complex L2 space with fixed domain and measure, and let Phi be a bounded real multiplication profile. The finite compression Y_ij=, with complex conjugation in the first slot, is Hermitian. For psi=sum_i c_i*eta_i, c^dagger*Y*c=; hence a normalized Rayleigh quotient and every eigenvalue of Y lie in Phi's supplied essential range. Under one common basis change eta'=eta*U, Y'=U^dagger*Y*U, so entries are basis dependent while the spectrum, trace, determinant, spectral multiplicities, and commonly transformed commutator data are invariant. If the two modes and multiplier have declared parities p_i,p_j,p_Phi in {+1,-1}, the whole-line entry vanishes when p_i*p_Phi*p_j=-1; in particular, changing only the width of a centered even profile cannot couple opposite parities. Two finite Hermitian compressions on one explicitly identified space admit a common unitary eigenbasis exactly when their commutator vanishes. Matrix inequality alone does not establish misalignment, and a relative eigenbasis additionally retains independent eigenvector phases, permutations, arbitrary U(m) rotations in m-fold degenerate subspaces, and any undeclared map between separately named spaces; its unitarity or nonidentity alone is not a physical observable. For C-QBL-003's actual normalized even sech(z)^2 and odd sech(z)*tanh(z) modes, z=kappa*x, compressing the separately supplied asymmetric profile A*sech(z)*(1+b*tanh(z)) gives exactly [[9*pi*A/32,sqrt(2)*A*b/5],[sqrt(2)*A*b/5,3*pi*A/16]]. The common positive kappa cancels, b=0 restores the parity-diagonal C-OVL-001 matrix, and the nonzero cross entry therefore requires the new free odd profile premise b rather than a width change. MH3's substituted sech even mode instead gives pi*A/16 times [[4,sqrt(3)*b],[sqrt(3)*b,3]], also independent of kappa; its two displayed textures differ because b is independently changed, not because their widths differ. These exact compression results derive no fermion or chirality, Yukawa or mass interaction, generation assignment, physical up/down sectors, charged-current operator, hierarchy, CKM or Cabibbo observable, CP quantity, absolute mass, or substrate mechanism. +- `C-RGE-003` — For two positive lengths ell0 and ell1 and a speed c, using dimension rows (L,T) and columns (ell0,ell1,c), the exact dimension matrix is [[1,1,1],[0,0,-1]], has rank 2, and has one-dimensional monomial kernel generated by (-1,1,0). Thus dimensional analysis supplies the ratio ell1/ell0 up to reciprocal orientation but neither its value nor physical labels. Under C-RGE-001's positive-input formal one-loop relation Lambda=mu0*exp(-X), X=8*pi^2/(b0*g^2), and separately declared positive inverse-energy conversions ell0=K0/mu0 and ell1=K1/Lambda, the exact ratios are Lambda/mu0=exp(-X) and ell1/ell0=(K1/K0)*exp(X). Only when K1=K0 are the energy and length ratios reciprocal. Inverting a supplied length ratio R gives g^2=8*pi^2/(b0*log(R/(K1/K0))) on the positive-coupling domain R>K1/K0; this is inference of a free input, not a parameter-free prediction. The fixed-ratio log equation [-1,1]*(log(ell0),log(ell1)) =log(R) has nullspace generated by (1,1), so it identifies only the relative log coordinate and leaves both absolute lengths free under common rescaling. AS1's executable assigns its reference-energy length to xi and lower-energy length to a, contrary to its opening UV-a and IR-xi labels, and then calls a/xi the earlier xi/a group; it also retains b0, g^2, mu0, and the conversion premises. The ledger establishes no lattice, soliton, QCD, confinement, hadron, operating point, absolute scale, or later-source closure. +- `C-GRV-001` — In M,L,T base-dimension order, for primitive columns consisting of a positive cutoff length a, speed c, and action scale hbar, the exact dimension matrix [[0,0,1],[1,1,2],[0,-1,-1]] has full rank. The unique monomial powers for Newton G are (2,3,-1), and those for 1/G are (-2,-3,1); hence dimensions permit G=q*a^2*c^3/hbar with an arbitrary dimensionless q but do not determine q or derive a cutoff mechanism. If a nonzero real coefficient s and cutoff energy E_cut=hbar*c/a are separately declared, the conditional leading inverse-coupling shift Delta(1/G)=s*hbar/(a^2*c^3)=s*E_cut^2/(hbar*c^5) is exact. Its sign, field and curvature-coupling content, regulator, cutoff identification, and coefficient remain premises. An independent baseline inverse coupling B adds before inversion, 1/G_total=B+Delta(1/G); dimensions do not set B=0, B=target-Delta realizes any supplied total, and B=-Delta cancels it. Only for B=0 and s>0 is the pure formula G=a^2*c^3/(s*hbar) available. With compatible dimensionless references, its log row on (log(a/a_ref),log(s/s_ref)) is (2,-1), whose nullspace spans (1,2); therefore one supplied G ratio identifies neither a nor s, and a=sqrt(s*hbar*G/c^3) is inverse inference after supplying G and s, not a prediction. For any separately declared source equation operator=kappa*source, [kappa]=[operator]-[source]. A direct kappa=alpha*G with dimensionless alpha is allowed only when this equals [G]; for a dimensionless scalar operator of dimension L^-2, mass-density and energy-density sources instead require alpha dimensions c^-2 and c^-4 respectively. C-OG-003 declares neither such source units nor a Newton normalization. AS3's exact power solve and cutoff substitution survive conditionally, but it leaves s free, omits the additive baseline and QFT data, imports kappa=8*pi*G from pending evidence, and does not pin a, kappa, or an over-determined physical scale. This ledger establishes no Sakharov mechanism, quantum spectrum, regulator, 3+1 Einstein dynamics, lattice ontology, medium map, observed constant, or absolute scale. +- `C-DIM-008` — In M,L,T base-dimension order, the declared primitive columns c=(0,1,-1) and hbar=(1,2,-1) form the matrix [[0,1],[1,2],[-1,-1]], whose coefficient rank is two; adjoining the pure-length target (0,1,0) raises augmented rank to three, so no monomial in c, hbar, and dimensionless numbers has length dimension. Appending that target itself as a primitive a gives the full-rank matrix [[0,1,0],[1,2,1],[-1,-1,0]] and the unique target powers (0,0,1), which supplies rather than derives the length primitive and leaves its value unconstrained. Conditional on C-RGE-001's positive formal relation Lambda=mu0*exp(-X), X=8*pi^2/(b0*g^2), and a separately supplied positive inverse-energy conversion K, the length ell=K/Lambda retains mu0, K, g^2, and b0. At fixed g^2, b0, and K, the finite reference change mu0->rho*mu0 gives Lambda->rho*Lambda and ell->ell/rho while Lambda/mu0 remains exp(-X). Any supplied positive target Lambda_t or ell_t can be reproduced by choosing mu0=Lambda_t*exp(X) or mu0=K*exp(X)/ell_t respectively, so these are inverse constructions with the target as input, not absolute predictions. Separately, for a fixed positive dimensionful quantity Q=N*u, rescaling the unit standard to rho*u changes its numeric coordinate to N/rho while reconstructing the same Q; unit-coordinate covariance neither creates Q nor selects its magnitude. AS5's target-span subclaim survives, but its displayed scale contains mu0, its no-import predicate does not test mu0, and its exp(-X) label for a/xi reverses the inverse-energy orientation inherited from AS1. This ledger derives no physical beta function, coupling, reference scale, conversion, QCD, confinement, lattice, soliton, chemistry, gravity, empirical scale, or preferred unit system. +- `C-SYM-002` — For explicitly positive real coordinates x and coefficients A, the map D_A(x)=A/x is an involution, every orbit has product x*D_A(x)=A, and its unique positive fixed coordinate is sqrt(A). Any supplied positive target t can be made fixed by the inverse construction A=t^2, so neither A nor t is selected by fixed-point algebra. If positive coupling coordinates beta and beta_tilde with separately supplied product P=beta*beta_tilde are exchanged and x=beta^2, the induced squared- coordinate map has A=P^2 and fixed x=P; in particular P=4*pi gives the conditional map x->16*pi^2/x and fixed coordinate 4*pi. Under a positive coordinate reparameterization x'=rho*x, conjugating the same reciprocal relation requires A'=rho^2*A, sends the numeric fixed coordinate to rho*sqrt(A), and preserves the dual-orbit structure. Generic positive x belongs to a valid off-fixed pair and is not thereby required to satisfy x=D_A(x). These exact facts establish no physical action or observable duality, no accepted coupling normalization or dual-product premise, no restriction to or mechanism selecting a self-dual subfamily, and no coupling value, beta coefficient, scale, particle interpretation, or substrate realization. +- `C-IDN-002` — Let G_r=G/G_ref, L_r=L/a_ref, C, and b0 be separately supplied positive exact dimensionless quantities, with compatible pure induced-gravity references, zero additive inverse-coupling baseline, and K=8*pi^2/b0. For u=log(a/a_ref), v=log(s/s_ref), and y=1/g^2, composing the declared pure-gravity row with the separately declared length relation L/a=C*exp(K*y) gives the exact system [[2,-1,0],[1,0,K]]*(u,v,y)^T=(log(G_r),log(L_r/C))^T. Its coefficient rank is two, its nullspace is generated by (-K,-2*K,1), and none of u, v, or y is identified by those two rows. Adding a separately supplied positive coefficient ratio sigma through v=log(sigma) gives a full-rank unique solve u=(log(G_r)+log(sigma))/2 and y=(log(L_r/C)-u)/K; it defines a positive inferred g^2=1/y only when y>0. Therefore substituting that inferred coupling back into the same length row has zero residual by construction and is inverse reconstruction, not an independent over-determination test. On the same zero-baseline branch, a supplied exact closed positive coefficient interval maps monotonically through a=sqrt(s*hbar*G/c^3), and every positive target cutoff a_t has coefficient preimage s_t=a_t^2*c^3/(hbar*G). In the general accepted additive family, B=1/G-s*hbar/(a^2*c^3) realizes any supplied positive total G for arbitrary positive a and nonzero real s. These ledgers establish no observed constant, coefficient prior, physical row independence, beta-function coefficient, conversion, coupling selection, absolute scale, Planck or hadron identity, cutoff ontology, or substrate granularity. +- `C-PRB-001` — For a separately supplied real initial frequency x in [0,1] and real dimensionless selection ratio S, define the continuous exponential fixation family by U(x,0)=x and, for S nonzero, U(x,S)=(1-exp(-S*x))/(1-exp(-S)). The value at zero is the unique continuous extension. For every real S, U(0,S)=0, U(1,S)=1, and U lies in [0,1]; for nonzero S its x derivative is S*exp(-S*x)/(1-exp(-S))>0, and U(x,S)+U(1-x,-S)=1. Conditional on the separately declared backward equation U_xx+S*U_x=0 with absorbing boundary values zero and one, this is its unique solution: for S nonzero the two-constant boundary matrix has determinant exp(-S)-1, while S=0 gives the unique linear solution U=x. For 00; its limits as S tends to negative and positive infinity are zero and one. Thus every supplied target in (0,1) has a unique implicit S preimage, which is inverse inference rather than a prediction. The neutral expansion is U=x+S*x*(1-x)/2+S^2*x*(1-x)*(1-2*x)/12 -S^3*x^2*(1-x)^2/24+O(S^4). Separately, for positive intensities I1,I2, total N=I1+I2 and x=I1/N, the raw contrast is I1-I2=N*(2*x-1). A declared S=kappa*(I1-I2) scales by lambda^2 under a common amplitude rescaling when kappa is held fixed, but kappa->kappa/lambda^2 preserves S, and unit-normalizing the intensities preserves S with kappa_unit=kappa*N. Hence raw-amplitude scale sensitivity requires a separately established physical normalization and coefficient convention. These exact facts establish no Wright-Fisher or Moran microscopic process, quantum state or Born postulate, measurement or actualization rule, empirical deviation, observable amplitude norm, parameter value, medium action quantum, cutoff, dimensional instability, or substrate granularity. +- `C-REP-001` — For a separately supplied nonempty finite table of positive integer multiplicities m_i and exact real generator eigenvalues t_i,y_i, and an exact real coefficient c, define Q_i=t_i+c*y_i and the weighted traces T_2=sum_i m_i*t_i^2, T_Y=sum_i m_i*y_i^2, T_X=sum_i m_i*t_i*y_i, and T_Q=sum_i m_i*Q_i^2. Then exactly T_Q=T_2+2*c*T_X+c^2*T_Y, and T_2/T_Q is a conditional table ratio when T_Q is provably nonzero. For every separately supplied positive rho and positive Abelian coupling g_Y, the coordinate change y_i'=rho*y_i, c'=c/rho, and g_Y'=g_Y/rho preserves every Q_i, every product g_Y*y_i, T_Q, T_2/T_Q, and g_Y^2*T_Y. Holding c fixed instead generally changes Q and the quotient. Every homogeneous Abelian moment H_p=sum_i m_i*y_i^p scales as rho^p, so H_p=0 cannot select the positive overall generator normalization. Separately, for positive supplied traces S_2,S_Y and couplings g_2,g_Y, the coupling coordinate A_g=g_Y^2/(g_2^2+g_Y^2) equals the trace coordinate A_T=S_2/(S_2+S_Y) if and only if g_Y^2/g_2^2=S_2/S_Y, equivalently the two inverse-trace coefficients 1/(g_i^2*S_i) are equal. Thus a common law 1/g_i^2=C*S_i is a sufficient separately supplied premise, not a consequence of the finite table. Applied only to WM1's declared fifteen-state table and Q=T3+Y, the exact table values are T_2=2, T_Y=10/3, T_X=0, T_Q=16/3, and T_2/T_Q=3/8; equality with a coupling angle additionally requires g_Y^2/g_2^2=3/5. These results establish no physical representation, anomaly derivation, gauge action, kinetic normalization, common induction mechanism, simple unification, boundary or running scale, weak mixing angle, observed value, Standard Model, or substrate realization. +- `C-RGE-004` — For a separately supplied finite family of at least two exact real inverse-coupling coordinates a_i and signed exact real coefficients b_i in the declared affine convention a_i=A+B*b_i, the rows (1,b_i) define an exact two-coordinate linear system. Composing C-LIN-001, the family has a common intersection exactly when coefficient and augmented ranks agree, and it uniquely identifies A and B exactly when the common rank is two. A pair with b_i!=b_j has the unique crossing B_ij=(a_i-a_j)/(b_i-b_j); equal coefficients with equal coordinates are coincident, while equal coefficients with unequal coordinates are parallel and disjoint. Thus a consistent rank-two family makes all distinct-pair crossings agree, whereas unequal pairwise crossings are an exact counterexample to common intersection. Under the declared reference shift a_i'=a_i-delta*b_i, A is invariant and B'=B-delta. Separately, for supplied positive electromagnetic inverse E, strong inverse S, hypercharge weight n, and signed b1,b2,b3 with D=b2+n*b1-(1+n)*b3 nonzero, the exact-matching equations a_i=A+B*b_i, a3=S, and E=a2+n*a1 have the unique inverse reconstruction B=(E-(1+n)*S)/D, A=S-B*b3, weak coordinate w=a2/E, and common-coupling boundary w_boundary=1/(1+n). This is conditional inverse inference from two supplied observations and an exact matching premise, not an ab-initio prediction. For a positive Abelian coordinate rescaling alpha1'=q*alpha1, the paired map a1'=a1/q, b1'=b1/q, and n'=q*n preserves the electromagnetic row but not an unqualified equality between the Abelian and non-Abelian coupling coordinates. Independent additive sector matching offsets can represent arbitrary supplied intercepts and therefore must remain explicit. Applied only to WM3's supplied exact readings E=1279/10, S=500/59, n=5/3, and (b1,b2,b3)=(41/10,-19/6,-7), the reconstruction gives A=1639681/39530, B=186383/39530, w=6296809/30335322, and boundary 3/8. These results derive no beta function, U1 or SU2 coefficient, physical gauge sector, simple-group embedding, normalization, matching boundary, threshold spectrum, observation, weak-angle scheme, reference scale, perturbative domain, Standard Model, or substrate realization. +- `C-COH-001` — Let N be a positive integer, I1 a positive exact per-source directional intensity, and z_j independent identically distributed unit complex phasors with common mean mu and pair coherence V=|mu|^2 in [0,1]. Then I1*E[|sum_j z_j|^2]=I1*(N+N*(N-1)*V). The V=0 and V=1 endpoints are N*I1 and N^2*I1, and the expected-to-incoherent ratio is 1+(N-1)*V. For centered iid Gaussian phase noise of variance sigma^2, V=exp(-sigma^2). These are directional ensemble statements at fixed per-source normalization: substituting I1=I_total/N changes the aligned result to N*I_total, and deterministic correlated phases need not obey the iid lower endpoint. Separately, declare the continuous positive-coordinate interpolation Theta(n,V)=theta*n*(1+(n-1)*V) with theta>0, n>0, and V in [0,1]. For a supplied positive barrier E its unique positive continuous crossing is n=E/theta at V=0 and n=(sqrt((1-V)^2+4*V*E/theta)-(1-V))/(2*V) for V>0; the fully aligned endpoint is sqrt(E/theta). Thus alignment lowers the endpoint threshold exactly when E/theta>1, is neutral at one, and reverses below one. Theta is strictly increasing in V for n>1 and independent of V at n=1. The separately conditional activated factor exp(-E/Theta) inherits that monotonicity but is dimensionless and is not a rate without a supplied dimensionful prefactor and kinetic model. These exact results derive no emitter population, phase preparation or dynamics, observable normalization, total or conserved energy, effective temperature, stochastic escape process, material barrier, precursor identity, nuclear interaction, reaction branch, deposition law, event payload, engine replacement, or substrate realization. +- `C-SG-015` — For the C-SG-001 rest breather with 00; at the core it is b_1(0)=8*eta/(1+omega). The expression 4*eta/omega is only the leading small-amplitude core term and becomes asymptotically equivalent as omega approaches one from below. A time-origin shift rotates sine and cosine phases while preserving zero mean and harmonic support. These statements concern the Fourier series of an undriven field trace. They establish no susceptibility, Green function, causal response, perturbing interaction, deposition or absorption spectrum, transfer efficiency, seeded population, derivative voltage coupling, breakdown law, engine replacement, event channel, or substrate realization. +- `C-SG-016` — For the exact undamped C-SG-001 rest-breather family with 0=omega*J=16*omega*acos(omega). Hence the family damping form factor is D(omega)=omega*J/E=omega*acos(omega)/sqrt(1-omega^2). Writing theta=acos(omega) gives D=theta*cot(theta): D is strictly increasing in omega, tends to zero as omega approaches zero, tends to one as omega approaches one, and equals pi/4 at omega=1/sqrt(2). For the bulk-damped equation phi_tt-phi_xx+sin(phi)=-Gamma*phi_t with Gamma>=0, C-SG-012 gives the exact energy balance dE/dt=boundary_flux-Gamma*integral_R phi_t^2 dx. Conditional on negligible boundary flux, period averaging, slow damping, and tracking the undamped family, dJ/dt=-Gamma*J, so J(t)=J0*exp(-Gamma*t), omega(t)=cos((J0/16)*exp(-Gamma*t)), and E(t)=16*sin((J0/16)*exp(-Gamma*t)). Thus 1/(Gamma*D(omega)) is the instantaneous tangent scale -E/(dE/dt), not the finite full-amplitude energy e-fold time. For Gamma>0 and theta0=acos(omega0), the reduced energy e-fold time is log(theta0/asin(sin(theta0)/e))/Gamma and tends to 1/Gamma only as omega0 approaches one. At omega0=1/sqrt(2), the two dimensionless times are about 1.27324 and 1.09344. For the normalized zero-source PDE with uniform Gamma=0.02, exact phase-zero omega0=1/sqrt(2) data, final time 50, domains [-60,60] and [-80,80], homogeneous Dirichlet endpoints, leapfrog dx=0.2,0.1,0.05 with dt=dx/2, and DOP853 at dx=0.2, the reduced law has fine-grid energy RMS error 0.00581 and action RMS error 0.00642 relative to their initial values; the energy-ledger residual falls to 5.26e-5 relative, the domain and method checks agree, and the frozen-D energy law has RMS error 0.03139. At Gamma=0.01 and equal slow time the energy and action errors improve to 0.00294 and 0.00324; Gamma=0 is a lossless numerical control. This is finite-time simulation evidence for the declared adiabatic regime, not an exact positive-Gamma breather, global exponential energy or amplitude law, fixed-frequency decay, physical medium lifetime, collision or density map, coherence probability, population, event channel, or substrate realization. +- `C-DYN-001` — For the declared real linearly damped oscillator q_tt+Gamma*q_t+omega_0^2*q=0 with omega_0>0 and Gamma>=0, the exact characteristic roots are -Gamma/2 +/- sqrt(Gamma^2-4*omega_0^2)/2. The oscillator is underdamped, critically damped, or overdamped according as Gamma is below, equal to, or above 2*omega_0. In the underdamped branch its angular frequency is omega_d=sqrt(omega_0^2-Gamma^2/4), its coordinate amplitude envelope is exp(-Gamma*t/2), and the square of that envelope is exp(-Gamma*t). Its instantaneous mechanical energy E=(q_t^2+omega_0^2*q^2)/2 instead satisfies the exact on-shell balance dE/dt=-Gamma*q_t^2, so it is not pointwise a single exponential. In a declared 1/Gamma quadratic-envelope window the actual oscillation count is omega_d/(2*pi*Gamma); omega_0/(2*pi*Gamma) is only a nominal natural-frequency convention. The actual count tends to zero at critical damping while the nominal count tends to 1/(4*pi). For the normalized damped sine-Gordon linearization psi_tt-psi_xx+psi=-Gamma*psi_t, a real Fourier mode k has omega_0(k)=sqrt(1+k^2)>=1 and critical damping 2*sqrt(1+k^2). A finite-amplitude breather frequency 0=0, t>=0, delta_0=0, and a standard real Wiener process, the exact integer-harmonic characteristic is E[exp(i*n*delta_t)]=exp(-n^2*D*t), and Var(delta_t)=2*D*t. Hence the one-ensemble mean phasor is exp(-D*t), while the iid same-time pair coherence is its square exp(-2*D*t); neither is a survival probability. For a positive uniform observation window T, their temporal averages are respectively (1-exp(-D*T))/(D*T) and (1-exp(-2*D*T))/(2*D*T), with continuous value one at D=0, and they are not their endpoint values. Separately, if an independent deterministic coordinate-amplitude envelope exp(-Gamma*t/2) with Gamma>=0 is declared, its product with the Brownian mean phasor is the coherent mean-field factor exp(-(Gamma/2+D)*t), whose quadratic factor is exp(-(Gamma+2*D)*t). These formulas derive no physical diffusion coefficient, fluctuation-dissipation normalization, effective temperature, oscillator or breather phase projection, total energy, lifetime, survival fraction, population, event rate, or material or discharge map. +- `C-MED-003` — Conditional on a dimensionless real field u, physical coordinates x and t, exact positive coefficients lambda, T, and mu, and the declared energy-per-length density L=lambda*u_t^2/2-T*u_x^2/2-mu*(1-cos(u)), the Euler-Lagrange equation is lambda*u_tt-T*u_xx+mu*sin(u)=0 and the positive scales are c=sqrt(T/lambda), omega_0=sqrt(mu/lambda), ell=sqrt(T/mu)=c/omega_0, E_scale=sqrt(T*mu), and J_scale=sqrt(lambda*T), with E_scale=omega_0*J_scale. Under base rows (energy,length,time), the coefficient columns (lambda,T,mu) are (1,-1,2), (1,1,0), and (1,-1,0), a full-rank matrix. The coordinate map X=x/ell and tau=omega_0*t sends the dimensional residual exactly to mu times U_tau_tau-U_XX+sin(U). The logarithmic map from coefficients to (c,omega_0,ell) has rank two and right nullspace spanned by (1,1,1): at fixed positive c and omega_0 the inverse coefficient family is (lambda,lambda*c^2,lambda*omega_0^2) for an arbitrary positive lambda. A common coefficient multiplier preserves c, omega_0, ell and the field solution set while multiplying physical energy and canonical action by that factor. This is a conditional continuum cosine model; dimensions, periodicity, ratio closure, or symbol absence does not derive a material, select coefficient values, or remove the common energy/action scale. +- `C-SG-017` — Under C-MED-003 and for every real normalized frequency 00+ with lambda and T fixed sends physical frequency and energy to zero and profile length to infinity; at fixed positive physical frequency the ratio omega_b/omega_0 instead leaves the breather domain before that limit. These statements are an exact lift of the accepted normalized family. They do not prove that every gapless medium lacks every localized periodic solution, select a material or coefficient scale, or authorize treating normalized 16*eta as a physical energy without its scale factor. +- `C-SG-018` — Under C-MED-003, exact differentiation of the physical sine-Gordon residual at every vacuum u=2*pi*n gives lambda*psi_tt-T*psi_xx+mu*psi=0. Its positive real-wavenumber branch is Omega(k)=sqrt(omega_0^2+c^2*k^2), with floor omega_0. For k>0 its phase and group velocities are v_p=Omega/k and v_g=c^2*k/Omega, so v_p*v_g=c^2, v_p>c, 00+, while both tend to c as k->infinity. For a real frequency Omega>=0, a separated field a(x)*exp(-i*Omega*t) obeys a''=((omega_0^2-Omega^2)/c^2)*a. Below the gap there is one decaying L2 branch on each exterior half-line with rate kappa=sqrt(omega_0^2-Omega^2)/c, but smooth whole-line matching has matrix [[1,-1],[-kappa,-kappa]] and determinant -2*kappa, so only the zero global L2 mode remains. At threshold the affine branches, and above the gap the oscillatory branches, are likewise non-L2 on a half-line unless zero. Thus the homogeneous constant-coefficient whole-line linearized equation has no nonzero L2 real-frequency separated mode in any branch. The C-SG-017 nonlinear breather independently supplies a core whose exterior rate eta/ell equals this sub-gap rate; exterior decay alone does not prove such a core exists. Above-gap oscillation alone does not imply outgoing radiation, because standing and directed waves on the same dispersion can have zero and nonzero mean flux. In the gapless limit the no-L2 separated-mode statement persists, but the wave equation admits localized finite-energy traveling packets such as sech(x-c*t). These conditional results derive no material, coefficient value, defect, finite-box mode, nonlinear existence theorem beyond C-SG-017, outgoing boundary condition, density of states, cutoff, lifetime, population, or absolute scale. +- `C-LAT-002` — Let N>=2 and let dimensionless real phases u_j(t) live on a one-dimensional periodic nearest-neighbour chain with exact positive physical phase inertia I, coupling energy K, on-site energy V0, and spacing a. The per-site-energy Lagrangian sum_j[I*dot(u_j)^2/2-K*(u_(j+1)-u_j)^2/2 -V0*(1-cos(u_j))] has exact site equation I*ddot(u_j)-K*(u_(j+1)-2*u_j+u_(j-1))+V0*sin(u_j)=0. Linearization about a vacuum has first-zone dispersion Omega^2=(V0+4*K*sin(k*a/2)^2)/I, gap sqrt(V0/I), zone-edge frequency sqrt((V0+4*K)/I), and long-wave speed a*sqrt(K/I). Under base rows (mass,length,time), the columns (I,K,V0,a,m,b) are (1,2,0), (1,2,-2), (1,2,-2), (0,1,0), (1,0,0), and (0,1,0). If a physical displacement is declared by q=b*u, then I=m*b^2; if its neighbour stiffness is kappa, then K=kappa*b^2. Thus sqrt(V0/m) has speed rather than frequency units when m is a bare mass. For two declared hosts A and B, Omega_A(0)/Omega_B(0)=sqrt(V_A*I_B/(I_A*V_B)), or after displacement lifts sqrt(V_A*m_B*b_B^2/(V_B*m_A*b_A^2)). The H-over-D value sqrt(2) follows only if V_D=V_H, b_D=b_H, and m_D=2*m_H exactly; changing either curvature or phase scale can remove that shift. At fixed I, V0->0+ closes the linear gap. These exact conditional results derive no lattice or material realization, coefficient value, collective effective mass, exact isotope prediction, nonlinear discrete-breather existence, lifetime, radiation, or absolute frequency. +- `C-MED-004` — Conditional on a dimensionless real field theta, a physical length coordinate z, a physical time coordinate tau, and an exact positive coefficient g with dimensions 1/(length*time), the mixed-coordinate equation theta_z_tau=g*sin(theta) linearizes at a vacuum to theta_z_tau=g*theta. A plane wave exp(i*(k*z-Omega*tau)) therefore has characteristic k*Omega=g; for positive k its branch Omega=g/k has phase velocity g/k^2, group velocity -g/k^2, tends to infinity as k->0+, and tends to zero as k->infinity, so it has no finite k-independent laboratory angular-frequency floor. With xi=z/L and eta=tau/T, the normalized coefficient is g*L*T. Imposing g*L*T=1 leaves the exact positive family T=1/(g*L) for arbitrary L; its logarithmic scale Jacobian with respect to (L,T) is [1,1], with reciprocal null direction (-1,1). The map X=xi+eta and S=xi-eta sends the equation exactly to theta_SS-theta_XX+sin(theta)=0. Under rows (length,time), the columns (g,alpha,rate,Omega_squared) are (-1,-1), (-1,0), (0,-1), and (0,-2): an inverse-length absorption alpha is neither g nor a laboratory frequency squared and requires a separately supplied inverse-time rate and dimensionless prefactor even to form g. The normalized static kink is a coordinate-map cross-check only. These exact conditional results derive no Maxwell-Bloch reduction, self-induced-transparency regime, optical coefficient, gas or isotope map, material, nonlinear pulse existence, laboratory gap, damping, or absolute scale. +- `C-COL-001` — Under C-MED-003, let a dimensionless real field be restricted on a fixed spatial domain D to a declared sufficiently differentiable one-parameter profile family u(x,t)=phi(x,q(t)). If partial_q phi is square-integrable and nonzero and lambda>0, exact substitution into the kinetic density gives the positive finite collective metric M(q)=lambda*integral_D((partial_q phi)^2 dx) and kinetic term M(q)*qdot^2/2. For a separately declared twice-differentiable reduced potential U(q), L_red=M(q)*qdot^2/2-U(q) has exact equation M*qddot+(partial_q M)*qdot^2/2+partial_q U=0. At a rest stationary point q0 with U'(q0)=0 and M(q0)>0, the linearization is M(q0)*delta_qddot+U''(q0)*delta_q=0: positive curvature gives a stable angular frequency sqrt(U''/M), zero curvature is linearly neutral, and negative curvature gives real exponential roots with rate sqrt(-U''/M). Under a smooth locally invertible reparameterization q=g(Q), the metric becomes M_Q=M(g(Q))*g'(Q)^2 and the potential Hessian is U_QQ=U_qq*g'^2+U_q*g''; hence at a stationary point U_QQ/M_Q is coordinate invariant while curvature and inertia separately are not. If q is a length coordinate, x is a length coordinate, and lambda has C-MED-003 dimension (E,L,T)=(1,-1,2), then M has (1,-2,2), U'' has (1,-2,0), and their ratio has (0,0,-2). Conditional on an independently established profile/action identification q=R and C-RG-001's capillary potential, R*=T/P has U''=-2*pi*P, so sqrt(2*pi*P/M(R*)) is an unstable exponential rate, not a stable oscillation frequency. These results do not supply the profile, prove its physical realization, couple the continuum and capillary coefficients, select their normalization or a material, identify a quantum or thermal onset, derive stochastic escape, or establish an ignition or observed event. +- `C-FPT-001` — Let a0, gamma>0, and let U be a declared real continuously differentiable potential on [a,b]. Conditional on the one-dimensional overdamped Ito diffusion dX=-(U'(X)/gamma)dt+sqrt(2*Theta/gamma)dW with reflection at a and absorption at b, the mean absorption time tau(x) is the unique solution of Theta*tau''-U'*tau'=-gamma on (a,b), tau'(a)=0, tau(b)=0, and is tau(x)=(gamma/Theta)*integral_x^b exp(U(y)/Theta) integral_a^y exp(-U(z)/Theta) dz dy. It is strictly positive for x0, L=b-a, and a start at a, the exact result is gamma*Theta*(exp(F*L/Theta)-1-F*L/Theta)/F^2, with zero-force limit gamma*L^2/(2*Theta). A finite-horizon mean over completed paths is a conditional completed-only statistic rather than the full MFPT, and a rule returning zero below a completion threshold is an operational classifier rather than a zero physical rate. Inverse MFPT is not in general a constant hazard: free reflected diffusion started at a has squared first-passage coefficient of variation 2/3 rather than the exponential value 1. These exact conditional results derive no physical coordinate, potential, bath, friction, mobility, stochastic convention in physical units, absorbing event, material, population coupling, ignition process, attempt frequency, Kramers or Langer prefactor, or observed rate. +- `C-RMAP-001` — Let S^2 carry its oriented unit round metric and normalized average =(1/(4*pi))*integral_{S^2} F dOmega, with stereographic coordinate z=tan(theta/2)*exp(i*phi). For a declared nonconstant coprime holomorphic rational map R(z)=p(z)/q(z) of exact algebraic degree B>=1, define its conformal Jacobian J_R=((1+|z|^2)/(1+|R|^2)*|dR/dz|)^2, understood in a homogeneous polynomial representation at target poles. The pullback-area identity gives =B. Hence the angular functional I[R]= obeys I[R]>=B^2, with exact deficit I[R]-B^2=<(J_R-B)^2>. Polynomial degree is assigned only after exact common-factor cancellation. For the axial map R(z)=z^B, explicit radial substitution and Euler-beta integration give I_B=(B^3/3)*(1+Gamma(2-1/B)*Gamma(2+1/B)); in particular I_1=1 and I_2=pi+8/3 exactly. These conditional sphere-geometric results do not select a global fixed-degree minimizer, derive a physical Skyrme action or rational-map ansatz, solve a radial profile, identify map degree with a physical baryon or nucleus, or supply a mass, binding energy, reaction, yield, material, or observation map. +- `C-RMAP-002` — Conditional on C-RMAP-001 and on the exact declared rational map R_4(z)=(z^4+2*i*sqrt(3)*z^2+1)/(z^4-2*i*sqrt(3)*z^2+1), exact polynomial reduction makes the numerator and denominator coprime and the map degree four. Homogeneous-Wronskian tensor Gauss-Legendre cubature on u=cos(theta),phi at orders 16x32 through 64x128 gives normalized pullback area converging to four and angular functional converging to 20.6496264884189 in IEEE-754 binary64/complex128. An independent nested adaptive integration split between direct and reciprocal stereographic charts at relative and absolute tolerances 1e-7, 1e-9, and 1e-11 agrees within 2e-12 relative. Axis rotations preserve the two integrals, while changing the imaginary quadratic coefficient to 3.2*i preserves degree area and changes I to about 20.7744. This is resolution-bounded evidence for one declared map only; it proves neither full cubic symmetry nor global degree-four minimality and supplies no radial solution, physical state, baryon or nucleus identification, energy, mass, binding, reaction, yield, material, or observation. +- `C-RPROF-001` — Conditional on C-RMAP-001's declared positive integer degree B and positive angular coefficient I, and on the separately declared dimensionless radial functional E=4*pi*integral_0^infinity [r^2*f'^2 +2*B*sin(f)^2*(1+f'^2)+I*sin(f)^4/r^2] dr, exact one-dimensional variation gives (r^2+2*B*sin(f)^2)*f''+2*r*f' +B*sin(2*f)*(f'^2-1)-I*sin(2*f)*sin(f)^2/r^2=0. The density separates as E=E2+E4 with E2=4*pi*integral [r^2*f'^2+2*B*sin(f)^2]dr and E4=4*pi*integral [2*B*sin(f)^2*f'^2+I*sin(f)^4/r^2]dr. Under the scale family f_s(r)=f(exp(s)*r), convergence and endpoint data that remove boundary terms give E(s)=exp(-s)*E2+exp(s)*E4; a stationary member therefore obeys E2=E4 and has positive curvature E2+E4 in this scale direction. Linearized regular-origin and decaying massless-tail powers are sigma=(sqrt(1+8*B)-1)/2 and p=(sqrt(1+8*B)+1)/2, satisfying sigma*(sigma+1)=2*B and p*(p-1)=2*B, with asymptotic residuals r*f'+sigma*(pi-f)=0 and r*f'+p*f=0. At B=1,I=1 the density and equation reduce exactly to C-MOD-001. These are conditional exact reduced-model identities. They do not derive a physical Skyrme action or rational-map ansatz, prove half-line existence or uniqueness, establish local or global minimization, solve a full three-dimensional field, identify degree with a baryon or nucleus, or supply a mass, binding energy, reaction, yield, material, or observation. +- `C-RPROF-002` — Conditional on C-RPROF-001 and on accepted angular inputs (B,I)=(1,1),(2,pi+8/3),(4,20.6496264884189), two independent float64 routes give resolution-bounded evidence for one monotone stationary branch at each input. Vacuum-complement DOP853 amplitude shooting on [10^-4,24] with regular-origin and massless-tail Robin data, rtol=3e-10, atol=3e-12, maximum step 0.05, 2401 samples, shared trapezoidal integration, and explicit leading endpoint-energy estimates gives conventional conditional coefficients E/(12*pi^2) of 1.2314456867, 2.4162704269, and 4.5460579996, hence per-degree values 1.2314456867, 1.2081352135, and 1.1365144999. The relative E2/E4 imbalances are 4.98e-9, 1.50e-11, and 3.42e-13. Independent solve_bvp collocation from a fresh two-power initial construction on the same cutoffs, tolerance 3e-7, adaptive residual below 3e-7, and Simpson integration gives 1.2314503696, 2.4162703856, and 4.5460579996. Isolated sampled-quadrature, origin-cutoff, outer-domain, IVP-tolerance, and maximum-step refinements preserve the values and ordering. Accepted, source-biased, I=B, and I=B^2 angular inputs give materially distinct energies; both simple mutations still preserve the selected ordering, so comparator rejection is not its oracle. This is numeric evidence for three declared stationary branches and their conditional selected ordering only. It proves no half-line existence or uniqueness theorem, local or global minimum, rational-map angular minimum, variational upper bound, full three-dimensional solution, physical baryon, deuteron, alpha particle or nucleus, fission threshold, binding hierarchy, reaction, yield, quantum state, mass scale, material, or observation. +- `C-RDIFF-001` — Let U and alpha be positive, let A and n be positive integers, and let b(A) and b(n*A) be real dimensionless coefficients. Conditional on the declared masses M(k)=alpha*b(k)*U and binding convention B_E(k)=k*M(1)-M(k), the signed difference satisfies exactly Q=n*M(A)-M(n*A)=B_E(n*A)-n*B_E(A) =alpha*U*(n*b(A)-b(n*A)). Hence kappa=Q/U has inverse b(n*A)=n*b(A)-kappa/alpha, and for positive alpha its sign is the sign of n*b(A)-b(n*A), with zero surface b(n*A)=n*b(A). If independent input intervals are b(A) in [l_A,u_A] and b(n*A) in [l_F,u_F], their sharp rectangular image is alpha*(n*l_A-u_F) <= kappa <= alpha*(n*u_A-l_F). Separate upper bounds on M(A) and M(n*A) do not in general bound their signed difference: their unknown nonnegative slacks enter with opposite signs. This is an exact conditional linear transformation. It derives no mass formula, scale, action, minimum, state identity, binding convention, reaction, yield, material, or observation. +- `C-RDIFF-002` — Conditional on C-RDIFF-001, C-RPROF-002's resolution-bounded total stationary-branch energy coefficients b(2)=2.4162704269425106 and b(4)=4.54605799958882, multiplicity n=2, and the separately declared normalization alpha=3*pi^2, the normalized signed difference is 2*b(2)-b(4)=0.2864828542962012 and the coefficient is kappa=8.482417318795285 in IEEE-754 binary64 evaluation. Applying the same transformation to P105's independent collocation values gives 8.482414868843847. Treating the componentwise extrema of those two methods as a rectangular sensitivity input gives [8.482414867768218,8.482417319870914], whose positive lower endpoint preserves the conditional sign. This envelope is method-spread sensitivity evidence, not a confidence interval or rigorous discretization enclosure. The result is a conditional reduced-model coordinate only: it is not a variational bound, physical mass, binding energy, baryon or nucleus map, deuteron or helium state, reaction, empirical comparison, quantum correction, BPS limit, or yield. +- `C-BPS-001` — Let X be a closed connected oriented three-manifold with volume form dvol, and let the target be the oriented unit round S^3 with volume form Omega normalized by integral_{S^3} Omega=2*pi^2. For a sufficiently regular map U:X->S^3 of nonzero signed degree B, define the normalized pullback density by U*Omega/(2*pi^2)=B0*dvol. Let lambda and mu be positive, let V:S^3->[0,infinity) have integrable square root, and set W=(1/(2*pi^2))*integral_{S^3} sqrt(V)*Omega. Then the declared energy E[U]=integral_X[(lambda*pi^2*B0)^2+mu^2*V(U)]dvol has the exact decomposition E=integral_X(lambda*pi^2*B0-sign(B)*mu*sqrt(V(U)))^2*dvol +2*lambda*mu*pi^2*abs(B)*W and therefore obeys E>=2*lambda*mu*pi^2*abs(B)*W. Equality holds if and only if the displayed square vanishes almost everywhere. This theorem does not establish that an equality configuration exists in any degree sector, select a potential or coupling, identify degree with a physical baryon or nucleus, or derive a mass, reaction, binding, or yield. +- `C-BPS-002` — Under C-BPS-001, define the energy M(B) of a nonzero degree sector as the infimum of the declared energy over a specified nonempty admissible class, and write K=2*lambda*mu*pi^2*W. Then M(B)>=K*abs(B). If an admissible configuration attains the C-BPS-001 equality in a specified sector, M(B)=K*abs(B) in that sector. Hence, for positive integers A and n with attainment in sectors A and n*A, n*M(A)-M(n*A)=0. If sectors 1 and B>0 attain, the declared binding B*M(1)-M(B) is zero. Linearity of the lower bound alone does not imply these conclusions: writing M(B)=K*abs(B)+s_B with unknown nonnegative sector slacks leaves n*M(A)-M(n*A)=n*s_A-s_(n*A), which can have either sign. This is a conditional attainment theorem, not an existence or physical zero-binding claim. +- `C-BPS-003` — Let A and n be positive integers and epsilon a positive dimensionless parameter tending to zero. Suppose the same positive-degree sectors used in C-BPS-002 have controlled expansions at D=A and D=n*A of the form M_epsilon(D)=K*D+epsilon*Delta_D+r_D(epsilon), with finite fixed-degree coefficients Delta_D. Then exactly n*M_epsilon(A)-M_epsilon(n*A) =epsilon*(n*Delta_A-Delta_(n*A)) +n*r_A(epsilon)-r_(n*A)(epsilon). If both remainders are o(epsilon), this is epsilon*(n*Delta_A-Delta_(n*A))+o(epsilon); if both are O(epsilon^2), the residual is O(epsilon^2). The first-order coefficient may be positive, zero, or negative. This theorem does not establish that a proposed deformation admits the expansion, derive the corrections or their sign, provide a global interpolation, or make epsilon or a physical binding coefficient numerically small. +- `C-SG-019` — Let H and L be independent real formal coordinates, let A, a_H, a_L, and phi_0 be real, and declare the classical local function V(H,L)=A*(1-cos(phi_0+a_H*H+a_L*L)). For nonnegative integers j and k, the coefficient of H^j*L^k is A*g^(j+k)(phi_0)*a_H^j*a_L^k/(j!*k!), where g(phi)=1-cos(phi). Equivalently, the raw mixed origin derivative is this coefficient multiplied by j!*k!. At phi_0=0 the constant coefficient is zero; every positive odd-total coefficient is zero; and for positive even m=j+k the coefficient is A*(-1)^(m/2+1)*a_H^j*a_L^k/(j!*k!). Hence the one-high coefficient [H*L^n]V vanishes for even n and equals A*(-1)^((n-1)/2)*a_H*a_L^n/n! for odd n. For unit amplitude and scales, its nonzero magnitudes 1/n! tend to zero along the odd subsequence even though the entire series has no finite terminal order. A finite Taylor polynomial is not the full cosine and requires separate remainder control for approximation claims. The parity support is expansion- background dependent, and every coefficient carries the displayed coordinate-scale powers. These are exact classical local-coordinate identities. They establish no selected frequency or normal-mode split, canonical normalization, quantization, matrix element, occupation factor, phase space, spacetime overlap, energy-momentum channel, resonance, transition rate, material realization, or nuclear process. +- `C-SPN-002` — Let N be a positive integer and let H=(C^2)^(tensor N) have the orthonormal computational basis of N declared two-state factors. For 0<=k<=N, let |D_N^k> be the normalized equal superposition of all binomial(N,k) basis vectors with exactly k excited factors. With local dimensionless raises sigma_i^+ and a declared real operator scale s, define J_+=s*sum_i sigma_i^+, J_-=J_+^dagger, and J_z=(s/2)*sum_i sigma_i^z. Then exactly J_+|D_N^k>=s*sqrt((N-k)*(k+1))*|D_N^(k+1)> for k=s*sqrt(k*(N-k+1))*|D_N^(k-1)> for k>0 and zero for k=0. In j=N/2, m=k-N/2 coordinates these are the standard irreducible su(2) ladder coefficients, with [J_+,J_-]=2*s*J_z and Casimir J_z^2+(J_+*J_-+J_-*J_+)/2=s^2*j*(j+1). The ground-edge coefficient is s*sqrt(N), while central-rung coefficients are order N and approach N*abs(s)/2 in magnitude for even N. More generally, for a weighted ground-state raise A_+=a*sum_i g_i*sigma_i^+ with declared complex g_i and a, its projection on |D_N^1> is a*sum_i(g_i)/sqrt(N), its total one-excitation norm squared is |a|^2*sum_i|g_i|^2, and the orthogonal dark norm squared is their nonnegative difference. Hence the familiar equal-coupling square-root enhancement requires equal magnitudes and phases; unequal phases can cancel the symmetric projection without canceling the full image. These are exact normalized finite-dimensional vector-space identities. A squared ladder coefficient is not a rate. The theorem establishes no physical two-level constituents, nuclear or phonon state, symmetric preparation, common mode, interaction Hamiltonian, resonance, spectral density, linewidth, decoherence, Fermi-Golden-Rule regime, supertransfer, material realization, or observed transition rate. +- `C-RES-001` — Let H_PP be a finite square complex endpoint block, H_QQ a finite square complex intermediate block, and H_PQ and H_QP compatible complex blocks. At a declared real spectral energy E for which E*I-H_QQ is invertible, define the exact finite effective block H_eff(E)=H_PP+H_PQ*(E*I-H_QQ)^-1*H_QP. For a two-state intermediate block with diagonal energies +Delta-i*Gamma/2 and -Delta-i*Gamma/2, real nonzero Delta, real Gamma>=0, and declared off-diagonal coupling products c_plus and c_minus, its paired endpoint contribution is exactly R(E)=c_plus/(E-Delta+i*Gamma/2)+c_minus/(E+Delta+i*Gamma/2). When both products equal c, this becomes 2*c*(E+i*Gamma/2)/((E+i*Gamma/2)^2-Delta^2), and at E=0 it is -i*c*Gamma/(Delta^2+Gamma^2/4). At zero loss and E=0 the contribution cancels exactly if and only if c_plus=c_minus; away from E=0 the equal- product zero-loss expression is 2*c*E/(E^2-Delta^2) where defined. For nonzero c, the magnitude of the equal-product E=0 contribution on Gamma>0 vanishes as Gamma tends to zero and infinity, has small-loss coefficient -i*c/Delta^2 and large-loss coefficient -4*i*c/Gamma, and has its unique positive maximum |c|/|Delta| at Gamma=2*|Delta|. A sum of L identical pairs grows by L when each pair product is fixed and is invariant when the total product weight is held fixed by using c/L per pair; changing L is model enlargement, not a numerical refinement. These are exact finite complex-matrix identities. A common imaginary shift is a declared phenomenological input and the theorem establishes no microscopic bath, complete open-system or Lindblad dynamics, physical loss mechanism, normalized probability, transition rate, nuclear or phonon channel, material realization, magnitude, or observation. +- `C-SCR-001` — Let E and G be positive real energies and U a nonnegative real energy in the same unit. Define the dimensionless bare inverse-square-root barrier factor B(E,G)=exp(-sqrt(G/E)), the shifted factor P(E,U,G)=exp(-sqrt(G/(E+U))), and the enhancement F=P/B. Then exactly F=exp(sqrt(G/E)-sqrt(G/(E+U))), with 00 and partial_G log(P)=-1/(2*sqrt(G)*sqrt(E+U))<0. For U>0, F>1 and decreases strictly with E. As E tends to zero from above, B tends to zero, P tends to the finite positive value exp(-sqrt(G/U)), and F tends to infinity; for U=0, P=B and both tend to zero. As E tends to infinity, P and F tend to one. Common positive rescaling (E,U,G)->rho*(E,U,G) leaves all three factors invariant. Consequently, if an independent premise establishes 0<=U<=U_max, then P(E,U,G)<=P(E,U_max,G); the theorem neither derives U_max nor establishes a material screening model. Direct evaluation of P avoids the separate numerical zero-times-infinity form B*F at very small E. These are exact conditional dimensionless identities. They do not derive a Coulomb or Gamow approximation, physical screening energy, universal material ceiling, cross section, astrophysical S factor, collision flux, attempt frequency, density, branching, transition rate, reaction yield, heat, coherent channel, material realization, or observation. +- `C-CMP-001` — Let m be a positive integer, let Delta_j be real nonzero detunings, and let c_j be real nonnegative coupling products with at least one c_j>0. For real Gamma>0 and omega>0, C-RES-001's common-loss zero-energy magnitude is K(Gamma)=Gamma*sum_(j=1)^m c_j/(Delta_j^2+Gamma^2/4). Composing it with C-DYN-001's nominal count Q_nom=omega/(2*pi*Gamma) in a declared 1/Gamma quadratic-envelope window gives exactly H_nom(Gamma)=omega/(2*pi)*sum_j c_j/(Delta_j^2+Gamma^2/4). This product is strictly positive and strictly decreasing for Gamma>0, with derivative -omega*Gamma/(4*pi)*sum_j c_j/(Delta_j^2+Gamma^2/4)^2, finite positive right limit omega/(2*pi)*sum_j c_j/Delta_j^2 at zero loss, and inverse-square large-loss coefficient (2*omega/pi)*sum_j c_j. On the source-style extension that assigns zero at Gamma=0 and at Gamma>=2*omega, the right limit at zero and left limit omega/(2*pi)*sum_j c_j/(Delta_j^2+omega^2) at 2*omega are both positive. The extension therefore has two jumps, has a nonattained supremum at zero loss, and has no positive-loss maximizer. Replacing the nominal count by the actual underdamped oscillator count Q_act=sqrt(omega^2-Gamma^2/4)/(2*pi*Gamma) gives an also strictly decreasing product on 0rho^2*c gives H->rho*H, while holding c fixed gives H->H/rho. These are conditional finite-matrix and window-count identities, not a phase-coherence, probability, transition-rate, nuclear-channel, material, magnitude, yield, heat, or observation theorem. +- `C-XOV-001` — Let f be a real continuous strictly increasing function on [0,infinity), let a=f(0), and suppose f has a finite limit b>a at infinity that it never attains at finite input. A horizontal level below a or above b has no crossing, level a has the unique crossing zero, every level strictly between a and b has exactly one finite positive crossing, and level b occurs only as an infinite-input limit. If S(E;E0)=1-exp(-E/E0) for real E>=0 and E0>0, then S has range [0,1) and its unique finite inverse for 0<=c<1 is E_x=-E0*log(1-c). On 00, partial_c^2 E_x=E0/(1-c)^2>0, and partial_E0 E_x=-log(1-c)=E_x/E0>0; E_x tends to infinity as c tends to one from below. For C-SCR-001's shifted factor P(E,U,G)=exp(-sqrt(G/(E+U))), with G>0 and U>0 in the same energy unit, the attained lower value is p0=exp(-sqrt(G/U)), the unattained upper limit is one, and every p00, partial_G E_x=1/log(c)^2>0, and partial_U E_x=-1. At U=0 the lower endpoint is zero in the limiting convention and the inverse is G/log(c)^2. Common positive rescaling of all energy inputs rescales each crossover energy by the same factor. Continuity, strict monotonicity, and actual range are independently load bearing: a discontinuity can skip a level, a plateau can give repeated crossings, and a nonmonotone response can give multiple crossings. These are conditional dimensionless level-crossing identities. A free level, response scale, or physical normalization makes any selected positive crossing nonidentifying, and formal curve ordering alone establishes no common observable, state, interaction, coherent or tunnelling channel, transition rate, material crossover, predicted energy, yield, heat, or observation. +- `C-BRN-001` — Let A and B be exact nonnegative real quantities in one common dimension with A+B>0. Define q_A=A/(A+B) and q_B=B/(A+B). Then exactly 0<=q_A,q_B<=1, q_A+q_B=1, (A,B)=(0,B) gives (q_A,q_B)=(0,1), and (A,B)=(A,0) gives (1,0); the double-zero point is undefined. For A,B>0, q_A/q_B=A/B, partial_A q_A=B/(A+B)^2>0, and partial_B q_A=-A/(A+B)^2<0. At fixed B>0, q_A tends from zero to one as A runs from zero to infinity. Common positive scaling (A,B)->(s*A,s*B) leaves both fractions invariant, while independent channel scaling generally changes them. If positive common-dimension baselines r_w and r_c, a positive dimensionless weight w, and a positive integer N are separately declared, then A=r_w*w*N, B=r_c, and rho=r_c/r_w give q_A=w*N/(w*N+rho) and q_B=rho/(w*N+rho). On the positive continuous N extension, partial_N q_B=-rho*w/(w*N+rho)^2<0. Relative to the same positive baselines at population one and positive baseline weight w_1, the ratio of A/B odds is w*N/w_1. A common positive gate cancels, but unequal positive gates C_A and C_B change q_A by A*B*(C_A-C_B)/((A+B)*(C_A*A+C_B*B)). Every target 00, S2'=rho^2*S2 and C2'=rho^2*C2 imply b'_a=rho_a^2*b_a and B'_ab=rho_a^2*rho_b^2*B_ab, so beta'_a=beta_a/rho_a. Applied only to a separately supplied three-factor table with squared Abelian normalization 3/5, three copies of five declared Weyl multiplets, and one declared complex scalar doublet, the exact specialization is b=(41/10,-19/6,-7) and B=((199/50,27/10,44/5),(9/10,35/6,12),(11/10,9/2,-26)). This theorem imports the loop weights and supplied invariants. It excludes the same-order Yukawa contribution, multiple-Abelian kinetic mixing, thresholds, matching, boundary data, field-content or anomaly derivation, a preferred Abelian normalization, unification, Standard-Model identity, observed running, and substrate realization. +- `C-RGE-006` — Let C-RGE-005 supply an exact three-factor gauge-only coefficient ledger (b,B), and let inverse couplings a_i=4*pi/g_i^2 remain positive. In the downward coordinate u=log(Lambda/mu), its declared beta convention gives da_i/du=b_i/(2*pi)+sum_j B_ij/(8*pi^2*a_j). Supply a positive reference scale mu0, positive high-boundary ratios S, a real rank-two 2-by-3 constraint matrix C with nonzero target vector d, a nonzero real readout vector w, and a positive readout normalization N. The conditional inverse problem is a(0)=A*S with unknown A>0 and T=log(Lambda/mu0)>0, subject to C*a(T)=d; its remaining declared coordinate is w dot a(T)/N. When B=0, q=T/(2*pi) and a(T)=A*S+q*b. If the exact design matrix with columns C*S and C*b is nonsingular and its solution (A,q) and resulting trajectory are positive, (A,q)=design^(-1)*d is the unique exact solution. For the separately supplied specialization b=(41/10,-19/6,-7), B=((199/50,27/10,44/5),(9/10,35/6,12),(11/10,9/2,-26)), mu0=227969/2500 in the supplied scale unit, S=(1,1,1), C rows (0,0,1) and (5/3,1,0), d=(500/59,1279/10), w=(0,1,0), and N=1279/10, the zero-matrix solution is exactly A=1639681/39530, q=186383/39530, and readout=6296809/30335322. The nonzero-matrix status-gated numerical solution is A=41.3445253, T=29.1415861, Lambda=4.13015e14 in the supplied scale unit, and readout=0.210641136; DOP853 tolerance refinement, Radau, and an independent direct-gauge- coupling formulation agree beyond the reported digits, close the two constraints below 6e-14, and remain positive. Rescaling mu0 by a common positive factor rescales Lambda by that factor and leaves A, T, the low inverse couplings, and readout unchanged. This is a conditional inverse solution on supplied coordinates, not an ab-initio prediction. It excludes physical input provenance and uncertainty, a preferred equal boundary, Yukawa terms, multiple-Abelian kinetic mixing, thresholds, matching, scheme conversion, global perturbative validity, observed running, unification, unknown higher-order tensors, an all-orders no-go, and substrate realization. A comparator-fitted replacement B->k*B is a target-dependent one-parameter inverse family and grants none of those excluded conclusions. +- `C-MAX-001` — Conditional on C-GAU-001's real connection and field-strength convention, a positive integer spatial dimension d, flat (d+1)-dimensional signature (+,-,...,-), a positive supplied kinetic coefficient kappa, and a smooth supplied contravariant current j^mu, the declared action density L=-kappa*F_mu_nu*F^mu_nu/4-j^mu*A_mu has, under compactly supported variations, the exact Euler equation kappa*partial_mu F^mu_nu=j^nu. The Bianchi identity follows from F=dA; the double divergence vanishes identically, so a solution requires partial_nu j^nu=0, which is also the bulk condition for gauge invariance of the source action after its boundary term is removed. Deleting the kinetic term gives only j^nu=0 and does not force A to be pure gauge. In the static convention A_0=phi, A_i=0, j^0=rho, the equation is -kappa*Delta(phi)=rho and E=-grad(phi). Let S_(d-1)=2*pi^(d/2)/Gamma(d/2), let r>0, and separately supply a point source Q with normalized flux kappa*S_(d-1)*r^(d-1)*E_r=Q. For integer d>2 and phi tending to zero at infinity, phi=Q/[kappa*(d-2)*S_(d-1)*r^(d-2)] and E_r=Q/[kappa*S_(d-1)*r^(d-1)]. For d=2, fixing phi=phi0 at positive r0 gives phi=phi0-Q*log(r/r0)/(2*pi*kappa); for the even d=1 full-line branch with phi(0)=phi0, phi=phi0-Q*r/(2*kappa). These branches are harmonic away from the source and have the same normalized radial flux. The potential decays at infinity for every d>2; within this integer family the radial force is inverse-square only for d=3. If a separate test-charge dictionary declares U=q*phi and F=q*E, then like source and test charges repel and opposite signs attract. This theorem imports the action, kappa, current, dimension, point source, boundary data, and force dictionary. It derives no kinetic coefficient, charged matter, electric ontology, preferred dimension, photon, physical electromagnetic or material sector, gravity coupling, observed force, or substrate mechanism. +- `C-VAC-001` — Conditional on C-GAU-001's local-U1 connection convention, independently declare N positive-integer identical complex scalars in Euclidean two dimensions with mass m>0, charge magnitude e>0, covariant derivative D_mu=partial_mu-i*e*A_mu, quadratic operator -D^2+m^2, one-loop contribution Gamma_loop=Tr log(-D^2+m^2), a translation- and gauge-preserving regulator, and the quadratic convention Gamma_loop^(2)=A_mu*Pi_mu_nu*A_nu/2. Including both the scalar bubble and seagull, their contracted tadpole coefficients are respectively +2*N*e^2 and -2*N*e^2, so the Ward identity follows from their exact cancellation under the regulator's momentum-shift identity rather than from imposing a transverse ansatz. For Euclidean Q=q^2>0, define P_mu_nu=delta_mu_nu-q_mu*q_nu/Q and z=sqrt(Q)/sqrt(Q+4*m^2). Then Pi_mu_nu=P_mu_nu*Pi_hat(Q) =(Q*delta_mu_nu-q_mu*q_nu)*Pi_scalar(Q), where Pi_hat(Q)=N*e^2/pi*(atanh(z)/z-1) and Pi_scalar(Q)=Pi_hat(Q)/Q. Equivalently, Pi_hat=N*e^2*Q/(4*pi) times the integral from zero to one of (1-2*x)^2/[m^2+Q*x*(1-x)] dx. The low-momentum expansion is Pi_hat=N*e^2*Q/(12*pi*m^2)-N*e^2*Q^2/(120*pi*m^4)+O(Q^3), so this loop's leading local Euclidean effective-Lagrangian coefficients are N*e^2/(48*pi*m^2) for F_mu_nu*F_mu_nu and N*e^2/(24*pi*m^2) for F_01^2. At fixed Q>0 the m->0+ scalar limit diverges to positive infinity, whereas the fixed-Q heavy-mass limit and the e->0 loop contribution vanish. The finite e^2/pi massless limit of the fermion-shaped x*(1-x) integrand is not this scalar theorem. The result imports the quantum field, statistics, determinant, mass, charge, multiplicity, regulator, and loop prescription. It neither quantizes C-U1-001's classical field nor identifies physical electric charge, and it supplies no bare Maxwell coefficient, counterterm choice, gauge-field measure, analytic continuation, gauge fixing, propagator pole, propagating photon, preferred dimension, dimensional lift, observation, or substrate gauge-sector mechanism. +- `C-KRN-002` — Let C-KRN-001 fix the inverse-angular Fourier convention and its subcritical Riesz kernel. Supply a positive real analytic dimension parameter d, positive radii r and r0, and a real nonzero inverse-kernel coefficient A. Put s=d/2-epsilon with epsilon>0, subtract the subcritical kernel's value at r0, and only then take epsilon to zero from above. The exact critical limit is Gcrit(r;r0)=2*log(r0/r)/[A*4^(d/2)*pi^(d/2)*Gamma(d/2)]. The unsubtracted kernel diverges through Gamma(epsilon); changing r0 changes Gcrit only by a radius-independent constant; and dGcrit/dr=-2/[A*4^(d/2)*pi^(d/2)*Gamma(d/2)*r]. At d=2 this becomes log(r0/r)/(2*pi*A), with the same unit radial-flux normalization as C-MAX-001's source-normalized logarithmic branch. C-MAX-001's ordinary d=1 full-line linear branch is a separately prescribed distributional inverse and is outside the subcritical Riesz integral. Separately, for the C-KRN-001 subcritical domain 00 and m>=0, so that K*(-Delta+m^2)*phi_a=J_a. Put equal triplet point-dipole sources of magnitude P>=0 at distinct X_A and X_B, with J_A^a=-P*A_ai*partial_i delta(x-X_A) and J_B^a=-P*B_ai*partial_i delta(x-X_B), A and B proper rotations. Let R=|X_B-X_A|>0, u=(X_B-X_A)/R, D=A^T*B in SO(3), and G_m(R)=exp(-mR)/(4*pi*R). After subtracting the two divergent isolated point-dipole self energies, the finite on-shell cross energy is exactly E_int=P^2/K*[(G_m'/R)*Tr(D)+(G_m''-G_m'/R)*u^T*D*u]. Write a=G_m'/R=-exp(-mR)*(1+mR)/(4*pi*R^3) and c=G_m''=exp(-mR)*(m^2*R^2+2*mR+2)/(4*pi*R^3). Over the full proper relative-orientation space, the global minimum is -P^2*c/K, attained by every pi rotation about an axis perpendicular to u; the global maximum is P^2*(c-2a)/K, attained by the pi rotation about u; and the identity orientation has energy P^2*m^2*G_m/K. At fixed most-attractive orientation the radial force is -P^2*exp(-mR)*(m^3*R^3+3*m^2*R^2+6*mR+6)/(4*pi*K*R^4), strictly inward for P>0. For m>0 the interaction and force decay faster than every power; as m tends to zero from above the minimum energy tends to -P^2/(2*pi*K*R^3) and the force to -3*P^2/(2*pi*K*R^4). This is an exact conditional source-coupled long-range point-dipole theorem. It does not derive the declared field or sources from a nonlinear Skyrme action, construct B=1 or two-center Skyrmion solutions, prove a product ansatz or short-range core, quantize a nucleon, or establish a nucleon-nucleon potential, binding, material, scale, observation, or substrate mechanism. +- `C-IRR-001` — In the standard fundamental SU(3) convention of C-LIE-001 with Y=2*T_8/sqrt(3), let p and q be arbitrary nonnegative integer Dynkin labels and use the U(3) Gelfand-Tsetlin top row (p+q,q,0). The exact Weyl dimension is (p+1)(q+1)(p+q+2)/2, the quadratic Casimir is (p^2+p*q+q^2+3*p+3*q)/3, and the C-LIE-002 center triality is p+2*q modulo three. Every interlacing pattern p+q>=m12>=q>=m22>=0 and m12>=m11>=m22 gives one basis state with I=(m12-m22)/2, I3=m11-(m12+m22)/2, and Y=m12+m22-2*(p+2*q)/3; these patterns are complete, their count equals the Weyl dimension, and aggregation gives the exact weight multiplicities and multiplicity-free SU(2)xU(1) branching rows. Conditional on an exact rational target hypercharge and explicitly supplied finite nonnegative bounds max_p and max_q, enumerating all labels in the rectangular domain returns every irrep containing that hypercharge and preserves all minimum-dimension ties. At target Y=1 the unique global minimum is (1,1) of dimension eight with I=1/2; the next distinct dimension is ten and contains both (0,3) with I=1/2 and (3,0) with I=3/2. This is a mathematical representation and kinematic-filter theorem only. It does not supply a collective-coordinate action, right generator constraint, WZW level or response, N_c, baryon number, statistics, Hamiltonian, symmetry breaking, particle dictionary, mass spectrum, physical octet/decuplet selection, or substrate mechanism. +- `C-VEC-001` — Let x be a nonempty exact real matrix with three columns, define L_i=i*x_i^a*sigma_a using the Pauli convention of C-CHI-001, let G_ij=x_i dot x_j, I1=Tr(G)^2, and I2=Tr(G^2), and use ordered sums over i,j. Then sum_ij |x_i cross x_j|^2=I1-I2 and sum_ij Tr([L_i,L_j]^2)=-8*(I1-I2). Separately declare positive exact kappa and dimensionless g. The quadratic density kappa*sum_ia(v_i^a-x_i^a/2)^2 has the unique stationary point v=x/2 and positive Hessian 2*kappa*I. If L obeys Maurer-Cartan flatness, the half connection Gamma_i=L_i/2 has F_ij(Gamma)=-[L_i,L_j]/4. Substitution into the separately declared curvature density -sum_ij Tr(F_ij*F_ij)/(2*g^2) gives -sum_ij Tr([L_i,L_j]^2)/(32*g^2)=(I1-I2)/(4*g^2), so the equally normalized leading Skyrme coupling is e=g. With L=O(p), this density is order p^4; the kinetic vector equation changes the field at order p^3/M^2 and first changes the action at order p^6/M^2 under the supplied heavy-scale counting. Conditional on the separately declared relation m_V^2=a*g^2*F^2 with positive dimensionless a and equal mass dimensions for m_V and F, e=g=m_V/(sqrt(a)*F) is dimensionless. This is an exact current-algebra and conditional leading-connection theorem. It does not derive HLS field content, a physical rho or pion, the KSRF premise or a=2, B1, c4, J1, a full-vector solution, Skyrmion stabilization, a medium response, particle values, or a substrate realization. +- `C-RAD-001` — Let A and c be exact positive real quantities, B an exact real source coupling, and q(t) an exact real point-source amplitude, and separately declare the one-dimensional scalar Lagrangian density A*(phi_t^2-c^2*phi_x^2)/2+B*phi*q(t)*delta(x). Its Euler equation is phi_tt-c^2*phi_xx=(B/A)*q(t)*delta(x). Under no-incoming retarded boundary data and a source primitive I with I'=q and vanishing past boundary term, the distributional solution is phi(t,x)=B*I(t-|x|/c)/(2*A*c). Its one-sided derivatives obey the outgoing characteristic relations, and their jump is -B*q/(A*c^2), exactly reproducing the delta source. From the same action, the canonical energy density is A*(phi_t^2+c^2*phi_x^2)/2 and the right-directed flux is -A*c^2*phi_t*phi_x. Each outgoing side carries B^2*q^2/(4*A*c), total outward power is B^2*q^2/(2*A*c), and that total equals the local source-work rate B*q*phi_t(t,0). The field rescaling phi'=s*phi, A'=A/s^2, B'=B/s preserves B^2/A and the power. For constant q, the static field -B*q*abs(x)/(2*A*c^2) obeys the same local equation and jump but has zero flux, so the retarded radiation conclusion requires the stated boundary and history data and does not follow from the sourced equation alone. This is an exact conditional scalar-action theorem. It does not derive a dilaton or gravitational action, propagating graviton, optical metric, sine-Gordon or breather source, accelerated solution, multipole expansion, coupling value, radiation reaction, physical gravity, observation, or substrate realization. +- `C-GOR-001` — Let eta_ab=diag(-1,1,1,1), let u^a be an exact real unit timelike four-velocity with eta_ab*u^a*u^b=-1, and let n be an exact positive refractive index. The signature-consistent mostly-plus Gordon effective inverse metric and metric are g^ab=eta^ab+(1-n^2)*u^a*u^b and g_ab=eta_ab+(1-1/n^2)*u_a*u_b. They are exact mutual inverses with determinants -n^2 and -1/n^2, so no positive-index sqrt(2) pole occurs; in the medium rest frame g_ab=diag(-1/n^2,1,1,1) and its null phase speed is 1/n. If the medium has constant z velocity v with |v|<1 and gamma^2=1/(1-v^2), while n=n(x) depends only on the transverse coordinate, define K=(n*n_xx-2*n_x^2)/n^2. In coordinates (t,x,y,z), the only nonzero covariant Einstein components are G_tt=-gamma^2*v^2*K, G_tz=G_zt=gamma^2*v*K, G_yy=-K, and G_zz=-gamma^2*K; G_xx=0 and the Ricci scalar is 2*K. Direct Christoffel reconstruction satisfies the contracted Bianchi identity componentwise. A constant index is flat. At v=1/2 and the local profile data n=2, n_x=1, n_xx=0, G_tt=1/6, G_tz=-2*G_tt, G_yy=3*G_tt, and G_zz=4*G_tt. This is an exact conditional effective-wave-geometry theorem. It does not supply a dielectric or matter action, conserved material stress, Einstein-source match, coupling, boundary-value solution, dynamical or physical gravity, observation, or substrate realization. +- `C-GOR-002` — Let n(x) be an exact positive twice-differentiable refractive index on a connected interval, let the C-GOR-001 mostly-plus Gordon metric have a uniform exact real z velocity v with |v|<1, and let an exact real canonical C-STG-001 scalar U depend only on t and x, with exact real potential value V and exact positive coupling kappa. At each scalar jet, every covariant algebraic Einstein equation G_ab=kappa*T_ab holds if and only if U_t=0, U_x=0, V=0, and K=(n*n_xx-2*n_x^2)/n^2=0. For v nonzero, the source-ray equations for U_t^2, U_x^2, and V have rank three; in the exact parameterization v=r/sqrt(1+r^2), r real and nonzero, one numerator-normalized minor is 8*n^2*r^2*sqrt(1+r^2), hence nonzero. At v=0, the zero tt and xx equations are n^2*U_t^2+U_x^2+2*V=0 and n^2*U_t^2+U_x^2-2*V=0; real-square nonnegativity forces the same scalar vacuum, and the remaining yy and zz equations force K=0. Conversely those four zero conditions make both tensors zero. Exactly (1/n)_xx=-K/n, so K vanishes on the interval if and only if the positive reciprocal index is affine there. This is a local algebraic compatibility theorem. A full Einstein-scalar solution additionally requires the scalar Euler equation, including V'(U)=0 on the constant branch, and suitable global boundary data. It does not derive a sine-Gordon breather embedding, nonvacuum source, material medium, physical gravity, observation, or substrate realization. +- `C-STG-001` — Let g_ab be an exact four-dimensional Lorentzian metric with mostly-plus signature, let phi be an exact real scalar, let V(phi) be an exact real potential, and let kappa be exact and positive. Declare the action S=integral sqrt(-g)*[R/(2*kappa)-g^ab*partial_a(phi)*partial_b(phi)/2-V(phi)] d^4x, with compact metric variations or the corresponding gravitational boundary term. Its matter stress is T_ab=partial_a(phi)*partial_b(phi)-g_ab*[(partial phi)^2/2+V], its equations are G_ab=kappa*T_ab and box(phi)=V'(phi), and nabla^a T_ab=(box(phi)-V'(phi))*partial_b(phi), so the stress is conserved on shell. For V=0, positive reference time t0 and scale a0, real phi0, branch s in {-1,1}, and domain t>0, the spatially flat FLRW metric ds^2=-dt^2+a(t)^2*(dx^2+dy^2+dz^2), with a(t)=a0*(t/t0)^(1/3) and phi(t)=phi0+s*sqrt(2/(3*kappa))*log(t/t0), is an exact nonvacuum solution. It has H=1/(3*t), rho=p=1/(3*kappa*t^2), Ricci scalar -2/(3*t^2), and Kretschmann scalar 20/(27*t^4). Thus t=0 is a curvature singularity and the solution approaches flatness as t tends to infinity. On noncompact spatial slices its homogeneous positive energy is extensive, not a localized finite-energy source. This theorem does not identify phi with a refractive index or sine-Gordon breather, derive a localized three-dimensional source, select a physical kappa, or supply a material, observational, gravitational-analog, or substrate realization. +- `C-RR-001` — Let u be a nonzero finite exact real generalized-rate column, let P be an exact nonnegative supplied power, and let G be a separately declared exact real symmetric positive-definite coordinate metric whose entries include any unit conversions required for the coordinate pairing. Then Q0=-P*G*u/(u^T*G*u) satisfies Q0^T*u=-P and uniquely minimizes Q^T*G^(-1)*Q over every force covector with that work. The complete balanced family is Q=Q0+z for arbitrary exact real z satisfying z^T*u=0. Thus one scalar power does not determine multiple generalized- force components without extra allocation data. In one coordinate with nonzero rate v, the force is uniquely -P/v. At v=0, positive P is inconsistent with instantaneous work balance, while P=0 leaves the force unconstrained. Separately, if an exact real symmetric positive- semidefinite damping matrix D is declared, the Rayleigh function R=u^T*D*u/2 gives Q_d=-D*u, dissipated power P_d=u^T*D*u=2*R>=0, and the open energy ledger E_dot=Q_ext^T*u-P_d. This is an exact conditional generalized-coordinate power-balance and effective-dissipation theorem. It does not derive P, G, D, a field or source action, source dynamics, self-field, regularization, causal response, near-field energy, radiation-reaction coefficient, internal-mode backreaction, physical coupling, observation, material mechanism, or substrate realization. +- `C-MED-005` — In SI base-dimension order (M,L,T,I), permittivity epsilon has (-1,-3,4,2), inverse permeability mu_inverse has (-1,-1,2,2), mechanical mass density rho has (1,-3,0,0), and stiffness or energy density K has (1,-1,-2,0). Conditional on exact positive quantities and separately declared multiplicative conversions rho=a*epsilon and K=b*mu_inverse, both a and b require dimension (2,0,-4,-2), and c_m^2=K/rho=(b/a)*mu_inverse/epsilon. Hence c_m^2 equals the electromagnetic wave-speed square mu_inverse/epsilon exactly iff a=b. Their common value remains a free calibration: common rescaling changes rho, K, and energy while leaving speed fixed. For separately declared exact dimensionless real strain xi, u=K*xi^2/2 and u/c_m^2=a*epsilon*xi^2/2. Thus bare SI epsilon/2 is not a mass density and bare mu_inverse/2 is not an energy density. This exact conditional constitutive theorem derives no material, calibration, field or strain state, gravity coupling, observation, or substrate realization. +- `C-BND-001` — Let phi be a sufficiently smooth real scalar, let u=phi_t(t,b) and v=phi_x(t,b), and let a, beta, and the scalar source J be exact real declared boundary data. Define R_(a,beta,J)=a*u+beta*v-J. Under scalar spatial parity phi_P(t,x)=phi(t,-x) and source pullback J_P(t,x)=J(t,-x), exact chain rule gives R_(a,beta,J_P)[phi_P](t,b)=R_(a,-beta,J)[phi](t,-b). Thus beta and minus beta form a parity-covariant parameter family. At a parity center, a fixed residual is internally invariant for arbitrary traces iff beta=0. The mixed residual is not a parity-odd eigenobject: its a*u-J part is even and beta*v part is odd; it is purely odd only when the even part is absent. Under the simultaneous parity map from a right half-line to a left half-line, the outward-normal derivative and its coefficient are unchanged. Finally one residual equation does not determine both u and v; for beta=0 it leaves v arbitrary, and for beta nonzero it admits the family v=(J-a*u)/beta. It therefore derives neither boundary sign correlation nor topological charge transfer or selection without additional dynamics and vacuum-boundary data. This exact conditional theorem supplies no boundary action, field evolution, charge map, fermion parity, chiral matter, weak interaction, or substrate realization. +- `C-REP-002` — Let H_I=C^2 have its standard inner product and Hermitian generators T_a=sigma_a/2. Let H_C be a nonzero finite-dimensional complex inner-product space and let P be a Hermitian idempotent on H_C, with P_R=I-P. Then G_a^L=T_a tensor P and G_a^R=T_a tensor P_R are Hermitian and each obeys [G_a,G_b]=i*epsilon_abc*G_c; G^L acts as the fundamental doublet on H_I tensor image(P) and vanishes on H_I tensor kernel(P), with the roles exchanged for G^R. The vector and axial combinations are G^V=G^L+G^R=T_a tensor I and G^A=G^L-G^R. If a declared unitary parity operator exchanges P and P_R while leaving H_I untouched, it exchanges G^L and G^R, leaves G^V even, and makes G^A odd; G^L alone is not a parity-odd eigenoperator. On H_I itself the full commutant of all T_a consists only of scalar matrices, so no nontrivial rank-one projector on that same irreducible carrier can be an independent chirality projector. In particular T_a*diag(1,0) is non-Hermitian for a=1,2 and fails the su2 algebra. Any Hermitian Abelian generator Y commuting with every T_a is y*I, so Q=T_3+c*Y has eigenvalues c*y+1/2 and c*y-1/2 with separation one. Assigned labels plus and minus one therefore cannot both be eigenvalues of one such Q; the rescaled declaration q_top/2=T3 with y=0 is algebraically compatible but is not selected by the representation. These exact finite-dimensional identities establish no physical state identification, topological transition, Lorentz chirality, fermion parity, gauge connection, interaction, anomaly statement, weak sector, or substrate realization. +- `C-KIN-001` — Let m1 and m2 be exact positive masses and theta an exact real rapidity. In 1+1 signature (+,-), take the center-of-mass threshold four-vector P=(m1+m2,0) and one observed on-shell vector p1=(m1*cosh(theta),m1*sinh(theta)). The residual p2=P-p1 closes total four-momentum exactly and has mass-shell defect p2^2-m2^2=2*m1*(m1+m2)*(1-cosh(theta)), which is nonpositive and vanishes if and only if theta=0. At that equality point p1=(m1,0) and p2=(m2,0), so both particles are at rest. Thus a nonzero two-particle recoil requires total energy above threshold; otherwise the residual cannot be a second free particle of mass m2 and requires a separately modeled channel such as boundary storage, work, or radiation. This exact kinematic theorem derives no scattering outcome, state production, detector response, charge selection, current, neutrino, weak interaction, or substrate realization. +- `C-SCT-001` — Let c and zeta be exact positive speeds and z=zeta/c. On the right half-line x>=0, let a real massless scalar obey phi_tt-c^2*phi_xx=0, with incoming harmonic proportional to exp(-i*omega*(t+x/c)), reflected harmonic proportional to exp(-i*omega*(t-x/c)), and passive boundary condition phi_t-zeta*phi_x=0. Exact substitution gives amplitude reflection r=(z-1)/(z+1), reflected power R=r^2, and, for a steady harmonic ledger with no other boundary storage or flux channel, absorbed fraction T=1-R=4*z/(z+1)^2. The half-line bulk energy rate contributed by the boundary is -c^2*zeta*phi_x(0,t)^2<=0. Under z->1/z, r changes sign while R and T are invariant, so power data alone do not identify which reciprocal impedance applies. If a separate reference channel is declared perfectly reflected, C-BRN-001 gives contrast A=(1-R)/(1+R)=2*z/(z^2+1)=T/(2-T); A is a deterministic transform, not an independent prediction, and is also reciprocal-invariant. This theorem derives no boundary law from a piston action, lambda or mu coefficient, physical chirality, parity violation, current, charge, detector, weak interaction, or substrate realization. +- `C-NAG-001` — Let H_I=C^2 carry the exact Pauli-half generators T_a, let H_C be a nonzero finite-dimensional carrier, and let P be an exact Hermitian projector with P_R=I-P. On H_I tensor H_C define G_a=T_a tensor P. For a smooth local U_I(x) in SU2 define U=U_I tensor P+I tensor P_R. Let g be exact positive, W_mu=W_mu^a G_a, and D_mu=partial_mu-i*g*W_mu. The finite transformation W_mu'=U*W_mu*U_dagger-(i/g)*(partial_mu U)*U_dagger gives D_mu'(U*psi)=U*D_mu psi. The curvature F_mu_nu=partial_mu W_nu-partial_nu W_mu-i*g*[W_mu,W_nu] obeys [D_mu,D_nu]psi=-i*g*F_mu_nu*psi and transforms as F_mu_nu'=U*F_mu_nu*U_dagger. The connection annihilates the P_R sector. With a separately declared invariant metric and cyclic trace, the quadratic trace density is gauge invariant and contains the noncommutative cubic and quartic algebra, but this does not derive an action coefficient or equation of motion. A declared parity exchange maps the left construction to a distinct right construction rather than making the left connection parity odd. These identities derive no physical matter field, charged current, charge-event equality, anomaly cancellation, Yang-Mills induction, weak boson, mass, coupling match, detector, or substrate realization. +- `C-BER-001` — Let k be an integer and phi range from zero to two pi. In a complex carrier of dimension at least two, define the normalized real lift n_k(phi)=(cos(k*phi/2),sin(k*phi/2),0,...). Its rank-one projector is periodic and n_k(2*pi)=(-1)^k*n_k(0). More generally, let psi be an exact normalized section of this closed rank-one projector path with psi(2*pi)=tau*psi(0), where |tau|=1. With convention A=i*psi_dagger*d_phi psi, the closed-ray Berry holonomy is H=tau*exp(i*integral_0^{2*pi} A*dphi). Under a smooth exact phase change psi'=exp(i*chi)*psi, A'=A-d_phi chi and tau'=exp(i*(chi(2*pi)-chi(0)))*tau, so H is invariant. The real lift has A=0 and tau=(-1)^k. Its periodic complex gauge exp(-i*k*phi/2)*n_k has A=k/2 and tau=1. Both give H=(-1)^k. In contrast, exp(-i*k*phi/2) times a fixed ray has a constant projector and H=1 even though its bare integral phase is (-1)^k. Thus the local one-form is not a unique invariant, and these identities derive no physical vector potential, curvature or core source, defect dynamics, electromagnetic field, fermion, material realization, coupling, or observation. +- `C-GSM-001` — Let a nonzero finite-dimensional complex inner-product carrier have a nonempty supplied family of exact Hermitian generators T_a, exact real couplings g_a, and an exact declared vacuum column phi_0. Conditional on the scalar covariant kinetic term with D_mu*phi_0=-i*sum_a(g_a*A_mu^a*T_a)*phi_0, write its real gauge-field quadratic part as one half A^T*M2*A. Then M2_ab=g_a*g_b*phi_0_dagger*{T_a,T_b}*phi_0, equivalently twice the real Gram matrix of u_a=g_a*T_a*phi_0. It is real symmetric positive semidefinite and x^T*M2*x=2*norm(sum_a x_a*u_a)^2 for every real coefficient vector x. Thus its kernel is exactly the coupled vacuum stabilizer. When the supplied generators are independent over real coefficients and every coupling is nonzero, its nullity is the stabilizer dimension in that generator span; dependent labels or zero couplings do not support that interpretation. If the real gauge fields instead have a separately declared symmetric positive-definite kinetic metric K, their quadratic mass parameters are the generalized eigenvalues of M2*x=lambda*K*x, not the raw eigenvalues of M2 unless K is identity. Under a real invertible field change A=S*A_new, both forms transform by congruence, M2_new=S^T*M2*S and K_new=S^T*K*S, preserving generalized eigenvalues and nullity. For T_i=sigma_i/2, T_4=I/2, couplings (g,g,g,g_prime), and phi_0=(0,v/sqrt(2)), with exact positive g, g_prime, and v, the canonical M2 has charged diagonal entries g^2*v^2/4 and neutral block v^2/4 times [[g^2,-g*g_prime],[-g*g_prime,g_prime^2]], rank three, a null vector proportional to (g_prime,g), nonzero neutral eigenvalue (g^2+g_prime^2)*v^2/4, and the conditional rho identity one. The congruence B mapped to -B flips the neutral off-diagonal sign while preserving these invariants. These are exact conditional quadratic identities. They do not derive the scalar action or potential, prove phi_0 is a stationary ground state or condensate, supply a gauge kinetic action, establish a spectral pole or Higgs particle, identify photon W or Z fields, select the Standard Model, or realize a substrate mass mechanism. +- `C-PRC-001` — Let A^mu be a real twice-differentiable vector field on 3+1 Minkowski space with eta=diag(-1,+1,+1,+1), define F_mu_nu=partial_mu*A_nu-partial_nu*A_mu, and separately declare the source-free density L=-F_mu_nu*F^mu_nu/4-m^2*A_mu*A^mu/2 with exact m>0 and boundary behavior licensing integration by parts. Exact variation gives partial_mu*F^(mu nu)-m^2*A^nu=0. Commuting derivatives and antisymmetry then derive partial_nu*A^nu=0 as a massive constraint rather than a gauge choice, and hence (box-m^2)*A^nu=0 for box=-partial_t^2+nabla^2. A plane wave proportional to exp(-i*omega*t+i*k dot x) has omega^2=|k|^2+m^2. For a static tangential component A_y(x) on x>=0 with A_y(0)=A0 and A_y tending to zero as x tends to infinity, the unique solution is A0*exp(-m*x); the general ODE also has a growing branch, and a longitudinal A_x(x) generally violates the derived divergence constraint. The inverse length is m and the penetration length is 1/m. More generally, a separately declared positive one-mode kinetic coefficient kappa and positive quadratic coefficient q give m^2=q/kappa. Conditional on C-GSM-001's lower-doublet coefficient q=g^2*v^2/4, exact positive g and v, and an independently declared canonical free kinetic action, m=g*v/2 and the length is 2/(g*v). At m=0 the divergence constraint is not derived and no finite positive penetration length follows. These are exact conditional action and boundary-value identities. They supply no London current, Maxwell-material response, Meissner magnetic observation, stationary condensate, physical W boson, Standard Model selection, or substrate mass mechanism. +- `C-HOL-001` — Let H be a nonzero finite-dimensional complex inner-product carrier and let B_1 through B_n be a nonempty chronological sequence of exact Hermitian integrated connection matrices. In the convention inherited from D=partial-i*g*W, define V(B_1,...,B_n)=exp(i*B_n)*...*exp(i*B_1), with later segments on the left. V is unitary and det(V)=exp(i*sum_j trace(B_j)); concatenation is ordered multiplication. The reversed path has chronological segments -B_n through -B_1 and transporter V_reverse=V_dagger=V_inverse. Pairwise commuting B_j collapse to exp(i*sum_j B_j). If supplied exact unitary node gauges U_0 through U_n transform segment transporters as E_j'=U_j*E_j*U_(j-1)_dagger, their product transforms as V'=U_n*V*U_0_dagger. For a declared closed path with U_n=U_0, V changes by conjugation, so its characteristic polynomial, spectrum, determinant, raw trace, and dimension-normalized trace are invariant; V itself is not a scalar invariant. A cyclic basepoint move conjugates the ordered product. In the Pauli-half fundamental carrier, exp(2*pi*i*T3)=-I_2 and exp(4*pi*i*T3)=I_2. For chronological segments a*T1 then a*T2, the ordered product minus exp(i*a*(T1+T2)) is not the zero matrix function and has leading term i*a^2*T3/2, while commuting axes collapse. The same 2*pi SU2 center element acts as -I_2 in the fundamental and +I_3 in the adjoint, with raw traces -2 and +3 and normalized traces -1 and +1. These are exact conditional finite-matrix transport identities. They supply no base-space defect, flux, curvature localization, gauge action, weak matter carrier, physical W sector, Aharonov-Bohm observation, detector response, or substrate map. +- `C-NVP-001` — Let Q>0, m>0, and g>0 be exact, let N_s be a positive integer, and let T_1,T_2,T_3 be same-size finite exact Hermitian matrices satisfying [T_1,T_2]=i*T_3 and cyclic permutations with tr_R(T_a*T_b)=T(R)*delta_ab for exact T(R)>0. Separately declare N_s identical massive complex-scalar multiplets in representation R on flat two-dimensional Euclidean space, with D_mu=partial_mu-i*g*W_mu^a*T_a, operator -D^2+m^2, the complex-scalar one-loop determinant, and a translation- and background-gauge-preserving regulator. In the convention Gamma^(2)=W_mu^a*Pi_mu_nu^ab*W_nu^b/2 and Pi_mu_nu^ab=P_mu_nu*Pi_hat^ab at nonzero momentum, define z=sqrt(Q)/sqrt(Q+4*m^2). Then Pi_hat^ab(Q)=delta_ab*N_s*T(R)*g^2/pi*(atanh(z)/z-1), equivalently the color transverse form factor is Pi_hat^ab/Q. Before transverse decomposition, contraction of the scalar bubble and seagull gives respectively +2*N_s*g^2*T(R)*delta_ab and -2*N_s*g^2*T(R)*delta_ab times the common tadpole and q_nu, so their sum vanishes. The projector coefficient vanishes as Q tends to zero and as m tends to infinity, while at fixed Q>0 it diverges as m tends to zero. Its leading low-momentum local Euclidean density is N_s*g^2*T(R)/(48*pi*m^2)*sum_a F_mu_nu^a*F_mu_nu^a, equivalently N_s*g^2/(48*pi*m^2)*tr_R(F_mu_nu*F_mu_nu). The independent background-field proper-time factors (4*pi)^(-1), 1/12, and integral_0^infinity exp(-m^2*s) ds=1/m^2 give the same trace-density coefficient and hence complete this leading term with the full non-Abelian curvature, including its connection-commutator pieces. This loop term is additive to separately declared bare and counterterm coefficients. These are exact conditional one-loop identities. They supply no unique total gauge coupling, massless Schwinger pole, propagating W sector, scalar-to-kink dictionary, weak matter content, dimensional lift, observation, or substrate gauge mechanism. +- `C-NVP-002` — Let Q>0, m>0, and g>0 be exact, let N_s be a positive integer, and let T_a for a=1,...,n be a nonempty same-size family of finite exact Hermitian matrices. Supply exact explicitly real fully antisymmetric f_abc such that [T_a,T_b]=i*sum_c f_abc*T_c and require tr_R(T_a*T_b)=T(R)*delta_ab for exact T(R)>0. Separately declare N_s identical massive complex-scalar multiplets in representation R on flat two-dimensional Euclidean space, with D_mu=partial_mu-i*g*W_mu^a*T_a, operator -D^2+m^2, the complex-scalar one-loop determinant, and a translation- and background-gauge-preserving regulator. In the quadratic convention of C-VAC-001, with z=sqrt(Q)/sqrt(Q+4*m^2), the nonzero-momentum color projector coefficient is Pi_hat^ab(Q)=tr_R(T_a*T_b)*N_s*g^2/pi*(atanh(z)/z-1), and the transverse form factor is Pi_hat^ab/Q. Before transverse decomposition, contraction of the scalar bubble and seagull gives respectively +2*N_s*g^2*tr_R(T_a*T_b) and -2*N_s*g^2*tr_R(T_a*T_b) times the common tadpole and q_nu, so their sum vanishes. The projector coefficient vanishes as Q tends to zero and as m tends to infinity, while at fixed Q>0 it diverges as m tends to zero. Its leading local Euclidean density is N_s*g^2*T(R)/(48*pi*m^2)*sum_a F_mu_nu^a*F_mu_nu^a, equivalently N_s*g^2/(48*pi*m^2)*tr_R(F_mu_nu*F_mu_nu), with the full non-Abelian curvature fixed by the C-NAG-001 convention. The independent proper-time factors (4*pi)^(-1), 1/12, and integral_0^infinity exp(-m^2*s) ds=1/m^2 give the same trace-density coefficient. This loop term is additive to separately declared bare and counterterm coefficients. It supplies no unique total coupling, physical quark or gluon, QCD sector, massless pole, preferred dimension, dimensional lift, observation, or substrate gauge mechanism. +- `C-PGA-001` — On C^3 tensor C^2, let T_a be the eight standard fundamental SU3 generators of C-LIE-001 and let t_i=sigma_i/2 be the three standard Pauli-half SU2 generators of C-REP-002. For an exact explicitly real nonzero supplied scalar y, define C_a=T_a tensor I_2, L_i=I_3 tensor t_i, and Y=y*I_6. Then every C_a and L_i is Hermitian, each factor obeys its accepted Lie bracket, all 35 cross-factor commutators among C_a, L_i, and Y vanish, and the twelve matrices are linearly independent. They therefore give a faithful finite-dimensional representation of the local Lie algebra su3 direct-sum su2 direct-sum u1. The full joint commutant of the C_a and L_i in Mat_6(C) is exactly the scalar span of I_6, so every Hermitian Abelian generator commuting with both non-Abelian factors has the form y*I_6. For separately supplied exact positive factor couplings and exact connection components, the algebra-valued connection component is their unique displayed linear sum in this fixed basis. If a compact U1 parameter is separately assigned period 2*pi, its full-turn action is exp(2*pi*i*y)*I_6 and descent to that compact parameterization additionally requires exp(2*pi*i*y)=1. These exact local matrix facts do not select a global direct product rather than a finite central quotient, a compact hypercharge normalization, physical matter representations, fields, a gauge transformation law, kinetic terms, an action, currents, couplings matched to observation, gauge bosons, a Standard Model identification, or a substrate mechanism. +- `C-REP-003` — For a separately supplied nonempty finite table of uniquely named rows, let row a have positive integer spectator multiplicity m_a, a nonempty finite sequence of exact real isospin weights t_aj, and one exact real Abelian coordinate y_a. For an exact real electric coefficient c, define q_aj=t_aj+c*y_a. The grouped spectra and state count N=sum_a m_a*n_a are exact, and flattening each component with multiplicity m_a gives precisely the C-REP-001 finite charge-trace ledger. Given separately supplied exact target charges qhat_aj of the same length and an exact provably nonzero c, one component fixes the unique candidate y_a=(qhat_a1-t_a1)/c; the row is consistent exactly when every reconstruction residual t_aj+c*y_a-qhat_aj vanishes. This uniqueness is conditional on the supplied weights, coefficient, and targets and does not select any of them. For the C-REP-002 Pauli-half doublet weights, every consistent common-coordinate spectrum has separation one. Charge conjugation maps every t_aj and y_a to its negative, retains m_a, and negates every q_aj; retaining a spectator dimension does not distinguish a representation from its conjugate. For separately supplied positive rho and positive Abelian coupling g, the simultaneous map y_a'=rho*y_a, c'=c/rho, and g'=g/rho preserves every grouped charge, every product g*y_a, the state count, and the C-REP-001 covariant trace data, while holding c fixed generally changes the spectra. These are exact supplied-data identities. They do not derive row labels, factor representations, chirality, target charges, the finite table, anomaly cancellation, Yukawa operators, generation completeness, conservation, a compact U1 period or charge lattice, a global gauge group, physical matter, an action, currents, dynamics, observation, a Standard Model identification, or a substrate mechanism. +- `C-ANO-001` — For a separately supplied nonempty finite table of uniquely named left-handed rows of G_a times SU(2) times U(1), let row r have positive integer factor dimensions d_ar and d_2r, exact real Abelian charge y_r, exact nonnegative quadratic indices T_ar and T_2r, an exact signed G_a cubic anomaly coefficient A_ar, and a supplied indicator that is true exactly for each fundamental SU(2) doublet in the scoped table. Under the explicitly imported four-dimensional chiral-anomaly criteria, the exact coefficients are sum_r d_2r*T_ar*y_r for G_a^2 U(1), sum_r d_ar*T_2r*y_r for SU(2)^2 U(1), sum_r d_ar*d_2r*y_r^3 for U(1)^3, sum_r d_ar*d_2r*y_r for gravity^2 U(1), and sum_r d_2r*A_ar for G_a^3. The supplied fundamental-doublet count is sum_r d_ar over marked rows; Witten's imported fundamental-doublet criterion requires this count to be even. Charge conjugation negates y_r and A_ar, preserves dimensions and quadratic indices, and reverses every odd local coefficient while preserving the doublet count. For the separately supplied five-row carrier (Q_L,u_R^c,d_R^c,L,e_R^c) with standard fundamental SU3 and SU2 indices, write its Abelian charges as (q,u,d,l,e). The four charge-dependent local zero conditions are equivalently 2q+u+d=0, 3q+l=0, 6q+3u+3d+2l+e=0, and 6q^3+3u^3+3d^3+2l^3+e^3=0. Their complete real affine zero set is the union of the three lines t*(1,-4,2,-3,6), t*(1,2,-4,-3,6), and t*(0,1,-1,0,0). The displayed SM3 values are the first line at t=1/6; the row-exchanged and vectorlike lines prove that anomaly freedom alone does not uniquely select those ratios up to scale. The fixed carrier's G_a^3 coefficient is zero and its fundamental-doublet count is four on every charge branch, so those charge-independent conditions select none of the three lines. This exact conditional theorem does not derive the carrier, physical row identities, observed charges, a unique hypercharge table, Yukawa or scalar constraints, completeness, higher- representation SU2 global-anomaly classification, a global U1 period or charge lattice, a global gauge group, an action, renormalizability, unitarity, general covariance, a Standard Model generation, a coupling normalization, running, observation, or a substrate mechanism. +- `C-DIM-009` — Let D be a separately supplied exact positive real spacetime-dimension bookkeeping parameter in natural units, with [d^D x]=-D and [partial]=1. In the canonical gauge-potential convention with dimensionless action, density kappa_A F(A)^2/4, dimensionless positive kappa_A, and covariant derivative partial-i*g*A, the exact mass exponents are [A]=(D-2)/2, [g]=(4-D)/2, [F(A)]=D/2, and [F(A)^2]=D. In a quadratic momentum action whose transverse projector is dimensionless, its scalar coefficient Pi_hat has exponent two. The special scale-free ansatz Pi_hat=g^2*c with nonzero dimensionless c is homogeneous exactly when D=2. This implication is not a universal polarization no-go: with a separately supplied positive mass scale M, g^2*M^(D-2)*c has exponent two for every D. In D=4, for momentum-square Q and positive M, every expression Q*f(Q/M^2) has exponent two for a dimensionless f; constant, Q/(Q+M^2), and log(1+Q/M^2) are exact distinct examples, so dimensional analysis alone selects neither a logarithm nor any form factor or coefficient. In the connection-field convention B=g*A and F(B)=g*F(A), [B]=1, [F(B)]=2, and the identical density is kappa_B F(B)^2/4 with kappa_B=kappa_A/g^2 and [kappa_B]=D-4. For a dimensionless fixed generator basis, the convention change T_a'=rho*T_a, g'=g/rho with supplied positive rho gives T(R)'=rho^2*T(R) and preserves g^2*T(R); a trace factor alone carries no spacetime dimension and is not a convention-preserving Abelian limit. These exact identities perform dimensional and normalization bookkeeping only. They do not derive a loop determinant, numerator, regulator, subtraction, counterterm, matching scale, kinetic coefficient value, total coupling, preferred dimension, logarithmic running, dimensional lift, propagating gauge particle, physical gauge group, observable, or substrate mechanism. +- `C-STG-002` — Let the canonical C-STG-001 scalar be phi=F*u with exact positive field scale F, exact positive mass scale mu, and potential V(phi)=mu^2*F^2*(1-cos(u)); let kappa be exact and positive. Define x=mu*r, tau=mu*t, m=mu*M_geo, and alpha=kappa*F^2. On a domain with x>0, N=exp(Phi)>0, and f=1-2m/x>0, use the dimensionless static areal metric ds^2=-N(x)^2*d tau^2+dx^2/f(x)+x^2*d Omega_2^2 and the real single-harmonic ansatz u=a(x)*cos(Omega*tau), with exact positive frequency Omega. Exact phase averaging gives rho=Omega^2*a^2/(4*N^2)+f*a_x^2/4+1-J_0(a), p_r=f*a_x^2/4+Omega^2*a^2/(4*N^2)-(1-J_0(a)), and p_t=Omega^2*a^2/(4*N^2)-f*a_x^2/4-(1-J_0(a)). The reduced Einstein and fundamental-projected scalar equations are m_x=alpha*x^2*rho/2, Phi_x=[m+alpha*x^3*p_r/2]/[x*(x-2m)], and a_xx+[Phi_x+f_x/(2f)+2/x]*a_x+ [Omega^2*a/N^2-2*J_1(a)]/f=0. Identically, (p_r)_x+(rho+p_r)*Phi_x+2*(p_r-p_t)/x equals f*a_x/2 times the displayed scalar residual; this is an induced averaged conservation identity, not an independent pointwise equation. At a regular origin with a(0)=A and Phi(0)=Phi_0, let rho_0 and p_r0 be the displayed stress with a_x=0. Then a_xx(0)=[2J_1(A)-Omega^2*A*exp(-2Phi_0)]/3, m=alpha*rho_0*x^3/6+O(x^5), and Phi_xx(0)=alpha*(rho_0/6+p_r0/2). The flat alpha=0, m=Phi=0 limit is exactly the C-PDE-005 single-harmonic radial equation, and A=0 is the vacuum limit. The first discarded scalar harmonic in the same residual convention is 2J_3(a) cos(3Omega*tau), with coefficient a^3/24+O(a^5), while the pointwise energy density generally retains the cos(2Omega*tau) coefficient -Omega^2*a^2/(4N^2)+f*a_x^2/4+2J_2(a). Therefore this exact object is a scaled phase-averaged single-harmonic reduction and is not generally a pointwise solution of the full time-dependent Einstein-scalar PDE. It establishes no solution existence or uniqueness, exact half-line breather, full oscillaton, nonminimal Horndeski or Gordon dynamics, selected physical kappa, F, or mu, observation, material model, gravity realization, or substrate mechanism. +- `C-PDE-013` — On the exact C-STG-002 reduced model, declare central amplitude A=3, dimensionless coupling alpha=0.03, IEEE float64, origin cutoff epsilon=0.001, finite wall R=40, and frequency constrained to 00, and the exterior Schwarzschild gauge match Phi(R)=log(f(R))/2. SciPy adaptive collocation with 400 initial points, tolerance 1e-10, and at most 100000 nodes converges to Omega=0.890839827775792, m(R)=0.290960714264522, and extrapolated central Phi_0=-0.182426921486489. It uses 3408 adaptive nodes, has maximum collocation RMS residual 9.99802e-11, zero boundary residual at reported precision, maximum off-grid relative first-order ODE residual 7.01282e-11, and minimum full-domain f=0.879013430362296. Axis-isolated initial-mesh 200/400/800, tolerance 1e-6/1e-8/1e-10, origin-cutoff 0.002/0.001/0.0005, and wall 30/40/60 studies all pass the frozen solver, residual, evanescent-tail, and horizon gates. Across each axis against its finest or largest reference, the maximum normalized state, frequency, and outer-mass differences are respectively 1.8344020768e-8, 3.1371716336e-10, and 4.7119512736e-9, below the frozen 2e-6, 2e-7, and 2e-7 gates. An independently written DOP853 plus two-variable root-shooting method with wall continuation gives Omega=0.890839827776100, m(R)=0.290960714264587, Phi_0=-0.182426921486121, and minimum f=0.879018721894243. Zero-coupling, wrong-central-amplitude, and wrong-J_1-sign mutations fail their preregistered gates. This is numeric evidence for one declared finite-wall solution of the phase-averaged reduced BVP. The amplitude and alpha are free branch coordinates, the wall is not spatial infinity, and the result establishes no uniqueness, exact half-line state, pointwise time-dependent Einstein-scalar solution, full oscillaton, nonminimal Horndeski theory, physical scale, observation, material gravity, or substrate mechanism. +- `C-RMOM-001` — On the oriented unit sphere and conformal-Jacobian convention of C-RMAP-001, declare the conditional angular-resolved density epsilon=f_prime^2+2*sin(f)^2*(1+f_prime^2)*J/r^2 +sin(f)^4*J^2/r^4 for an exact real radial profile f. If the declared map has normalized sphere averages =B and =I, exact sphere integration of r^2*epsilon gives precisely C-RPROF-001's radial density r^2*f_prime^2+2*B*sin(f)^2*(1+f_prime^2) +I*sin(f)^4/r^2. Its normalized second STF moment factorizes as I_STF=H1*A1+H2*A2, where H1=integral_0^infinity 2*r^2*sin(f)^2*(1+f_prime^2) dr, H2=integral_0^infinity sin(f)^4 dr, and Ak=integral_S2 J^k*(n*n^T-delta/3) dOmega. The isotropic radial term contributes no STF part. For R(z)=z, J=1 and A1=A2=0 exactly. For the axial map R(z)=z^2, writing u=cos(theta) gives J=4*(1-u^2)/(1+u^2)^2, =2, and =pi+8/3. Both angular tensors are diagonal with A_xx=A_yy=-A_zz/2 and exact axial coefficients A1_zz=8*pi*(3*pi-10)/3<0 and A2_zz=8*pi*(3*pi-16)/9<0. Hence every nontrivial integrable declared profile with positive H1 or H2 has I_STF=diag(q,q,-2*q) with q>0. In the C-MOM-001 convention the source quadrupole is Q=3*I_STF. These are exact conditional reduced-ansatz moment identities. I_STF/M has radial-coordinate-squared units and is not a scale-free physical observable. The result establishes no full three-dimensional field solution, local conserved physical stress, stationary-profile existence or minimum, physical baryon or nucleus, absolute scale, rotation, gravity, waveform, radiation, observation, or substrate realization. +- `C-RMOM-002` — Conditional on C-RMOM-001 and the corrected C-RPROF-002 degree-two stationary branch with (B,I)=(2,pi+8/3), IEEE-754 binary64 DOP853 vacuum-complement amplitude shooting on [10^-4,24], 2401 samples, rtol=3e-10, atol=3e-12, maximum step 0.05, canonical trapezoidal integration, and explicit leading origin and massless-tail estimates gives monopole M=286.171598879686 and normalized tensor I_STF=diag(48.4848609876855,48.4848609876855,-96.969721975371). Therefore I_STF_zz/M=-0.338851662271835 in declared dimensionless radial-coordinate-squared units, while the C-MOM-001 triple convention gives Q_zz/M=-1.016554986815505. Independent solve_bvp collocation from a two-power initial profile with tolerance 3e-7, boundary tolerance 3e-8, at most 50000 nodes, and Simpson integration gives M=286.171593977632 and I_STF_zz/M=-0.338851565198, a relative method difference 2.865e-7. Isolated outer-radius 16/24/32/48, origin-cutoff 2e-4/1e-4/5e-5, sample-count 1201/2401/4801, IVP-rtol 1e-8/3e-10/1e-11, and maximum-step 0.1/0.05/0.025 studies pass solver, endpoint, finite-data, monotonicity, trace, convention, and frozen convergence gates. Independent tensor Gauss-Legendre cubature at 24x48 through 96x192 converges to C-RMOM-001's exact B2 angular tensors. This is resolution-bounded evidence for one conditional stationary reduced-ansatz branch. It proves no half-line existence, uniqueness, local or global minimum, full three-dimensional field solution, local conserved stress, physical state or mass, absolute length, rotation, stability, gravitational coupling, waveform, radiation, observation, or substrate realization. +- `C-GW-009` — Let q and Omega be real, let s be a declared nonzero quadrupole convention scale, and let the normalized body-frame STF moment be I_body=diag(q,q,-2*q), with Q_s=s*I_body. Every proper orthogonal conjugation R*Q_s*R^T has characteristic polynomial (lambda-s*q)^2*(lambda+2*s*q), so it remains axisymmetric about the rotated eigenvector even when three Cartesian diagonal entries are pairwise distinct. Rotation by Omega*t about the body symmetry axis leaves Q_s constant. For the right-handed perpendicular x-axis rotation of an initial z symmetry axis, the nonzero components are Q_xx=s*q, Q_yy=s*q*(-1/2+3*cos(2*Omega*t)/2), Q_zz=s*q*(-1/2-3*cos(2*Omega*t)/2), and Q_yz=Q_zy=3*s*q*sin(2*Omega*t)/2. Thus the tensor is DC plus a pure twice-frequency harmonic. Its second- and third-derivative Frobenius norms are 72*s^2*q^2*Omega^4 and 288*s^2*q^2*Omega^6. When q*Omega is nonzero, the third derivative has three distinct eigenvalues 0 and plus/minus 12*s*q*Omega^3 even though the full moment retains its repeated eigenvalue. Under exactly the conditional C-GW-001/002/008 premises, the same field has convention-independent total power 288*G*q^2*Omega^6/5 and, along the rotation axis with the declared y-z transverse frame, conventional readouts h_plus=-12*G*q*Omega^2*cos(2*Omega*t)/r_obs and h_cross=-12*G*q*Omega^2*sin(2*Omega*t)/r_obs. For a symmetry axis at fixed tilt beta from a z rotation axis, the exact tensor contains an Omega harmonic proportional to sin(beta)*cos(beta) and a 2*Omega harmonic proportional to sin(beta)^2; its third-derivative norm is 18*s^2*q^2*Omega^6*sin(beta)^2*(cos(beta)^2+16*sin(beta)^2), monotone in sin(beta)^2 and maximal at perpendicular tilt. These are exact prescribed moment kinematics and conditional TT identities. They establish no full-field rotating or tumbling solution, local conserved source stress, collective-coordinate action or inertia, dynamically selected Omega or axis, stability, gravitational action or coupling, physical retarded waveform or flux, absolute scale, detector signal, observation, or substrate realization. +- `C-GW-010` — Apply C-GW-009's prescribed perpendicular rotation to a normalized axisymmetric STF moment with repeated transverse eigenvalue q and let Q_s=s*I_STF for declared nonzero convention scale s. Write the dimensionless orbit phase as psi, let Omega be the prescribed angular speed, and parameterize a unit sightline from the x rotation axis by n=(cos(i),sin(i)*cos(phi),sin(i)*sin(phi)). In the oriented natural transverse frame p=(sin(i),-cos(i)*cos(phi),-cos(i)*sin(phi)) and v=(0,sin(phi),-cos(phi)), define c=cos(i), chi=2*(psi-phi), and H=12*G*q*Omega^2/R. Under exactly the conditional C-GW-001/002/008 waveform premises, the conventional matrix readouts are h_plus=-H*(1+c^2)*cos(chi)/2 and h_cross=-H*c*sin(chi); normalized C-GW-002 coordinates are sqrt(2) times these. Rotating the transverse frame by alpha sends the harmonic coefficient matrix to M=R(2*alpha)*diag(-H*(1+c^2)/2,-H*c), where R is the oriented real two-dimensional rotation. Therefore M^T*M=diag(H^2*(1+c^2)^2/4,H^2*c^2) and det(M)=H^2*c*(1+c^2)/2. When H is nonzero, the temporal coefficient rank is two for c nonzero and one for the edge-on c=0 view; the rank-two curve is circular exactly when |c|=1 and otherwise elliptical, with minor-to-major semiaxis ratio 2*|c|/(1+c^2). The TX3 sample n=(1,1,1)/sqrt(3) has ratio sqrt(3)/2. Both traces have the same pure 2*Omega frequency, not incommensurate frequencies. On phases where h_plus is nonzero, h_cross/h_plus=2*c*tan(chi)/(1+c^2), so q, G, R, s, and the common Omega^2 amplitude scale cancel at fixed dimensionless phase; the ratio has poles at plus zeros, and after psi=Omega*t it retains Omega dependence at fixed physical time. The full waveform also retains Omega^2 amplitude and 2*Omega frequency. These are exact prescribed-moment and conditional-TT identities. They establish no dynamically selected Omega, full-field rotating solution, local conserved source stress, collective inertia, stability, gravitational action or coupling, physical retarded waveform or flux, helicity or graviton population, detector signal, observation, or substrate realization. +- `C-FLO-001` — Let B and K be exact constant complex square matrices of equal finite dimension, let Q(t)=exp(t*K), and consider the declared laboratory linear system x_dot=Q(t)*B*Q(t)^(-1)*x. Under x=Q(t)*y, exact chain rule gives y_dot=(B-K)*y and the laboratory fundamental matrix Phi(t)=Q(t)*exp(t*(B-K)). If T is exact and positive with Q(T)=I, the laboratory and transformed monodromies both equal exp(T*(B-K)). For any exact finite monodromy M, its integer powers are bounded exactly when every eigenvalue lies in the closed unit disk and every eigenvalue on the unit circle is semisimple, equivalently its algebraic and geometric multiplicities agree there. Thus a time-independent transformed generator or unit-modulus multipliers alone does not prove stability: a nontrivial boundary Jordan block gives secularly unbounded powers. This is an exact conditional finite-dimensional linear theorem. It supplies no field action, periodic field solution, unbounded-operator domain, constraint or gauge slice, boundary data, complete field spectrum, nonlinear estimate, energy-momentum coercivity, physical source, observation, or substrate realization. +- `C-ROT-001` — Declare a free axisymmetric Euler top in body angular-velocity space with principal inertias diag(A,A,C), exact C>A>0, and exact Omega>0. The circle E_Omega={(Omega*cos(phi),Omega*sin(phi),0): phi real} consists of equilibria. At the member (Omega,0,0), the exact linearization is nonzero rank-one nilpotent, its fundamental matrix is I+t*J, and every period monodromy eigenvalue is one but nonsemisimple. The exact nearby family (r*cos(phi+(C-A)*epsilon*t/A), r*sin(phi+(C-A)*epsilon*t/A),epsilon) solves Euler's equations. With r=Omega and arbitrarily small positive epsilon it reaches squared distance 2*Omega^2+epsilon^2 from the fixed member, so that member is not Lyapunov stable. By contrast, squared Euclidean distance to the entire circle is (r-Omega)^2+epsilon^2 and is constant, so E_Omega is stable as a set in this declared state space. Separately, for an ordinary axisymmetric density with radial second moment R2 and axial moment Z, I_zz-I_xx=-(3/2)*(I_STF)_zz exactly. This ordinary inertia identity does not identify a field theory's collective rotational metric. The theorem establishes no orientation-space stability, forced or dissipative rotor, Skyrme action or inertia, rotating field solution, full-field stability, selected Omega, physical state, fission, gravity, radiation, observation, or substrate realization. +- `C-RMAP-003` — For the oriented-sphere angular functional I[R] and conventions of C-RMAP-001, use the local degree-two chart R=(z^2+a1*z+a0)/(b2*z^2+b1*z+b0) about R=z^2 with real coordinate order (Re(a1),Im(a1),Re(a0),Im(a0),Re(b2),Im(b2),Re(b1),Im(b1), Re(b0-1),Im(b0-1)). Exact differentiation under the full-sphere integral gives I[z^2]=pi+8/3 and zero gradient. The exact real symmetric Hessian has rank five, nullity five, and eigenvalues zero five times, pi once, pi+16/3 twice, and 7*pi+64/3 twice. Its kernel is exactly the span of the five independently derived infinitesimal domain-Mobius, target-Mobius, and phase directions displayed by the canonical API; the displayed complementary five-dimensional quadratic form is positive definite. Hence R=z^2 has an exact positive second variation modulo those symmetry directions inside this coefficient chart. This is a chart-local fixed-degree rational-map angular theorem. It establishes no collective kinetic metric, radial or full-field Hessian, global minimum, full three-dimensional solution, dynamical or Floquet stability, physical Skyrmion, fission barrier, gravity, radiation, observation, or substrate realization. +- `C-SKY-002` — Let G be an exact real 3-by-4 matrix with rows g_i, let D=G*G^T, and declare the pointwise quadratic and quartic static densities e2=tr(D) and e4=((tr(D))^2-tr(D^2))/2. Then e2 is the sum of the twelve squared gradient components and e4 is exactly the sum over i=2*I_4, and the coefficient two is sharp because gradients and tangent parallel to one component saturate it. For positive integrated two- and four-derivative energies, the declared full-space scale family phi_alpha(x)=phi(x/alpha) has E(alpha)=alpha*E2+E4/alpha, slope E2-E4 and curvature 2*E4 at alpha=1; positive scale curvature does not supply the separate stationarity equation E2=E4. These are exact conditional pointwise and scaling identities. They do not derive a physical Skyrme action or normalization, a unit-vector constraint consequence, topology or boundary conditions, a stationary finite or continuum field, static Hessian positivity, a strict local minimum, a rotating relative equilibrium, gyroscopic terms, linear or nonlinear stability, fission, gravity, radiation, observation, or substrate realization. +- `C-VAC-002` — Conditional on C-GAU-001's connection convention and C-DIM-009's mass- dimension and normalization bookkeeping, independently declare one free complex charged-Dirac field, its fermionic determinant at one loop, the tensor convention Pi_mn=(q^2*g_mn-q_m*q_n)*Pi2(q^2), and a common shift- invariant gauge-preserving regulator. For exact invertible free propagator matrices A=S(p) and B=S(p+q), a spectator vertex G, and the inverse-propagator identity qslash=B^-1-A^-1, exact trace cyclicity gives Tr(qslash*B*G*A)=Tr(G*A)-Tr(G*B); the regulated momentum translation then derives the integrated Ward contraction rather than imposing a transverse ansatz. For exact positive spacelike Q=-q^2, nonnegative mass square M2, positive charge magnitude e with [e^2]=4-d, positive integration dimension d away from unevaluated Gamma poles, and a separately declared positive-integer spinor trace n_gamma, define Delta=M2+x*(1-x)*Q. The exact source-convention form factor is Pi2(-Q)=-2*n_gamma*e^2*Gamma(2-d/2)/(4*pi)^(d/2) times the integral from zero to one of x*(1-x)*Delta^(d/2-2) dx. Pi2 is dimensionless, while the mixed-projector coefficient q^2*Pi2=-Q*Pi2 has mass dimension two. No analytic n_gamma(d) is asserted. The endpoint d=2, n_gamma=2, M2=0 at Q>0 gives Pi2(-Q)=-e^2/(pi*Q) and q^2*Pi2=e^2/pi for one Dirac fermion; it is not the massless complex-scalar limit of C-VAC-001. Separately, hold n_gamma=4, set d=4-2*epsilon, and multiply the master by mu2^epsilon. At zero momentum the bare form factor is -e^2*Gamma(epsilon)*(4*pi*mu2/M2)^epsilon/(12*pi^2), with Laurent residue -e^2/(12*pi^2). Adding the displayed MS-bar pole counterterm and an arbitrary finite local constant c_fin gives Pi2_MSbar(0)=e^2*log(M2/mu2)/(12*pi^2)+c_fin; its log(M2) and log(m) slopes are e^2/(12*pi^2) and e^2/(6*pi^2), while the scale slopes have the opposite signs. For real timelike w=q^2/M2 with 0<=w<4, the finite subtraction is e^2/(2*pi^2) times the integral from zero to one of x*(1-x)*log(1-w*x*(1-x)) dx. Its series begins -e^2/(2*pi^2)*(w/30+w^2/280+w^3/1890), with convergence radius and first branch point four; above threshold requires a separately typed -i0 boundary value. A generator convention change T->c*T, g->g/c preserves only g^2*tr(T*T), not the trace alone. These exact conditional identities derive no physical charged excitation, bare or total Maxwell coefficient, finite matching condition, selected coupling, physical representation or gauge group, preferred dimension, on-shell polarization count, dimensional lift, observation, or substrate mechanism. +- `C-VAC-003` — Conditional on C-GAU-001's connection convention, C-DIM-009's mass- dimension and normalization bookkeeping, C-VAC-002's independently declared charged-Dirac D4 comparator, and C-MAX-001's exact no-kinetic variational ceiling, independently declare one or more free complex charged scalars, the scalar determinant Tr log(-D^2+M2) at one loop, the tensor convention Pi_mn=(q^2*g_mn-q_m*q_n)*Pi2(q^2), and one common shift-invariant gauge-preserving regulator for the oriented scalar bubble and seagull. With D_p=p^2+M2 and D_pq=(p+q)^2+M2, q*(2p+q)=D_pq-D_p. After the declared common momentum translation, the bubble Ward contraction is +2*q_nu times the tadpole integral and the seagull is its negative, deriving transversality rather than imposing a transverse ansatz. For positive spacelike Q=-q^2, nonnegative M2, positive charge magnitude e with [e^2]=4-d, positive integration dimension d away from unevaluated Gamma poles, and positive integer species count N, define Delta=M2+x*(1-x)*Q. The exact source-convention scalar form factor is Pi2(-Q)=-N*e^2*Gamma(2-d/2)/(4*pi)^(d/2) times the integral from zero to one of (1-2*x)^2*Delta^(d/2-2) dx, and is dimensionless. Separately set d=4-2*epsilon, multiply by mu2^epsilon, require M2>0, and take zero momentum. The bare form factor is -N*e^2*Gamma(epsilon)*(4*pi*mu2/M2)^epsilon/(48*pi^2), with Laurent residue -N*e^2/(48*pi^2). Adding the displayed MS-bar pole counterterm and arbitrary finite local c_fin gives Pi2_MSbar(0)=N*e^2*log(M2/mu2)/(48*pi^2)+c_fin. Its log(M2), log(m), and log(mu) slopes are respectively N*e^2/(48*pi^2), N*e^2/(24*pi^2), and -N*e^2/(24*pi^2); in the same convention the C-VAC-002 Dirac slopes are exactly four times these one-species scalar slopes. For the connection field B=e*A and kinetic action -Z_B*F(B)^2/4, separately supplied nonnegative exact complex-scalar and Dirac invariant weights W_s and W_f compose as b=W_s/3+4*W_f/3, with mu*dZ_B/dmu=-b/(8*pi^2). For positive mu and mu_ref and an independent real reference value Z_ref, the complete exact family is Z_B(mu)=Z_ref+b*log(mu_ref/mu)/(8*pi^2). Under mu_ref'=kappa*mu_ref, the same function has Z_ref'=Z_ref-b*log(kappa)/(8*pi^2). The separately declared zero-matching specialization Z_ref=0 has positive Z_B exactly when b>0 and mu_ref>mu; general positivity depends on Z_ref. Neither an absent kinetic operator in a no-matter theory nor C-MAX-001's current constraint fixes Z_ref in a theory after charged matter is introduced. This theorem does not fix Z_ref, a bare coefficient, finite matching condition, reference scale, physical charged spectrum, multiplicity, representation, gauge group, total coupling, preferred dimension, observation, or substrate realization. +- `C-VAC-004` — Conditional on C-RGE-003's positive formal one-loop energy and paired inverse-energy length map and C-VAC-003's exact affine matter-induced connection-field kinetic family, separately supply positive exact lengths ell0 and ell1, positive exact conversions K0 and K1, exact real local and finite matching coordinates with sum Z_ref, and nonnegative exact invariant complex-scalar and Dirac weights W_s and W_f. Define E0=K0/ell0, E1=K1/ell1, R_ell=ell1/ell0, R_K=K1/K0, and b=W_s/3+4*W_f/3. Then E0/E1=R_ell/R_K, the exact scale logarithm is L=log(R_ell/R_K), and the complete affine composition is Z(E1)=Z_ref+b*L/(8*pi^2). A common positive rescaling of ell0 and ell1 leaves R_ell, L, and Z unchanged while shifting both absolute energies; at fixed lengths, changing either conversion changes L. On C-RGE-003's consistently paired specialization E1=E0*exp(-X), X=8*pi^2/(b0*g^2), ell0=K0/E0, and ell1=K1/E1, one has R_ell=R_K*exp(X), so L=X without requiring K0=K1 and Z(E1)=Z_ref+b/(b0*g^2). Only under the separately declared zero-matching specialization Z_ref=0 and a provably positive b does the positive inverse kinetic coordinate equal b0*g^2/b. Holding physical lengths fixed while changing a conversion, inserting an unpaired positive soliton or threshold factor, reversing an unpaired orientation, or changing Z_ref changes the relevant logarithm or total. The composition imports the logarithm from the accepted one-loop kinetic theorem; dimensional power zero is a constant rather than a logarithm, and dimensions do not select a unique form factor. This theorem does not identify either length or energy with a lattice, granularity, soliton, Compton scale, QCD, confinement, hadron, cutoff, observation, or substrate object; select b0, g^2, a conversion, matter content, group, matching boundary, absolute scale, or perturbative domain; or turn the inverse kinetic coordinate into a physical gauge coupling. +- `C-CMB-001` — For every positive integer n, define q_n=1/(n!)^2. Then q_n is a positive exact rational, q_1=1, q_(n+1)/q_n=1/(n+1)^2, and the sequence is strictly decreasing. The positive exponential series gives e^n>n^n/n!, hence q_n<(e/n)^(2*n). For every fixed nonnegative integer p, if a_n=n^p*q_n then a_(n+1)/a_n=((n+1)/n)^p/(n+1)^2<=2^p/(n+1)^2; from n>=ceil(sqrt(2^(p+1)))-1 onward this ratio is at most one-half, so n^p*q_n tends to zero. Thus q_n decays faster than every fixed inverse power. The exact exponential-series bounds e<49/18<11/4 and integer comparison (11/4)^20<10^9 further imply that for every positive integer d, at n=10^d one has q_n<10^(-((20*d-9)*n/10)). In particular the exact exponent ceilings for d=7,9,11 are respectively -131000000, -17100000000, and -2110000000000. Applied to C-SG-019's exact zero-background one-high cosine coefficient with declared real amplitude A and coordinate scales a_H and a_L, the coefficient square is zero for even low order n and is A^2*a_H^2*a_L^(2*n)*q_n for odd n. The exact q_n remains positive at every positive integer even when a finite machine representation underflows or overflows. A classical coefficient square is not by itself a matrix element, transition rate, probability, branching weight, physical subdivision law, or material prediction. +- `C-CMB-002` — Let z be an exact positive dimensionless real and let the positive integers carry counting measure. Define m_z(n)=1_odd(n)*z^(2*n)/(n!)^2, so the support is the positive odd integers. Its exact positive total is M(z)=sum_(n positive odd) m_z(n)=(I_0(2*z)-J_0(2*z))/2, and p_z(n)=m_z(n)/M(z) is a normalized mathematical mass function. For an exact positive rational z and positive odd cutoff K, put L_K=sum_(n odd,n<=K)m_z(n), f_K=m_z(K+2), and r_K=z^4/((K+3)^2*(K+4)^2). Whenever r_K<1, the monotone consecutive-odd ratio gives L_K972/1000 and p_1(1)+p_1(3)>9999/10000, and the odd masses have mode one. The mode is normalization-dependent: at z=4 the order-three to order-one ratio is 64/9 and the order-five to order-three ratio is 16/25, so the unique mode is three. This normalization declares a mathematical sample space and does not by itself derive occurrence probabilities, normalized quantum states, a complete interaction, transition rates, branching channels, a physical subdivision band, or a material prediction. +- `C-OSC-001` — Let H_F be a one-mode bosonic Fock Hilbert space with orthonormal basis |n> for nonnegative integers n, and let D_alg be its algebraic finite-support span. Define a|0>=0, a|n>=sqrt(n)|n-1> for n>=1, and a_dagger|n>=sqrt(n+1)|n+1>. Then D_alg is a common invariant domain, a_dagger*a|n>=n|n>, a*a_dagger|n>=(n+1)|n>, and [a,a_dagger]=I on D_alg. Moreover (a_dagger)^n|0>=sqrt(n!)|n>, with squared norm n!. In the D-level matrix truncation the exact commutator is I_D-D|D-1>=q_0^n*sqrt(n!): in exactly n ladder actions only the all-creation word reaches |n> from |0>. Thus the conditional low-sector element of the formal H-linear coefficient is zero for even n and, for positive odd n, equals U*(-1)^((n-1)/2)*h*q_0^n/sqrt(n!), whose real square is U^2*h^2*S^n/n! with S=q_0^2>0. This is not a complete transition amplitude because a high-sector operator and state element remain separate. For the mathematical factorial-one mass S^n/n!, the exact totals on all nonnegative integers, positive integers, and positive odd integers are respectively exp(S), exp(S)-1, and sinh(S), with zero off the declared support and normalization by the corresponding total. Consecutive all-order masses have ratio S/(n+1), while consecutive odd masses have ratio S^2/((n+1)*(n+2)); these ratios determine modes and retain every exact integer tie. At S=25 the positive-integer family has tied modes 24 and 25, whereas the positive-odd family has unique mode 25. The declared Fock structure, ladder norm, conditional low-sector element, and normalized mathematical masses do not by themselves derive substrate quanta, a count or density of distinct final states, a high-sector matrix element, interaction Hamiltonian, energy rule, spectral density, physical probability, transition rate, branching channel, medium parameter, or material prediction. +- `C-CMB-003` — Let S be one exact positive dimensionless real and define the normalized all-nonnegative factorial-one mass p_S(n)=exp(-S)*S^n/n! for every nonnegative integer n. For every n>=1 its interior log-concavity quotient is p_S(n)^2/(p_S(n-1)*p_S(n+1))=(n+1)/n>1. This strict log-concavity retains the exact C-OSC-001 mode set: noninteger S has the unique mode floor(S), while positive integer S has the adjacent tied modes S-1 and S. Its probability-generating function is G_S(t)=exp(S*(t-1)), and every nonnegative integer falling-factorial moment is E[(N)_r]=S^r; in particular its mean and variance are both S. For exact alpha>0 and integer N>=0 satisfying N+1>=S*exp(alpha), one has p_S(N+k)<=p_S(N)*exp(-alpha*k) for every integer k>=0 and sum_{n=N+1}^infinity p_S(n)<=p_S(N)/(exp(alpha)-1). Moreover, for every fixed nonnegative integer r and every exact rational q in (0,1), any integer N>=1 with N+1>=S*2^r/q makes the consecutive ratio of n^r*p_S(n) at most q for all n>=N, hence n^r*p_S(n) tends to zero. These are exact mathematical distribution and tail statements on counting measure. They do not derive a physical Poisson process, occurrence rate, time interval, phase-space law, energy-gap suppression, power-law interpolation, subdivision mechanism, medium mean, material parameter, branching channel, or prediction. +- `C-OSC-002` — Let x be real, V(x)=1-cos(x), Q(x)=x^2/2, and E(x)=Q(x)-V(x). Then 0<=E(x)<=x^4/24 for every real x. For x!=0 the error relative to the quadratic approximation obeys 0<=E(x)/Q(x)<=x^2/12, so for every declared epsilon>0 the symmetric domain |x|<=sqrt(12*epsilon) is sufficient to guarantee E(x)/Q(x)<=epsilon. At the cosine barrier x=pi this relative error is exactly 1-4/pi^2>0.59, so the barrier is not a universal small-oscillation accuracy boundary. Separately, let P and delta be real, let omega be real and nonzero, and define phi(t)=P*cos(omega*t+delta). Its mean square over any full period T=2*pi/|omega| is P^2/2 and its RMS amplitude is |P|/sqrt(2). Hence if A_RMS>=0 denotes RMS amplitude then the mean square is A_RMS^2 and the harmonic peak is sqrt(2)*A_RMS, whereas if A_peak>=0 denotes peak amplitude then the mean square is A_peak^2/2. These conventions are distinct from C-OSC-001's separately declared one-mode Fock-coordinate intensity S=q_0^2 unless an explicit map is supplied. These exact approximation and cycle-average identities derive no material amplitude, quantum state, multimode composition, density of states, topological winding, transition probability, rate, reaction, or substrate realization. +- `C-DOS-001` — Let d and b be separately supplied positive integers, let V and c be positive reals, let omega_0 be a nonnegative real, and use the declared isotropic continuum phase-space measure b*V*d^d k/(2*pi)^d with dispersion omega(k)=sqrt(omega_0^2+c^2*k^2). Write S_(d-1)=2*pi^(d/2)/Gamma(d/2). On the open band omega>omega_0 the density of states is g_(d,b)(omega)=b*V*S_(d-1)*omega*(omega^2-omega_0^2)^((d-2)/2) /((2*pi)^d*c^d), and it is zero below the gap. Its integral from the threshold to omega(K) is exactly N_(d,b)(K)=b*V*S_(d-1)*K^d/(d*(2*pi)^d), independent of omega_0. Therefore a separately supplied positive continuum target N has the unique positive matching cutoff K=(N*d*(2*pi)^d/(b*V*S_(d-1)))^(1/d). In d=1 the open-band density has an integrable threshold singularity, in d=2 it has a finite threshold limit, and in d>=3 it tends to zero. For d=3 the per-branch density is V*omega*sqrt(omega^2-omega_0^2)/(2*pi^2*c^3); choosing b=3 and separately supplying N=3*V/a^3 gives the conditional corollary K=(6*pi^2)^(1/3)/a. The continuum integral is not an exact finite periodic lattice-point count or dynamical-matrix rank, the matched cutoff is not a derived microscopic Brillouin-zone boundary, and none of these identities derives a cell complex, branch degeneracy, participation fraction, coupling, state, channel, rate, material, or substrate realization. +- `C-QFL-001` — Let hbar, V, kappa, c, omega_0, and K be positive reals and let b be a separately supplied positive integer. Declare b independent scalar harmonic branches in three spatial dimensions, each with dispersion omega(k)=sqrt(omega_0^2+c^2*k^2), effective oscillator mass V*kappa/c^2 per wavevector, product ground state, and C-DOS-001's continuum measure V*d^3k/(2*pi)^3. One mode then has coordinate variance hbar*c^2/(2*V*kappa*omega(k)). Integrating every declared branch over 0<=|k|<=K gives exactly Sigma=b*hbar/(8*pi^2*kappa)*(K*sqrt(omega_0^2+c^2*K^2) -omega_0^2*asinh(c*K/omega_0)/c). With ell=c/omega_0, X=K*ell, beta^2=hbar*c/(kappa*ell^2), and J(X)=(X*sqrt(1+X^2)-asinh(X))/2, the same result is Sigma=b*beta^2*J(X)/(4*pi^2). J is the integral from zero to X of x^2/sqrt(1+x^2), is strictly increasing, and obeys J(X)0 with J(X)/(X^2/2)->1 as X->infinity. At fixed positive omega_0 the small-cutoff leading term is b*hbar*c^2*K^3/(12*pi^2*kappa*omega_0); the continuous zero-gap limit is b*hbar*c*K^2/(8*pi^2*kappa); and dSigma/dK is the positive cutoff shell b*hbar*c^2*K^2/(4*pi^2*kappa*sqrt(omega_0^2+c^2*K^2)). Separately, for any fixed nonempty finite sequence of nonnegative mode variances v_i, M=len(v), Sigma_set=sum_i v_i, and v_bar=Sigma_set/M obey M*v_bar=Sigma_set. This fixed-set identity does not imply that changing the admitted sequence leaves Sigma_set fixed: adding a zero term changes M alone, while adding a positive term changes both M and Sigma_set. These identities derive no quantization of the accepted classical medium, three-dimensional lift, vacuum preparation, beta value, stiffness, branch count, microscopic cutoff, granularity, material variance, participating-mode set, growth law, channel, probability, rate, or substrate realization. +- `C-VOP-001` — On C-OSC-001's standard one-mode bosonic Fock space with orthonormal number basis |n>, let alpha be a separately declared complex number and let S=conjugate(alpha)*alpha. The norm-convergent vector |alpha>=exp(-S/2)*sum_{n=0}^infinity alpha^n/sqrt(n!)*|n> is normalized, obeys a|alpha>=alpha|alpha>, and has exact overlap =exp(-(S_alpha+S_beta)/2+conjugate(alpha)*beta). A measurement of the declared number operator on |alpha> has Born probability p_alpha(n)=exp(-S)*S^n/n!, with mean and variance S. At alpha=0 only the vacuum has nonzero probability; for alpha!=0 every nonnegative occupation has positive probability. C-CMB-003 therefore supplies the complete mode and tail corollaries: noninteger positive S has mode floor(S), while positive integer S has both adjacent modes S-1 and S. In the standard infinite Weyl representation, D(alpha)=exp(alpha*a_dagger-conjugate(alpha)*a) maps |0> to |alpha>; for real lambda and alpha=i*lambda its generator is i*lambda*(a+a_dagger). This conditional identification is not justified by treating a finite truncated commutator as globally central. The number law is a probability only for the declared coherent state and number measurement. It does not quantize or prepare the accepted classical medium, identify S with a classical peak, RMS phase, material variance, or supplied energy band, change the minima of a cosine potential, derive a Huang-Rhys or Franck-Condon material Hamiltonian, or supply an interaction, transition, branching, reaction, channel, physical occurrence probability, rate, or substrate realization. +- `C-MKV-001` — Let N_t be a separately declared continuous-time Markov chain on the nonnegative integers, let S>0 be a dimensionless stationary mean, and let r>0 have inverse-time units. For birth rates lambda_n=r*S and death rates mu_n=r*n, with mu_0=0, the exact generator is Lf(n)=r*S*(f(n+1)-f(n))+r*n*(f(n-1)-f(n)). The normalized factorial-one mass pi_n=exp(-S)*S^n/n! from C-CMB-003 is reversible because pi_n*lambda_n=pi_(n+1)*mu_(n+1). The identity-function local drift is r*(S-n). For a separately declared initial law with finite mean m0, the mean is S+(m0-S)*exp(-r*t). If the initial probability generating function is G0, then G(z,t)=G0(1+(z-1)*exp(-r*t))*exp(S*(z-1)*(1-exp(-r*t))). For deterministic initial state n0, the transition law is the independent sum of Binomial(n0,exp(-r*t)) and Poisson(S*(1-exp(-r*t))). The stationary mass alone does not select this generator: detailed balance fixes only lambda_n/mu_(n+1)=S/(n+1), and the distinct positive rates lambda_n=r*S/(n+1), mu_n=r for n>=1 with mu_0=0 share the same stationary mass while having different holding rates, drift, and transients. A positive local drift does not imply monotone sample paths because death jumps remain possible at every positive state. The state space, generator, initial law, S, and r are independent model declarations. Neither a static factorial-one mass nor a coherent-state number measurement, continuum mode count, vacuum variance, or accepted medium supplies them. No material state preparation, granularity map, participation law, physical growth, open or rescued channel, transition matrix element, branching fraction, isotope effect, reaction, rate, or substrate realization follows. +- `C-BRN-002` — Let N>0 be a continuous population parameter, let rho>0 be a declared dimensionless ratio of two common-dimension positive rate normalizations, and let w(N)>0 be a differentiable dimensionless weight. For the comparison fraction B_c(N)=rho/(N*w(N)+rho), the exact total derivative is B_c'(N)=-rho*(w(N)+N*w'(N))/(N*w(N)+rho)^2. Consequently B_c is locally decreasing, stationary, or increasing exactly as w+N*w' is positive, zero, or negative. A constant positive weight recovers C-BRN-001's strictly decreasing specialization. Positive weight alone is insufficient: w=N^(-1/2), w=N^(-1), and w=N^(-2) realize the three respective verdicts. The theorem concerns a declared differentiable continuation. For integer-only counts, discrete monotonicity instead follows from adjacent values of N*w(N); no derivative is silently imported. The weight law, common rate dimensions, exhaustive channel set, and physical meaning of the inputs remain separate premises. No material population law, state preparation, interaction, isotope map, reaction, branching observable, rate, yield, heat, or substrate realization follows. +- `C-QBL-004` — Declare on flat 3+1 spacetime with signature (+---) a dimensionless smooth complex scalar Psi, rho=conjugate(Psi)*Psi, and action density partial_mu(conjugate(Psi))*partial^mu(Psi)-U(rho), where U(rho)=1-cos(sqrt(rho)) is defined by its analytic rho-series at zero. The Euler-Lagrange equation is box Psi+(dU/drho)*Psi=0. For the real nonnegative stationary radial ansatz Psi(t,r)=f(r)*exp(-i*omega*t), f''+2*f'/r=sin(f)/2-omega^2*f, with f'(0)=0 and f tending to zero. A localized tail requires 00. Thus the local identity neither states that a nonzero shallow one-dimensional well has no bound state nor makes small absolute overlap and binding contradictory. Independently, C-OVL-002's exact Poschl-Teller ground eigenvalue is negative for every positive well depth. Replacing c by a nonlinear or independent multiplier, rescaling its lambda, or deforming the potential changes the relation. These statements derive no stability, physical condensate or Yukawa interaction, generation, mass hierarchy, mixing, multisoliton solution, Standard-Model map, or substrate realization. +- `C-MIX-003` — Let R_a be finite real matrices with a common row dimension N, let theta_a be real scalars, and set Y_a=exp(i*theta_a)*R_a for a=1,2. Each left Gram is H_a=Y_a*Y_a^dagger=R_a*R_a^T and each right Gram is Y_a^dagger*Y_a=R_a^T*R_a; both are real symmetric positive semidefinite and independent of the phases. Real orthogonal matrices O_a may be chosen to diagonalize the left Grams, so V=O_1^T*O_2 is real orthogonal and every quartet V_ij*V_kl*conjugate(V_il)*conjugate(V_kj) has zero imaginary part. The commutator [H_1,H_2] is real antisymmetric, every odd-power trace vanishes, and its determinant vanishes when N is odd. Repeated Gram eigenvalues retain enlarged unitary basis freedom: arbitrary complex bases inside degenerate subspaces can display nonzero coordinate quartets even though a real representative and the commutator null identities remain. The hypothesis is the globally phased real matrix form itself; a scalar field or condensate label does not enforce real coupling matrices, real modes, or a spatially constant phase. These statements establish no Yukawa interaction, CKM matrix, physical CP operation or violation, condensate ontology, generation count, observed phase, Standard-Model map, or substrate realization. +- `C-QBL-006` — Conditional on C-QBL-001's equal positive profiles f_1=A*sech(kappa*(x+d/2)) and f_2=A*sech(kappa*(x-d/2)) with A,kappa,d>0, separately declare the complex trial superposition Phi=f_1+exp(i*delta)*f_2 and the whole-line functional E[Phi]=integral dx*[|Phi_x|^2+kappa^2*|Phi|^2-|Phi|^4/24]. With s=kappa*d and c=cos(delta), define J31(s)=2*(sinh(s)*cosh(s)-s)/sinh(s)^3 and J22(s)=4*(s*cosh(s)-sinh(s))/sinh(s)^3. Exact expansion, isolated-profile subtraction, and the profile equation give I31=A^4*J31/kappa, I22=A^4*J22/kappa, and E_int=E[Phi]-E[f_1]-E[f_2]=-c*I31/6-(1+2*c^2)*I22/12. Thus the finite interaction contains phase-independent, linear-cosine, and cosine-squared terms and is not a pure cosine. At c=0 it is exactly -I22/12. The exact tail limits are J31~4*exp(-s) and J22~16*(s-1)*exp(-2*s), so for fixed nonzero c the leading interaction is -2*A^4*c*exp(-s)/(3*kappa), whereas the perpendicular case has the doubled exponential rate and linear prefactor. This is the energy of a two-profile trial superposition. It establishes no common nonlinear two-soliton or multi-profile solution, merger or persistence dynamics, equilibrium separation, spectral orbital or nonlinear stability, selected object count, physical condensate, generation, Standard-Model map, or substrate realization. +- `C-PHS-001` — Let N>=2 scalar phases theta_1,...,theta_N be points on one circle and let W=max_{i=1, let psi be a complex L2(R) mode normalized by integral_R |psi|^2 dx=1, and let phi be a bounded real continuous multiplier that vanishes at spatial infinity. For each separation parameter t, let R_1(t),...,R_n(t) be real centers whose minimum pairwise distance tends to infinity, define psi_a(x)=psi(x-R_a(t)) and phi_c(x)=phi(x-R_c(t)), choose fixed real phases theta_c, and set Y_ab(t)=sum_c exp(i*theta_c)*integral_R conjugate(psi_a)*psi_b*phi_c dx. If the common matched self-overlap alpha=integral_R |psi|^2*phi dx is positive, then Y(t) tends in every finite matrix norm to alpha*diag(exp(i*theta_1),...,exp(i*theta_n)). Consequently every singular value tends to alpha. More sharply, whenever epsilon(t)=||Y(t)-alpha*diag(exp(i*theta_c))||_2 is less than alpha, every singular value lies in [alpha-epsilon,alpha+epsilon] and the spectral condition number is at most (alpha+epsilon)/(alpha-epsilon). The limit follows because matched self-overlaps are translation invariant, displaced density-multiplier convolutions vanish, and off-diagonal translated L2 correlations vanish. Unequal profiles, amplitudes, widths, nonunit weights, nonseparating centers, or a nondecaying multiplier change the conclusion. This theorem establishes no Yukawa interaction, physical mass or hierarchy, field species, role, generation, selected center or count, Standard-Model map, or substrate realization. +- `C-PHS-002` — For N>=2 real scalar phases theta_1,...,theta_N, define the equal-weight complete-graph surrogate S=sum_{a=-N/2 with equality exactly when Z=0. A regular N-gon attains the minimum for every N>=2. For N=2 the relative phase at the minimum is pi, and for N=3 the zero-resultant unit phasors form the regular Z3 configuration up to global phase and permutation. For N=4 there is a continuous antipodal-pair family (0,pi,beta,beta+pi) of global minima; the square beta=pi/2 has every pairwise cosine nonpositive and worst cosine zero, while a generic member has a positive pair. Thus one numerically selected positive-pair four-phase minimum is not universal. The identity concerns a static equal-weight complete scalar-phase surrogate. It establishes no physical interaction energy, phase dynamics or relaxation, equilibrium, merger, stability, CP operation or violation, occupied condensate or generation count, multiplicity ratio, Standard-Model map, or substrate realization. +- `C-MIX-004` — Let a nonempty finite family Y_a of nonempty same-size square complex matrices and a same-length family of complex scalar weights v_a be separately supplied, and define M=sum_a v_a*Y_a. Let U_L and U_R be separately supplied unitary bases for which D=U_L^dagger*M*U_R is diagonal, and define Gamma_a=U_L^dagger*Y_a*U_R. Then exactly D=sum_a v_a*Gamma_a. A diagonal weighted sum does not force the individual Gamma_a to be diagonal: their off-diagonal entries can cancel, and a zero-weight matrix can remain off diagonal while absent from M. Conditional on a separately declared multi-scalar interaction that identifies this complete Gamma_a family as the neutral-scalar couplings in the fixed mass basis, all such couplings are flavor diagonal exactly when every Gamma_a is diagonal. If Y_a=c_a*Y for separately supplied complex c_a and C=sum_a v_a*c_a is nonzero, common alignment is sufficient and Gamma_a=c_a*D/C is diagonal. If C=0, then M=0 and alignment alone does not force individual diagonality in an arbitrary degenerate mass basis. For a complex-symmetric M with Takagi form M=U*D*U^T, the corresponding right basis is conjugate(U), so the mass-basis transform is U^dagger*Y_a*conjugate(U), not U^dagger*Y_a*U. Repeated singular values and null blocks retain the basis freedoms of C-MIX-001 and require a separate degeneracy audit. These exact finite-matrix identities derive no Yukawa interaction, scalar field content, vacuum expectation values, physical mass or generation map, localization geometry, flavor-changing rate or bound, natural flavor conservation, Standard-Model identity, observation, or substrate realization. +- `C-VEC-002` — In the Hermitian Pauli-half basis T_0=I_2/2 and T_a=sigma_a/2 for a=1,2,3, every real symmetric Ad(U(2))-invariant bilinear form Q on u(2) has component Gram matrix diag(beta,alpha,alpha,alpha), equivalently Q(X,Y)=2*alpha*Tr(X*Y)+(beta-alpha)*Tr(X)*Tr(Y). It is positive definite exactly when alpha and beta are positive. The fundamental single-trace specialization and singlet-triplet degeneracy occur exactly at alpha=beta; U(2) invariance alone does not force that equality. Separately, for positive exact m and g, a supplied real current B, and the declared algebraic action L=m^2*w^2/2+g*w*B, stationary elimination gives w=-g*B/m^2 and L_eff=-g^2*B^2/(2*m^2). Thus the convention L6=-lambda_A^2*B^2 has lambda_A=g/(sqrt(2)*m), while C-BPS-001's convention L6=-lambda_BPS^2*pi^4*B^2 has lambda_BPS=lambda_A/pi^2. Restoring a vector kinetic differential operator replaces m^2 by a nonlocal kernel, so the displayed local term is only a leading low-momentum specialization under separately supplied power counting. This exact classification and conditional elimination derive no HLS field content, physical omega rho pion or baryon current, WZW or anomaly coefficient, N_c, universality, KSRF relation, mass, coupling value, decay scale, medium response, or substrate realization. +- `C-GSK-001` — Let B be a positive integer, let I be a separately supplied positive rational-map angular integral, let c6 and c0 be separately supplied nonnegative dimensionless coefficients, and let f(r) be a real radial profile for r>0. Declare E=4*pi*integral_0^infinity L dr with L=r^2*f'^2+2*B*sin(f)^2*(1+f'^2)+I*sin(f)^4/r^2 +c6*I*sin(f)^4*f'^2/r^2+c0*r^2*(1-cos(f)). Every term is nonnegative. One half of d/dr(dL/df')-dL/df is exactly K*f''+(2*r-2*c6*I*sin(f)^4/r^3)*f' +B*sin(2*f)*(f'^2-1) +c6*I*sin(f)^2*sin(2*f)*f'^2/r^2 -I*sin(f)^2*sin(2*f)/r^2-c0*r^2*sin(f)/2, where K=r^2+2*B*sin(f)^2+c6*I*sin(f)^4/r^2. For f_s(r)=f(r/s), E(s)=s*E2+s^-1*E4+s^-3*E6+s^3*E0, so a stationary scale satisfies E2-E4-3*E6+3*E0=0. At c6=c0=0 the density and equation reduce exactly to C-RPROF-001. Linearization gives origin exponent (sqrt(1+8*B)-1)/2; when c0>0 the decaying tail is proportional to r^-1/2*K_nu(sqrt(c0/2)*r), with nu=sqrt(1/4+2*B), while c0=0 recovers the massless power tail. Conditional on separately supplied positive lambda_BPS, mu, e, and F in the accepted conventions, the reduced coefficients are c6=lambda_BPS^2*e^4*F^2/8 and c0=32*mu^2/(e^2*F^4); equivalently c6=lambda_A^2*e^4*F^2/(8*pi^4) when lambda_A=pi^2*lambda_BPS. This exact conditional reduced model derives no coefficient value, physical action, rational map, existence or uniqueness theorem, minimizer, full three-dimensional field, particle, nucleus, binding energy, observation, or substrate mechanism. +- `C-GSK-002` — Conditional on C-GSK-001 with independently supplied (c6,c0)=(1/2,1/4) and angular inputs (B,I)=(1,1), (2,pi+8/3), and (4,20.6496264884189), adaptive double-precision collocation with regular-origin and massive-tail Robin data finds one monotone stationary branch for each input on [10^-4,20]. With solver tolerance 10^-6, 401 initial points, eight coefficient-continuation steps, at most 200000 adaptive nodes, and 8001 independent output quadrature points, the dimensionless energy coefficients E/(12*pi^2*B) are respectively 1.4326169552, 2.7988849886, and 5.1973886988, and 3*pi^2*(2*b_2-b_4)=11.85481448. The maximum reported collocation RMS residual is below 1.1e-6, the maximum endpoint residual below 2e-11, and the maximum relative Derrick residual below 2e-6. Individual coefficients and the signed difference stabilize as the outer radius changes from 14 to 20 to 26; the B=2 residual decreases from about 1.99e-6 to 4.94e-7 as tolerance tightens from 2e-6 to 5e-7; and the B=4 coefficient changes by less than 3e-11 as the inner cutoff changes through 2e-4, 1e-4, and 5e-5. An independent vacuum-complement DOP853 shooting route with Simpson quadrature at outer radius 14 agrees within 3e-7 per coefficient and 1e-5 in the signed difference. Replacing I by B^2 or setting c6=0 moves the signed difference to about 8.12019 or 8.85035 respectively. This is resolution-bounded evidence for the declared truncated-domain stationary branches only; it proves no half-line existence, uniqueness, global or local minimality, full three-dimensional solution, physical coefficient, baryon, nucleus, binding prediction, observation, or substrate mechanism. +- `C-VAR-002` — Let X be a nonempty set, let n be a positive integer, and let E_1,...,E_n be real-valued functionals on X whose component infima m_i=inf_{x in X} E_i(x) are finite real numbers. Then the joint functional has a finite infimum M=inf_{x in X} sum_i E_i(x) and M>=sum_i m_i. Equality holds if and only if, for every epsilon>0, there exists one common x_epsilon in X such that 0<=E_i(x_epsilon)-m_i0; global domain and endpoint coverage remain separate requirements. Under a positive pointwise conformal factor Omega, the massless density is invariant for g->Omega^2*g and e->Omega^2*e, while for m>0 the mass term changes by -(Omega^2-1)*e*(m*c0)^2/2 and is nonzero at each point where positive Omega differs from one. These are worldline action identities, not spacetime field-theory conformal symmetry. +- `C-LOR-001` — In mostly-plus flat spacetime, the future-directed unit timelike-vector orbit has exact induced metric diag(1,sinh(eta)^2) on H^2 in 2+1 dimensions and diag(1,sinh(eta)^2,sinh(eta)^2*sin(theta)^2) on H^3 in 3+1 dimensions, obtained by pulling back the ambient metric along the standard hyperboloid embeddings. This claim concerns one unit timelike vector only; it is not a timelike-plane or tube-worldsheet orbit, an invariant probability measure, or a physical ensemble. +- `C-LOR-002` — In the displayed mostly-plus matrix bases, the exact Lorentz-algebra stabilizer of the standard 2+1-dimensional massive momentum is generated by J12 and the displayed null stabilizer by K2+J12. In 3+1 dimensions, the displayed massive generators close as so(3) with [J1,J2]=-J3, [J2,J3]=-J1, and [J3,J1]=-J2. The displayed null generators T1=K1+J2, T2=K2-J1, and J3 fix (1,0,0,1) and obey [T1,T2]=0, [J3,T1]=-T2, and [J3,T2]=T1, the stated iso(2) convention. This is finite-dimensional matrix Lie algebra only and proves no global topology, unitary representation, spin, anyon, continuous-spin, helicity, photon, graviton, or interaction statement. +- `C-IGR-001` — Conditional on C-GRV-001's independent additive inverse-coupling baseline ledger, separately declare a positive self-adjoint boundaryless four-dimensional Euclidean real-scalar Laplace-type operator D_E=-nabla_E^2+xi*R_E+m2 with spacetime-constant m2>=0, adequate infrared convergence or reference subtraction, Gamma_E=(1/2)*ln(det(D_E)), heat-kernel prefactor (4*pi)^-2, and Euclidean Einstein-Hilbert matching factor 16*pi. For a positive sharp proper-time cutoff Lambda with tau0=Lambda^-2 and z=m2/Lambda^2, the exact local curvature and vacuum coefficient integrals are I2=Lambda^2*(exp(-z)-z*E1(z)) and I3=Lambda^4*exp(-z)/2-m2*I2/2, continuously extended at m2=0. They obey dI3/dm2=-I2, have massless values Lambda^2 and Lambda^4/2, and decay in the large-mass limit. For positive integer N and exact xi, the declared conditional additive coefficients are Delta(1/G)=N*(1-6*xi)*I2/(12*pi) and Delta(rho)=-(N/2)*(4*pi)^-2*I3. These are exact local coefficient families, not a full or nonlocal determinant, varying-mass theorem, selected regulator, cutoff ontology, total Newton constant, attractive- gravity result, sourced geometry, or empirical prediction. +- `C-IGR-002` — Under C-IGR-001's declared operator, determinant, local-coefficient, matching, constant-mass, and baseline assumptions, replace the sharp cutoff by the explicitly declared smooth proper-time weight exp(-1/(Lambda^2*tau)). For positive Lambda, m2>=0, and z=m2/Lambda^2, the exact coefficient integrals are I2=2*Lambda^2*sqrt(z)*K1(2*sqrt(z)) and I3=2*Lambda^4*z*K2(2*sqrt(z)), continuously extended at m2=0. They obey dI3/dm2=-I2, have massless values Lambda^2 and Lambda^4, and decay for large mass. The conditional additive coefficients are the same typed compositions Delta(1/G)=N*(1-6*xi)*I2/(12*pi) and Delta(rho)=-(N/2)*(4*pi)^-2*I3. The exact massless I3 value is twice C-IGR-001's sharp value at the same Lambda, so this claim exposes rather than removes regulator dependence and establishes no physical regulator selection or total gravitational normalization. +- `C-IGR-003` — Under C-IGR-001's declared operator, determinant, local-coefficient, matching, constant-mass, and baseline assumptions, declare instead the power-subtracted sharp-cutoff finite-part prescription at a positive scale mu. For constant m2>=0 its exact continuously extended coefficient family is I2=m2*(log(m2/mu^2)+EulerGamma-1) and I3=-(m2^2/2)*(log(m2/mu^2)+EulerGamma-3/2). These are the finite limits after the P230-frozen power and logarithmic subtractions, obey dI3/dm2=-I2, mu*dI2/dmu=-2*m2, and mu*dI3/dmu=m2^2, and vanish continuously at m2=0. The conditional additive shifts use the same typed factors as C-IGR-001. At unit scale and cutoff with m2=0, the sharp, smooth, and power-subtracted I3 values are exactly 1/2, 1, and 0; moreover the finite-part I2 can change sign with m2/mu^2. Therefore P230 derives exact scheme and scale dependence, not a regulator-free coefficient, selected subtraction, finite counterterm, total Newton constant, or physical normalization. diff --git a/src/substrate_framework/scalar_one_loop_mass.py b/src/substrate_framework/scalar_one_loop_mass.py index dcf08151..1fc72453 100644 --- a/src/substrate_framework/scalar_one_loop_mass.py +++ b/src/substrate_framework/scalar_one_loop_mass.py @@ -1,15 +1,20 @@ -"""Exact-mass one-loop proper-time coefficients for preregistered regulators. +"""Exact constant-mass one-loop coefficients for declared regulators. -This is the Route 1 (induced / Sakharov gravity) rung that follows -``covariant_sine_gordon_action``. The landed action rung hands the -fluctuation operator +Claims ``C-IGR-001`` through ``C-IGR-003`` concern the local coefficient +families of a declared positive self-adjoint, boundaryless, +four-dimensional Euclidean real-scalar operator D_E = -nabla_E**2 + xi*R_E + m**2, m**2 = V''(phi_bg), -to the determinant. This module evaluates the two leading proper-time -integral classes of ``Gamma_E = (1/2)*ln(det(D_E))`` with the mass retained -exactly --- no expansion of ``exp(-tau*m**2)`` inside the regulated integral ---- for three preregistered regulators: +only when ``m**2`` is a spacetime-constant nonnegative exact input. The +displayed ``V''`` equality is therefore an optional constant-vacuum +composition, not permission to factor a varying background endomorphism out +of the heat kernel. The module evaluates the two leading proper-time integral +classes of ``Gamma_E = (1/2)*ln(det(D_E))`` with the constant mass retained +exactly---no expansion of ``exp(-tau*m**2)`` inside the regulated integral--- +for three preregistered prescriptions. These are exact coefficient +integrals within the displayed heat-kernel organization, not an exact +truncation of the full determinant or its nonlocal terms: ``SHARP_PROPER_TIME_REGULATOR`` (reused from ``scalar_induced_newton``) A sharp lower proper-time cutoff ``tau_0 = Lambda**-2``. Writing @@ -32,20 +37,20 @@ I_3 = 2*Lambda**4*z*BesselK_2(2*sqrt(z)). ``ZETA_POWER_SUBTRACTED_REGULATOR`` - The Mellin finite-part scheme in which the power divergences are - subtracted; it requires a declared renormalization scale ``mu`` (the - cutoff argument is rejected) and yields the pure logarithmic running + A declared power-subtracted cutoff finite-part prescription, compatible + with the corresponding Mellin/zeta family. It requires a declared scale + ``mu`` (the cutoff argument is rejected) and yields I_2 = m**2*(ln(m**2/mu**2) + EulerGamma - 1), I_3 = -(m**4/2)*(ln(m**2/mu**2) + EulerGamma - 3/2). -Authority note: ``scalar_induced_newton`` and -``covariant_sine_gordon_action`` are landed conditional (unpromoted) -prior-work APIs -- the PR #14 and PR #25 harvests promoted no claims -- and -the accepted ``C-GRV-001`` supplies only the conditional dimensional and -additive-baseline ledger, leaving the coefficient, field content, and -regulator as premises. Every public symbol in this module is likewise -conditional, unpromoted infrastructure linked to open goal #76. +Authority note: accepted ``C-GRV-001`` supplies only the conditional +dimensional and additive-baseline ledger. ``C-IGR-001..003`` independently +derive the coefficient families here while leaving the scalar field content, +determinant convention, regulator, cutoff or subtraction scale, and any +physical scale identification as explicit premises. The previously landed +``scalar_induced_newton`` and ``covariant_sine_gordon_action`` APIs promoted no +claims and are not used as scientific authority. Composition with that landed conditional scheme factor (one real scalar, determinant weight 1/2, heat-kernel prefactor ``(4*pi)**-2``, @@ -54,19 +59,18 @@ Delta(1/G) = N * coefficient_per_field(xi) * I_2(regulator), -where ``coefficient_per_field`` is taken from the landed conditional -``scalar_induced_newton.leading_scalar_newton_shift_coefficient`` API, so -the massless sharp limit reproduces that conditional API's +where ``coefficient_per_field`` is rederived in this module from the declared +determinant, heat-kernel, curvature, and Einstein-Hilbert matching factors. +The massless sharp limit also reproduces the earlier conditional API's ``s*Lambda**2 = N*(1-6*xi)*Lambda**2/(12*pi)`` exactly. The vacuum sector of the same mass-resummed expansion is Delta(rho_Gamma) = -(N/2)*(4*pi)**-2 * I_3(m**2) -per scheme. The ``-m**2`` entry of the landed conditional -``scalar_heat_kernel_a2`` weights belongs to the *unresummed* organization, -in which the exponential is expanded and the mass survives only as that -coefficient; applying it on top of the resummed ``I_3`` double-counts the -mass. The two organizations agree to first order through the exact +per scheme. A separate ``-m**2`` coefficient belongs to the *unresummed* +organization, in which the exponential is expanded; applying it on top of the +resummed ``I_3`` double-counts the mass. The two organizations agree to first +order through the exact derivative identity ``d I_3/d m**2 = -I_2`` (which holds for all three schemes and is tested), i.e. ``I_3(m**2) = I_3(0) - m**2*I_2(0) + O(m**4)``. @@ -85,11 +89,12 @@ ``lim_{tau_0 -> 0} [I(tau_0) - power terms - log terms]`` evaluated symbolically. -This module does NOT identify the cutoff with any substrate scale, choose a -renormalization condition among the three schemes, derive a total Newton -constant (the accepted C-GRV-001 additive baseline remains independent), -evaluate the tau**-1 class (curvature-squared and m**2*R structures), or -confront any empirical comparator. Those are later rungs of issue #76. +This module does NOT handle a varying effective mass, identify the cutoff with +any substrate scale, choose a renormalization condition among the three +schemes, derive a total Newton constant (the accepted C-GRV-001 additive +baseline remains independent), evaluate the tau**-1 class +(curvature-squared and m**2*R structures), or confront an empirical +comparator. Those are later rungs of issue #76. Scheme differences are the deliverable: at z = 1 the smooth regulator induces about 1.88 times the sharp curvature coefficient, and the power-subtracted scheme changes both the parametric structure and the sign @@ -113,7 +118,6 @@ from .exact_symbolic import exact_real as _exact_real from .exact_symbolic import positive_exact as _positive_exact from .scalar_induced_newton import SHARP_PROPER_TIME_REGULATOR -from .scalar_induced_newton import leading_scalar_newton_shift_coefficient # conditional landed API SMOOTH_PROPER_TIME_REGULATOR = "proper_time_smooth_essential" ZETA_POWER_SUBTRACTED_REGULATOR = "zeta_power_subtracted" @@ -125,6 +129,13 @@ ) _FOUR_DIMENSIONAL_HEAT_KERNEL_PREFACTOR = (4 * sp.pi) ** -2 +_REAL_SCALAR_DETERMINANT_WEIGHT = sp.Rational(1, 2) +_EINSTEIN_HILBERT_INVERSE_COUPLING_FACTOR = 16 * sp.pi +_INVERSE_NEWTON_SCHEME_FACTOR = sp.simplify( + _EINSTEIN_HILBERT_INVERSE_COUPLING_FACTOR + * _REAL_SCALAR_DETERMINANT_WEIGHT + * _FOUR_DIMENSIONAL_HEAT_KERNEL_PREFACTOR +) def _nonnegative_exact(value: Any, name: str) -> sp.Expr: @@ -248,9 +259,9 @@ def curvature_proper_time_integral( ) -> sp.Expr: """Return the exact tau**-2-class proper-time integral for one scheme. - This is the integral that multiplies the declared heat-kernel weights - ``((1/6 - xi)*R_E - m**2)`` in ``Gamma_E``; with the landed conditional - scheme factor it also multiplies the induced inverse-Newton shift. + For constant ``mass_squared`` this is the integral that multiplies the + declared curvature weight ``(1/6-xi)*R_E`` in ``Gamma_E``. The mass is + already retained in the exponential and is not an additional weight. """ resolved_cutoff, scale = _resolve_regulator(regulator, cutoff, renormalization_scale) @@ -286,7 +297,7 @@ def vacuum_proper_time_integral( @dataclass(frozen=True) class ExactMassNewtonShift: - """Exact-mass induced inverse-Newton shift data for one scheme.""" + """Conditional constant-mass inverse-Newton shift for one scheme.""" regulator: str field_count: sp.Expr @@ -314,12 +325,9 @@ def exact_mass_inverse_newton_shift( ) -> ExactMassNewtonShift: """Return ``Delta(1/G) = N * coefficient_per_field * I_2`` exactly. - ``coefficient_per_field`` is the per-field scheme factor - ``scheme_factor*(1/6 - xi)`` read from the landed conditional - ``scalar_induced_newton.leading_scalar_newton_shift_coefficient`` API - (its massless sharp regulator tag is used only to read that factor, - which is regulator-independent). The massless sharp limit reproduces - that module's ``N*(1-6*xi)*Lambda**2/(12*pi)`` exactly. + ``coefficient_per_field`` is rederived here as + ``16*pi*(1/2)*(4*pi)**-2*(1/6-xi)``. The massless sharp limit reproduces + ``N*(1-6*xi)*Lambda**2/(12*pi)`` exactly. ``curvature_weight_sign`` is the decidable sign of ``1/6 - xi`` only. ``value_sign`` is the decidable sign of the full returned value, or @@ -333,10 +341,9 @@ def exact_mass_inverse_newton_shift( count = _positive_integer(field_count, "field_count") xi = _exact_real(non_minimal_coupling, "non_minimal_coupling") - landed = leading_scalar_newton_shift_coefficient( - 1, xi, regulator=SHARP_PROPER_TIME_REGULATOR + coefficient_per_field = sp.simplify( + _INVERSE_NEWTON_SCHEME_FACTOR * (sp.Rational(1, 6) - xi) ) - coefficient_per_field = landed.coefficient_per_field if regulator == SHARP_PROPER_TIME_REGULATOR: proper_time_value = _sharp_curvature_integral(resolved_cutoff, mass) @@ -396,7 +403,7 @@ def exact_mass_inverse_newton_shift( @dataclass(frozen=True) class ExactMassVacuumShift: - """Exact-mass one-loop vacuum-sector data for one scheme.""" + """Conditional constant-mass vacuum-sector data for one scheme.""" regulator: str field_count: sp.Expr @@ -425,10 +432,9 @@ def exact_mass_vacuum_density_shift( action density in the mass-resummed organization: the trace integrand is ``exp(-tau*m**2)*[tau**-2 + tau**-1*(1/6-xi)*R_E + ...]``, so the vacuum sector is the tau**-3 class with the exponential retained. The - ``-m**2`` heat-kernel weight of the landed conditional - ``scalar_heat_kernel_a2`` is the first-order remnant of that same - exponential in the unresummed organization and must not be added again; - the exact bridge is the derivative identity ``d I_3/d m**2 = -I_2``. + A separate ``-m**2`` heat-kernel weight is the first-order remnant of that + same exponential in the unresummed organization and must not be added + again; the exact bridge is ``d I_3/d m**2 = -I_2``. The sector is exhibited rather than omitted whenever the curvature-sector shift is quoted. """ diff --git a/tests/test_scalar_one_loop_mass.py b/tests/test_scalar_one_loop_mass.py index 69bef40d..8103fc20 100644 --- a/tests/test_scalar_one_loop_mass.py +++ b/tests/test_scalar_one_loop_mass.py @@ -21,7 +21,6 @@ import substrate_framework as framework from substrate_framework.scalar_induced_newton import ( SHARP_PROPER_TIME_REGULATOR, - leading_scalar_newton_shift_coefficient, ) from substrate_framework.scalar_one_loop_mass import ( KNOWN_ONE_LOOP_REGULATORS, @@ -164,7 +163,7 @@ def test_zeta_closed_forms_are_the_power_subtraction_limits() -> None: assert sp.simplify(zeta3 + (m2**2 / 2) * (sp.log(m2 / mu**2) + sp.EulerGamma - sp.Rational(3, 2))) == 0 -def test_massless_sharp_limit_reproduces_the_landed_conditional_shift() -> None: +def test_massless_sharp_limit_reproduces_the_independent_leading_shift() -> None: N, xi = sp.Integer(3), sp.Rational(1, 12) shift = exact_mass_inverse_newton_shift( N, @@ -173,10 +172,16 @@ def test_massless_sharp_limit_reproduces_the_landed_conditional_shift() -> None: cutoff=Lam, mass_squared=0, ) - landed = leading_scalar_newton_shift_coefficient( - N, xi, regulator=SHARP_PROPER_TIME_REGULATOR + independently_derived_coefficient = sp.simplify( + N + * (16 * sp.pi) + * sp.Rational(1, 2) + * (4 * sp.pi) ** -2 + * (sp.Rational(1, 6) - xi) ) - assert sp.simplify(shift.value - landed.coefficient * Lam**2) == 0 + assert sp.simplify( + shift.value - independently_derived_coefficient * Lam**2 + ) == 0 assert sp.simplify(shift.finite_mass_factor - 1) == 0 smooth = exact_mass_inverse_newton_shift( @@ -187,7 +192,9 @@ def test_massless_sharp_limit_reproduces_the_landed_conditional_shift() -> None: mass_squared=0, ) # The smooth weight also integrates to exactly Lambda^2 when massless. - assert sp.simplify(smooth.value - landed.coefficient * Lam**2) == 0 + assert sp.simplify( + smooth.value - independently_derived_coefficient * Lam**2 + ) == 0 zeta = exact_mass_inverse_newton_shift( N, @@ -227,7 +234,7 @@ def test_scheme_ledger_reports_exact_contrasts() -> None: assert abs(factor - sp.Float("0.1484955068", 10)) < sp.Float("1e-9") -def test_exact_mass_shift_composes_the_landed_conditional_scheme_factor() -> None: +def test_exact_mass_shift_rederives_the_conditional_scheme_factor() -> None: N, xi = sp.Integer(2), sp.Rational(1, 12) shift = exact_mass_inverse_newton_shift( N, @@ -236,11 +243,51 @@ def test_exact_mass_shift_composes_the_landed_conditional_scheme_factor() -> Non cutoff=Lam, mass_squared=m2, ) - landed = leading_scalar_newton_shift_coefficient( - 1, xi, regulator=SHARP_PROPER_TIME_REGULATOR + independently_derived_per_field = sp.simplify( + (16 * sp.pi) + * sp.Rational(1, 2) + * (4 * sp.pi) ** -2 + * (sp.Rational(1, 6) - xi) + ) + assert sp.simplify( + shift.coefficient_per_field - independently_derived_per_field + ) == 0 + assert sp.simplify( + shift.value + - N * independently_derived_per_field * shift.proper_time_value + ) == 0 + + +def test_cutoff_families_have_exact_large_mass_and_massless_limits() -> None: + sharp2 = sp.exp(-z) - z * sp.expint(1, z) + smooth2 = 2 * sp.sqrt(z) * sp.besselk(1, 2 * sp.sqrt(z)) + smooth3 = 2 * z * sp.besselk(2, 2 * sp.sqrt(z)) + assert sp.limit(sharp2, z, 0, "+") == 1 + assert sp.limit(smooth2, z, 0, "+") == 1 + assert sp.limit(smooth3, z, 0, "+") == 1 + # SymPy leaves the E1 cancellation limit unevaluated. Use the defining + # integral instead: after t=1+u, t^-2<=1 for u>=0, so + # 0 0. + u = sp.Symbol("u", nonnegative=True) + assert sp.factor(1 - 1 / (1 + u) ** 2).is_nonnegative is True + assert sp.limit(sp.exp(-z) / z, z, sp.oo) == 0 + assert sp.limit(smooth2, z, sp.oo) == 0 + assert sp.limit(smooth3, z, sp.oo) == 0 + + +def test_power_subtracted_scale_derivatives_are_exact() -> None: + i2 = curvature_proper_time_integral( + ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=m2, + renormalization_scale=mu, + ) + i3 = vacuum_proper_time_integral( + ZETA_POWER_SUBTRACTED_REGULATOR, + mass_squared=m2, + renormalization_scale=mu, ) - assert sp.simplify(shift.coefficient_per_field - landed.coefficient_per_field) == 0 - assert sp.simplify(shift.value - N * landed.coefficient_per_field * shift.proper_time_value) == 0 + assert sp.simplify(mu * sp.diff(i2, mu) + 2 * m2) == 0 + assert sp.simplify(mu * sp.diff(i3, mu) - m2**2) == 0 def test_vacuum_shift_is_the_mass_resummed_determinant_integrand() -> None: @@ -431,6 +478,9 @@ def test_module_is_target_blind() -> None: source = inspect.getsource(scalar_one_loop_mass) for forbidden in ("6.674", "Planck", "M_pl", "6.708", "observed"): assert forbidden not in source + assert "spacetime-constant nonnegative" in source + assert "varying effective mass" in source + assert "leading_scalar_newton_shift_coefficient" not in source def test_known_regulators_are_the_three_preregistered_schemes() -> None: