node_181_structure.md

The graph at node [181]: complete structural accounting

This document lists every fact that holds for the minimal counterexample $G$ when the proof reaches node [181], the arm taken at every diamond on the way, the technique (register T01–T19) that produced each fact and the register property (A01–I06) it evaluated, and the exact typed data at the leaf. Nothing is projected, summarized, or dropped; the closure of [181] is the theorem in §6, whose hypotheses are exactly the facts listed here.

Sources: to_formalize/erdos_64_proof.tex (labels in backticks; node numbers in brackets), closure_proofs.md (Theorems 1.3–1.5, 3.1–3.4), and the register web/frontend/src/structural-survey/data.ts.

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1. The path from the root to [181]

Each row: node(s) → arm taken → fact retained on the branch state → technique → register rows evaluated.

Node(s)	Arm taken	Fact retained	Technique	Rows
[1]–[2]	yes	$G$ finite simple, $\delta(G)\ge3$, no cycle of length $2^j$ (def:counterexample)	—	A04, C03
[4]	—	$G$ is the lexicographically minimal counterexample: minimum $\lvert V\rvert$, then $\lvert E\rvert$, then lexicographic order	T02	E01
[5]–[7]	no Mersenne return	$R_e(G)\cap\mathrm{Mers}=\varnothing$ for every oriented edge $e$, $\mathrm{Mers}={2^k-1:k\ge2}$ (lem:return-equivalence)	T08	C02
[8]	—	every proper subgraph $H\subsetneq G$ has $\delta(H)\le2$ (lem:no-proper-core)	T02	E02, A07
[9]–[10]	—	every edge has an endpoint of degree $3$; $V_{\ge4}(G)$ is independent (lem:deletion-critical)	T03, T02	E03, A06
—	—	$G$ is bridgeless (lem:bridgeless, by contraction of a bridge)	T03, T02	B02
[11]	—	boundaried pieces $X\oplus_TY$ with boundary degree profile $\mathbf d_\partial$ (def:boundaried-gluing, lem:degree-profile-fibres)	T05	B05, B06
[12]	—	context universality: a target-complete identification agrees against every $T$-context; an identification valid only for the actual outside is target-defective (lem:context-universality)	T05	B07, E06
[13]	—	replacement: no $T$-boundaried $X'\preceq_TX$ with the same $\mathbf d_\partial$, no internal power-of-two cycle, internal degrees $\ge3$, strictly smaller (lem:replacement)	T02, T03	E05
[14]	—	hereditary target-uncompressibility: no proper boundaried piece admits a nontrivial target-complete compression (cor:uncompressible)	T02, T05	E05
[15]–[16]	no	$G$ contains an induced $P_{13}$ (cor:p13-exists, via the black box thm:p13free: $P_{13}$-free $+\ \delta\ge3\Rightarrow$ power-of-two cycle)	T18	I06, C08
[17]	—	$\mathcal P$: a maximum-cardinality family of vertex-disjoint induced $P_{13}$'s, $p_{13}=\lvert\mathcal P\rvert=\theta n$, chosen lexicographically first among maximum ones; $W=\bigcup V(P)$, $R=G-W$	T06, T16	C09, C10
[18]	—	$P_{13}$ label algebra: $399$ legal labels (sizes $13,60,122,122,63,17,2$), relations $C_s$, obstruction tensor $\Omega_2$ (lem:labels)	T17	D01, G02
[19]/[20]; [125]–[144]	no non-near-cubic surplus survives	near-cubic spine: $m=\tfrac32n+O(\sqrt n)$, $\sigma(G)=2m-3n=O(\sqrt n)$, $\lvert V_{\ge4}\rvert\le\sigma(G)$ (def:near-cubic-spine, prop:nonnear-cubic-sharp-overload-routing, thm:tokenized-surplus-accounting-closure)	T13, T14, T15	A02, A05, A14
[21]	—	finite constants: $c_\Omega=2.28922315244$, $c_{13}=118.108581006$; two-step obstruction enumeration $543958,432672,111286$ (lem:curv-enum, lem:p13-window-package)	T17	I05, A09, G02
[158]	yes	the joint window package of $\mathcal P$ is realized by the labelled skeleton class: $\ge2^{c_{13}p_{13}\log_2n}$ target-complete states assigned canonically to skeletons in $\mathcal G_{n,m}$ (def:window-realization-test)	T12	G01, G03, G06
[22]/[145]–[157]	the live-hot entropy comparison does not close; the cold machinery returns to [24] on its bounded arm	thm:cold-branch-quantitative-closure is stated conditional on absence of node [181] (clause (v) of def:surviving-cold-branch); on the [181] branch its outputs are not available as facts. What is retained is only the return at [24]	T12, T13	G03, H08
[24]	—	$\theta\le\theta_{\rm win}+o(1)$, $\theta_{\rm win}=1.5/c_{13}=0.0127002$; sharpened on the hot arm to $\theta\le\theta^\ast+o(1)$, $\theta^\ast=0.75/(c_{13}-39/4-15)=0.0080335$ (Theorem 1.5)	T12, T16	G09, I02
[25]–[27]	—	$\lvert R\rvert\ge(1-13\theta)n$; every component of $R$ is $P_{13}$-free, hence of diameter $\le11$ and $\le6142$ vertices, and has empty internal $3$-core: every $P_{13}$-free induced subgraph of $G$ has a vertex of degree $\le2$ (lem:remainder-empty-internal-3-core, black box)	T06, T18, T04	C10, C08, A07
[28]–[29]	—	$\defp(X)=\sum_v\max(0,3-d_X(v))$; $\defp(R)\le e(R,W)\le15p_{13}+o(n)$; exact split $e(R,W)+2e_\times(W)=15p_{13}+\sigma_W$; $(\defp(R)-\sigma_R)/\lvert R\rvert\le15\theta/(1-13\theta)+o(1)$, i.e. $\le\tau_{\rm win}=0.2281749$, sharpened to $\tau^\ast=0.13456$ (lem:stub-positive, Theorem 1.5)	T01, T15	A10, A11, H01
[30]	—	$W_2(C)\ge3\lvert V(C)\rvert-2\defp(C)$ per component; $W_2(R)\ge\omega_{\rm win}\lvert R\rvert-o(\lvert R\rvert)$, $\omega_{\rm win}=2.54365$ (high entropy $2.57407$) (lem:wedge-lower)	T01	A09, F01
[31]–[47]	no rank drop	full obstruction rank $r_\Omega(R)\ge W_2(R)-o(W_2)$; every rank-reducing dependence is target-defective, a proper compression (forbidden), a proper-support dependence (forbidden, lem:proper-smearing), or a whole-graph dependence that is target-defective, has a smaller closed representative, or is exact on labels (lem:no-silent-global-smearing); repair identity $s=p-2+2\beta_Z-\sigma_Z$ (lem:smearing-support-repair); separated identical wedges are context-universal or defective (lem:separated-testers)	T11, T10, T05	F01–F07, A12
[48]	—	forced obstruction cost $c_\Omega r_\Omega(R)\ge K_{\rm win}\lvert R\rvert-o(\lvert R\rvert)$, $K_{\rm win}=5.82298$ (high entropy $K=5.89263$) (cor:forced-curvature-cost)	T12	G03, H09
[49]–[50]	high entropy	$\eta(R)=\log_2\lvert\mathcal G(R)\rvert/\lvert R\rvert\ge(1-\tau)\log_2\lvert R\rvert-O(1)>\tfrac1{10}\log_2n$: the low-entropy arms (b),(c) of prop:two-budget are empty (Corollary 1.4, relabeling orbits)	T12, T16	G01, G04, I02
[51]–[53]	remaining non-obstruction budget not $<K\lvert R\rvert$	large-budget branch: the skeleton budget minus the forced obstruction cost is at least $K\lvert R\rvert$; the entropy cap prop:entropy-high-theta ($\theta>\Theta(n)$) does not apply; $\Theta(n)=(1.4-K/\log_2n)/(116.808581006-13K/\log_2n)$	T12	H09, G08
[55]–[56]; [173]	—	Residual C: $\Delta_{\rm net}(R)=(\defp(R)-\sigma_R)/\lvert R\rvert\le\tau_{\rm win}+o(1)<\tfrac14$ (now $\le\tau^\ast$), decided exactly on the object at [173] (lem:exact-collision-test)	T01, T13	H01, I03
[57]–[61]	$\No(R)<0$	net charge $\No(X)=\defp(X)-\sigma(X)-\tfrac14\lvert V(X)\rvert$; $\sum_i\No(X_i)=\defp(R)-\sigma(R)-\tfrac14\lvert R\rvert\le-(\tfrac14-\tau)\lvert R\rvert$; some connected canonical support has $\No(X)<0$ (def:net-charge, lem:netcharge-superadd, prop:negative-net-charge); canonical decomposition of $R$ into components with surplus assigned to the piece containing the high-degree vertex (def:canonical-decomp)	T13, T04	H01–H03, D08
[62]; [64]–[85]	Type B closed	high-degree supports: centers independent, fan neighbours cubic, certificate-marked cap $d_G(h)\le8$, the fan-window ledger, B2 disjointness; every Type B support with $\No<0$ outside the bridge residual has a route-8 profile or a positive-deficit fan residual; the bridge residual mass is $M_B\le16\sigma(G)=o(\lvert R\rvert)$ (lem:typeB-exclusion, prop:typeB-bridge-sublinear, thm:branch-kill(b))	T07, T13, T14, T15	D03, D04, H05, H06, H08
[63], [86]–[88]	Type A	the negative supports of linear mass are Type A: $\sigma(X)=0$, so every $v\in V(X)$ has $d_G(v)=3$; $X$ is connected, subcubic, $P_{13}$-free, $\operatorname{diam}X\le11$, $\lvert X\rvert\le6142$, empty internal $3$-core, contextually target-safe, hereditarily uncompressible, every deficient vertex supplied from $W$; $\defp(X)<\lvert X\rvert/4$; receivers $w$ with $d_X(w)\le2$, $q(w)=3-d_X(w)$ ports; canonical traces $T_u$ and loads $L(w)$ (def:typeA-support, def:typeA-receiver-load)	T04, T05, T07	A03, A11, B05, C08, D08
[89]	some receiver saturated	$L(w)\ge4q(w)$ for some receiver (else lem:typeA-unsaturated-discharge: $\defp(X)\ge\tfrac14\lvert X\rvert$, $n_3\le3n_2+7n_1+11n_0$, closing)	T13	H04, H05
[93]	no	no completion port carries four visible receiver-entry returns (else exits (1)–(7), lem:typeA-visible-entry)	T07, T08	C01, C02
[94]	—	visible-first excess: $S^{\rm exc}_{\rm sil}(X)=\sum_w\lvert\mathcal U(w)\rvert\ge n_3-3n_2-7n_1-11n_0=4D_A(X)$, $D_A(X)=\tfrac14\lvert X\rvert-\defp(X)$ (lem:typeA-silent-excess-count, def:typeA-excess-basin)	T13, T15	H05, G07
[95]–[108]	exits (1),(2),(3),(5),(6) closed; (7) absent; (4) peels	at every saturated receiver of every $X\in\tilde{\mathcal X}$: no anchored return of Mersenne length through a port; no two internally disjoint receiver-entry returns through one port with lengths summing to a power of two (lem:typeA-common-port-return-cycle); no violated label relation $C_s$ on a shared window; no nontrivial target-complete response compression; no delocalizing response equality; no decorated handoff fan (exit (7)) since $X\in\tilde{\mathcal X}$ produces none; continuation routing at a port gives exits (4)–(6) or a surviving first separator of degree $\ge4$ (lem:typeA-continuation-routing, lem:typeA-cubic-switch-absorption, lem:typeA-high-degree-handoff)	T08, T09, T05, T07	C01–C05, D01, E05, E06, D03
[109]–[113]	—	the unified negative collection $\tilde{\mathcal X}={X:\sigma(X)=0,\No(X)<0,\text{no handoff}}$ with $\tilde D_A=\sum(\tfrac14\lvert X\rvert-\defp(X))\ge(\tfrac14-\tau)\lvert R\rvert-o(\lvert R\rvert)$; entries $\tilde\Xi={(X,w,u,B_u):u\in\mathcal U_X(w)}$, $\tilde N\ge4\tilde D_A$ (def:typeA-unified-negative, lem:typeA-unified-deficit, lem:typeA-unified-burden)	T13, T15	H03, H08, G07
[114]–[116]	—	every entry passes to its canonical minimal target-complete response-support core $\mathcal C_{\rm ess}(\xi)\subseteq\partial_EX$, $\alpha(\xi)=\lvert\mathcal C_{\rm ess}\rvert\ge2$ (lem:typeA-unified-carriers; entries with $\alpha\le1$ realize exits (4)–(7))	T11, T05	F02, F04, B08
[117]; [119]–[122]	two-support entry exists	if every entry had $\pi(\xi)\ge3$ private essential incidences then $3\tilde N\le\defp(R)$ against $\tilde N\ge12(\tfrac14-\tau)\lvert R\rvert$, impossible; so some $\xi$ has $\pi(\xi)\le2$ (prop:typeA-unified-reduction)	T15, T01	G07, H06
[118], [124]	route-8 two-support closed	no terminal two-support route-8 obstruction (thm:typeA-two-carrier-nogo); Theorem 3.2: every two-support entry realizes exit (4), so route-8 two-support entries do not occur	T05, T11	E06, F05
[101]–[102], [123]	target-defect two-support: peel	each such entry is peeled: its load leaves the receiver sum, $\Lambda_4=\sum_w\lvert\mathcal L(w)\setminus P_4(w)\rvert$ decreases by one, the deficit by $\tfrac14$, no invariant weakened (lem:typeA-exit4-discharge, lem:typeA-exit4-finite-descent); iterate while $\tilde D_A^{P_4}\ge(\tfrac14-\tau_{\rm win})\lvert R\rvert-o(\lvert R\rvert)$	T19	E08, H10
[181]	the reduced-rate test fails	the leaf (§4)	—	—

Arms not on the path (for completeness): [3] not a counterexample; [16] $P_{13}$-free; [20] non-near-cubic surplus (routed back to the spine); [23] live-hot overflow; [159]–[172] dense-packing residual ($\theta>\theta_{\rm win}$, the no-arm of [158]) and its nodes [163]–[172], [178]–[180], [182]; [60] net-cap contradiction; [90]–[92] unsaturated; [96], [98], [100], [104], [106] closed exits; [108] handoff; [124] closed.

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2. The complete hypothesis ledger, by register category

Every fact below is on the branch state $\mathcal B_{181}$. "Produced by" names the technique; "Row" the register property it evaluates.

A — size, degree, sparsity, local incidence

#	Fact	Source	Produced by	Row
A-1	$\delta(G)\ge3$; $n=\lvert V(G)\rvert\to\infty$ along the branch	def:counterexample	—	A04, A01
A-2	$m=\tfrac32n+O(\sqrt n)$; $\sigma(G)=2m-3n=O(\sqrt n)$; $\lvert V_{\ge4}(G)\rvert\le\sigma(G)$	def:near-cubic-spine	T13/T14/T15 (surplus ledger)	A02, A05, A14
A-3	$m\ge\lceil3n/2\rceil$, $m\le2n-2$; $\beta=m-n+1\ge n/2+1$; $\beta+\lambda=n-2$; $\sigma=2\beta-n-2$	invariants 9–13	T01	A02, A12, A13
A-4	$V_{\ge4}(G)$ independent; every edge has a degree-3 endpoint	lem:deletion-critical	T03/T02	A06, E03
A-5	every vertex of every Type A support has $d_G=3$; a vertex of internal degree $3-q$ has $q$ stubs to $W$; $\defp(X)=\sum q$; $\defp(X)<\lvert X\rvert/4$ on $\tilde{\mathcal X}$	def:typeA-support	T05	A03, A11
A-6	$\defp(R)\le e(R,W)\le15p_{13}+o(n)$; $e(R,W)+2e_\times(W)=15p_{13}+\sigma_W$; each window carries at most $15$ stubs (interior vertex one, end two)	lem:stub-positive	T01/T15	A10, A11
A-7	$(\defp(R)-\sigma_R)/\lvert R\rvert\le\tau^\ast+o(1)$, $\tau^\ast=15\theta^\ast/(1-13\theta^\ast)=0.13456$	Theorem 1.5	T12/T16	A11, H01
A-8	$W_2(C)\ge3\lvert V(C)\rvert-2\defp(C)$; $W_2(R)\ge2.54365\lvert R\rvert-o(\lvert R\rvert)$	lem:wedge-lower	T01	A09
A-9	every $P_{13}$-free induced subgraph of $G$ has a vertex of degree $\le2$	lem:remainder-empty-internal-3-core	T18	A07
A-10	every component of $G-V(X)$ sends $\ge2$ edges to $X$; $\defp(X)\ge2$	lem:bridgeless	T03/T02	A11, B02

B — connectivity, cuts, boundaries, contexts

#	Fact	Source	Produced by	Row
B-1	$G$ is bridgeless; every edge lies on a cycle	lem:bridgeless	T03/T02	B02
B-2	every proper subgraph has $\delta\le2$	lem:no-proper-core	T02	E02, B01
B-3	$R=G-W$; its components are the canonical supports; no edges of $R$ between distinct components; surplus units of $V_{\ge4}\cap V(R)$ assigned to their component	def:canonical-decomp	T04	B01, D08
B-4	boundaried pieces, boundary degree profiles, gluing $X\oplus_TY$; quotients fibrewise over $\mathbf d_\partial$	def:boundaried-gluing, lem:degree-profile-fibres	T05	B05, B06
B-5	context universality (target-complete identifications agree against every context)	lem:context-universality	T05	B07
B-6	trace basins $B_u$: lexicographically first inclusion-minimal trace-complete connected subgraph containing $T_u$; trace-response state $\rho_u(B_u)=(\mathbf d_\partial(B_u),\mathcal R_u(B_u),\profile)$	def:typeA-trace-basin	T05/T10	B08, B06
B-7	boundary incidences $\partial_EX$, $\lvert\partial_EX\rvert=\defp(X)$; every $u$-supported coordinate leaving $X$ records one	def:typeA-route8-carriers	T05	B05, B09
B-8	the demand ledger on $\tilde\Xi$: partition $\Xi_3\sqcup\Xi_2\sqcup\Xi_{\rm res}$ with disjoint incidence sets $A(\xi)$ ($3$ or $2$ per entry), lexicographically first maximizing $N_3$ then $N_2$; $\mathsf P_{\rm ext}=N_2+3N_{\rm res}$; $3N_3+2N_2\le\defp(R)$	def:typeA-pressure-ledger, lem:typeA-pressure-ledger-no-overcount	T14/T15	B09, H06
B-9	absorbers: (A1) unused boundary incidences of the same support, (A2) profile-dependence certificates; $\mathsf P_{\rm open}=\lvert\mathcal U_{\rm press}\setminus\mathcal U_{\rm abs}\rvert$; $3\tilde N-\mathsf P_{\rm open}\le\defp(R)$ once (A2) has routed	def:typeA-pressure-absorbers, lem:typeA-pressure-absorber-no-overcount	T14/T15	B09, H06
B-10	window blockers: each open unit is assigned one incidence $c(\upsilon)$ and its unique window $P(\upsilon)$; $\mathsf P_{\rm open}=\sum_PB_{\rm open}(P)$	def:typeA-open-window-blocker, lem:typeA-open-window-blocker-count	T15	B09, A10
B-11	$\mathsf P_{\rm open}\ge(3-13\tau)\lvert R\rvert-o(\lvert R\rvert)$ and $\mathsf P^{+}{\rm zero}\ge\varepsilon{\rm prim}\lvert R\rvert-o(\lvert R\rvert)$ on the branch (so the offered consumers of [181] are vacuous)	Theorem 3.1	T01	B09, H09

C — paths, cycles, lengths

#	Fact	Source	Produced by	Row
C-1	no cycle of $G$ has length in $\mathrm{Pow}={2^j}$; $R_e(G)\cap\mathrm{Mers}=\varnothing$ for every oriented edge	lem:return-equivalence	T08	C02, C03
C-2	every completion port has at least one actual anchored return	lem:typeA-port-return	T08	C02
C-3	receiver-entry returns are actual simple connector–channel paths, and the finite schedule contains every such return	def:typeA-visible-load; VisibleReceiverEntry.lean	T08/T16	C01, C02
C-4	connector/channel arithmetic: for a receiver-entry return $\Gamma\circ Q$ through $(w,h)$ with connector length $g$, $g+\lambda\notin\mathrm{Mers}$ for all $\lambda\in\Lambda_X(r,w)$; interval form with $I_X(r,w)$	lem:typeA-spectral-pressure, def:typeA-channel-spectrum	T09	C01, C04
C-5	theta closure: all branch-pair sums in a theta avoid $\mathrm{Pow}$; ear closure; symmetric difference of overlapping cycles avoids $\mathrm{Pow}$	invariants 31–33	T08/T09	C05–C07
C-6	two-path criterion: two internally disjoint returns through one port with lengths summing to $2^k$ give a forbidden cycle	lem:typeA-common-port-return-cycle, invariant 30	T08	C05
C-7	every cycle length has an odd prime divisor; no single odd prime divides all cycle lengths; overlap formula $q_p(E)=q_p(C)+q_p(D)-2t$ flat and non-killing	invariants 36–38	T09/T11	C04
C-8	$G$ has an induced $P_{13}$; $R$ has none; every component of $R$ has diameter $\le11$ and $\le6142$ vertices	cor:p13-exists, lem:remainder-empty-internal-3-core	T18/T06	C08
C-9	$p_{13}$ is the maximum number of vertex-disjoint induced $P_{13}$'s; $\theta=p_{13}/n\le\theta^\ast+o(1)$	[17], Theorem 1.5	T06/T12	C09, G09

D — local configurations, motifs, overlap, symmetry

#	Fact	Source	Produced by	Row
D-1	$399$ legal $P_{13}$ labels; relations $C_s$; the zero-defect quotient through path lengths $1,2,3$ is the identity	lem:labels	T17	D01, G02
D-2	$0.795414$ of locally safe wedges are obstructing; $c_\Omega=2.2892$ bits per independent obstruction coordinate	lem:curv-enum	T17	D02, G02
D-3	Type B: fan-safe graphs, certificate labellings, $d_G(h)\le8$ for certificate-marked fans, degree-4 profiles, B2 disjointness, bridge residual sublinear	[64]–[85]	T07/T13/T14	D03, D04
D-4	canonical decomposition, canonical traces (lexicographically first receiver-reaching paths in $X_3$), canonical payable set $A(w)$ (visible-first order), lexicographically first ledgers and assignments	def:canonical-decomp, def:typeA-receiver-load, def:typeA-excess-basin, def:typeA-pressure-ledger	T16	D08, I02
D-5	all auxiliary objects are functions of the labelled adjacency matrix under a fixed tie-break; states are $\mathrm{Sym}(R)$-invariant relative to $W$	lem:skeleton-dominates, Theorem 1.3	T16/T12	D09, G06

E — extremality, criticality, replacement, quotients

#	Fact	Source	Produced by	Row
E-1	lexicographic minimality of $G$ ($\lvert V\rvert$, then $\lvert E\rvert$, then lexicographic)	[4]	T02	E01
E-2	no proper subgraph with $\delta\ge3$; deletion criticality	[8]–[9]	T02/T03	E02, E03
E-3	replacement lemma and hereditary uncompressibility (I5)	lem:replacement, cor:uncompressible	T02/T05	E05
E-4	a quotient is valid only if target-complete against every context; otherwise target-defective	lem:context-universality	T05	E06
E-5	exit-(4) peeling is a well-founded descent on $\Lambda_4$ preserving every invariant	lem:typeA-exit4-finite-descent	T19	E08, H10
E-6	exits (5) and (6) never occur at any saturated receiver of $\tilde{\mathcal X}$ (standing-invariant contradictions); exit (7) is absent	lem:typeA-exits-discharged, lem:typeA-unified-burden	T02/T05	E05, E09

F — local tests, rank, dependence

#	Fact	Source	Produced by	Row
F-1	full obstruction rank $r_\Omega(R)\ge W_2(R)-o(W_2)$	lem:full-rank	T11	F02, F07
F-2	rank drop routes to target defect, proper compression, or support enlargement; proper enlargements $Z\subsetneq G$ are impossible; whole-graph dependence cannot silently reduce rank	lem:curvature-dependence-routing, lem:proper-smearing, lem:no-silent-global-smearing	T10/T11	F03–F07
F-3	separated identical wedges are context-universal or target-defective	lem:separated-testers	T05/T11	F05
F-4	every entry's trace basin fails target-complete-minimality only through alternative (a) — a trace-local quotient forgetting a coordinate on an internal edge of $B_u$ is distinguished by a compatible context; (b),(c),(d) do not occur	def:typeA-trace-basin, Theorem 3.2	T05/T16	F04, E06
F-5	$\lvert\mathcal C_{\rm ess}(\xi)\rvert\ge2$; every $c\in\mathcal C_{\rm ess}$ has a declared deletion witness (internal/mixed) with boundary-incidence support	lem:typeA-unified-carriers, def:typeA-carrier-deletion-witness, lem:typeA-deletion-witness-declared	T05/T11	F04, F05

G — counting and information

#	Fact	Source	Produced by	Row
G-1	$\lvert\mathcal G_{n,m}\rvert=\binom{\binom n2}{m}$; skeleton budget $\tfrac32n\log_2n+o(n\log n)$	lem:skeleton-dominates, lem:near-cubic-budget	T12	G01
G-2	the joint window package is realized: $\ge2^{c_{13}p_{13}\log_2n}$ states	[158] yes	T12	G03
G-3	orbit count: $\log_2\lvert\Phi(\mathcal S)\rvert\le\log_2\lvert\mathcal S\rvert-(\tfrac34\lvert R\rvert-15p_{13})\log_2\lvert R\rvert+O(n)$	Theorem 1.3	T12/T16	G01, G04
G-4	$\eta(R)\ge(1-\tau)\log_2\lvert R\rvert-O(1)$; high-entropy arm of prop:two-budget	Corollary 1.4	T12	G04
G-5	forced cost $c_\Omega r_\Omega\ge K_{\rm win}\lvert R\rvert$; the large-budget arm: remaining budget $\ge K\lvert R\rvert$	[48], [53]	T12	G03, H09
G-6	no double counting: demand incidences pairwise disjoint; absorbers single-use; the exact stage identity $4\tilde D_A=4\tilde D_A^{P_4}+p_4$	lem:typeA-pressure-ledger-no-overcount, lem:typeA-peeling-stage-accounting	T15	G07
G-7	asymptotics: all bounds with $o(n)$, $o(\lvert R\rvert)$ error terms; exact collision decided at [173]	lem:exact-collision-test	T12/T17	G08

H — charging and discharging

#	Fact	Source	Produced by	Row
H-1	$\No(X)=\defp(X)-\sigma(X)-\tfrac14\lvert V(X)\rvert$; superadditivity; some connected support has $\No<0$	def:net-charge, lem:netcharge-superadd, prop:negative-net-charge	T13	H01–H03
H-2	Type A: each cubic vertex charges $\tfrac14$ to its receiver; receiver charge $q(w)-\tfrac14-\tfrac14L(w)$; unsaturated receivers ($L\le4q-1$) pay; thresholds $H_0\le4,H_1\le8,H_2\le12$	lem:typeA-threshold-algebra, lem:typeA-unsaturated-discharge, lem:typeA-exit4-peeling-charge	T13	H04, H05
H-3	saturated receivers with silent excess: $S^{\rm exc}_{\rm sil}\ge4D_A(X)$; the unified deficit $\tilde D_A\ge(\tfrac14-\tau)\lvert R\rvert$; $\tilde N\ge4\tilde D_A$	[94], [111]–[113]	T13/T15	H05, H08
H-4	private-support budget: three private incidences per entry would force $3\tilde N\le\defp(R)$, contradiction; hence two-support entries exist	prop:typeA-unified-reduction	T15	H06
H-5	the demand ledger, absorbers and blockers (B-8–B-11)	—	T14/T15	H06
H-6	Type B bridge mass $o(\lvert R\rvert)$	prop:typeB-bridge-sublinear	T13/T15	H08
H-7	the required rate: with $\tau^\ast$, a single-use charging of $c$ incidence-units per entry contradicts iff $c>0.2914$; the branch is empty if every component has $\lvert X\rvert\le7\defp(X)$ (diagnostic)	Theorem 3.4	T13	H09
H-8	finite descent $\Lambda_4$	lem:typeA-exit4-finite-descent	T19	H10

I — finite certification and external inputs

#	Fact	Source	Produced by	Row
I-1	the black box thm:p13free (HSS): $P_{13}$-free $+\ \delta\ge3\Rightarrow$ power-of-two cycle; consumed as A-9 and C-8	[15]–[16]	T18	I06
I-2	Bondy–Vince / Gao–Ma are citable with exact hypotheses; the appendix's derived input "$\lvert Y\rvert<5b(Y)^2$ for quiet almost-cubic blocks" is assumed, not proved (lem:app-typeA-quiet-bound, lem:app-dense-window-closure)	appendix	T18 (not yet invoked)	I06
I-3	finite constants $c_\Omega$, $c_{13}$, label counts; the $91$-barrier computation; the two-strand table	app:curv-code, lem:labels, [167]	T17	I05, I01
I-4	exact small-order collision decided on the object	[173]	T17	I03, I04

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3. Techniques already used upstream, and the structural properties each one consumed

Technique	Where used	Properties consumed (register rows)	What it left behind
T01 Direct invariant calculation	[28]–[30], [56], [119]–[122], Theorems 3.1, 3.4	A02, A09–A12, H01, H06, H09	the inequalities of A-3, A-6–A-8, H-3, H-4, B-11
T05 Boundary-interface analysis	[11]–[14], trace basins, response states, cores, deletion witnesses, contexts	B05–B08, E05, E06, F04, F05	B-4–B-7, E-3, E-4, F-4, F-5
T08 Path–cycle and cycle-space analysis	[5]–[7], invariants 30–33, lem:typeA-port-return, lem:typeA-common-port-return-cycle	C02, C03, C05–C07	C-1, C-3, C-5, C-6
T10 Uncrossing and minimal obstruction	[31]–[47] (dependence localization), trace-basin minimality, continuation routing	F03–F07, B08, D05 (cold branch only)	F-2; on the [181] branch the corridor/overlap consumers of [169]–[172] are not available (they live on the dense-packing residual)
T11 Linear-algebraic rank	[31]–[47], response-support cores	F01–F07, A12	F-1, F-2, F-5
T12 Counting and information	[21], [48]–[55], [158], Theorems 1.3–1.5, Corollary 1.4	G01–G09, H09	G-1–G-5, A-7, C-9; the low-entropy arms are empty
T13 Potential and discharging	[56]–[62], Type A charging, Type B ledger	H01–H05, H08	H-1–H-3, H-6
T14 Demand–supply and flow	Type B B1/B2, the $2/3$-demand ledger, absorbers, surplus token ledger	B09, H05, H06, D04	B-8–B-10; the failure of the matching is the leaf
T16 Symmetry and canonicalization	lexicographic tie-breaks everywhere, $\mathrm{Sym}(R)$-invariance (Theorem 1.3), canonical traces/ledgers	D08, I02, E07, G04	D-4, D-5, G-3; the refined-order swap [165]–[166] is used only on the dense residual
T18 External structural theorem	[15]–[16] (HSS)	I06, C08	I-1; Bondy–Vince/Gao–Ma present but not invoked (I-2)
T19 Peeling and finite descent	[101]–[102], [123]	E08, H10	E-5, H-8, and the leaf's identity $4\tilde D_A=4\tilde D_A^{P_4}+p_4$

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4. The leaf: the typed data of [181]

def:typeA-peeled-demand-residual, after the procedure of thm:large-budget-route8-only:

• (R1) a valid family $P_4=(P_4(w))w$ of exit-(4) peeling sets at which $\tilde D_A^{P_4}<(\tfrac14-\tau{\rm win})\lvert R\rvert-o(\lvert R\rvert)$;
• (R2) the disjoint partition $\tilde\Xi=\tilde\Xi^{P_4}\mathbin{\dot\cup}\tilde P_4$, $p_4=\lvert\tilde P_4\rvert=\sum_w\lvert P_4^{\rm un}(w)\rvert$, the exact identity $4\tilde D_A=4\tilde D_A^{P_4}+p_4$ (so $p_4\ge4\tilde D_A-(1-4\tau_{\rm win})\lvert R\rvert+o(\lvert R\rvert)$, linear), and for each peeled entry its recorded exit-(4) witness;
• (R3) the maximal $2/3$-demand ledger on $\tilde\Xi$, its maximal absorption ledger, and the unique-window blocker partition $\mathsf P_{\rm open}=\sum_PB_{\rm open}(P)$.

Each peeled entry $\xi=(X,w,u,B_u)$ carries: a Type A support $X\in\tilde{\mathcal X}$ with all of §2; a saturated receiver $w$ ($L(w)\ge4q(w)$, $q(w)\in{1,2}$); a silent unpaid routed load $u$ (cubic; its trace $T_u$ ends at $w$; no receiver-entry return through a port of $w$ has a channel containing $T_u$); its trace basin $B_u$ with state $\rho_u(B_u)$; two-support: $\pi(\xi)\le2$ private essential incidences, $\lvert\mathcal C_{\rm ess}(\xi)\rvert\ge2$, declared deletion witnesses for each $c\in\mathcal C_{\rm ess}$; alternative (a) only: a trace-local quotient $q$ forgetting a $u$-supported coordinate on an internal edge of $B_u$, with the demand token $(\xi,q,S_0,S_1,Y,E)$ — $S_0$ the actual realization, $S_1$ an alternative in the same boundary-degree fibre with $q(S_0)=q(S_1)$, $Y$ a compatible outside context in which $S_0\oplus Y$ and $S_1\oplus Y$ differ in target predicate, $E$ the first witnessing event, using an internal edge of $B_u$ and at least two boundary incidences of $X$ ($K(\mathfrak t)$, $\lvert K\rvert\ge2$).

What each certificate does and does not say: the token and the deletion witnesses certify that specific quotients are not target-complete; $S_1$, $Y$, $E$ are hypothetical; only $S_0$, $u$, $T_u$, $B_u$, the ports and their actual returns are objects of $G$.

Derived facts at the leaf: Theorem 3.1 (the offered consumers are vacuous); Theorem 3.2 (every two-support entry realizes exit (4); no true route-8 two-support entry; at least one peel is performed); Corollary 3.3 ([181] is the only exit of [123]); Theorem 3.4 (diagnostic rate).

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5. Present rows not consumed by any step of §1

From the inventory (repair_and_closure.md §4.7): B02–B04 beyond bridgelessness (cyclic edge cuts, blocks, disjoint connections); A08 (chains of degree-2 receivers); C04/F08 on this branch (arithmetic class and periodicity of the entries' actual channel and connector lengths); C09 (cardinality maximality, consumed only through $P_{13}$-freeness); D05/D07 for entries (overlap of the basins of one receiver's loads; equal response states among linearly many entries over a bounded alphabet); E04 (safe suppression of degree-2 receivers); H07 (the Hall obstruction of the demand ledger as a located set of entries and ports); I06 (Bondy–Vince/Gao–Ma with exact hypotheses). The consumers of D05/D06/C11 ([169]–[172]) and of D07/E07 ([163]–[166]) are stated on the dense-packing residual and are not available on this branch.

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6. The closure theorem for [181]

Theorem [181]. Let (G) be a finite simple graph and suppose that all facts of §2 (A-1 through I-4), with the arms of §1 as stated, and all leaf data (R1)–(R3) of §4 hold. Then (G) contains a cycle whose length is a power of two.

Equivalently, the complete declared-support residual routed from [123] after [124] is empty. The proof must act on the target-defect two-support entries themselves; it may not replace their declared carriers by the weaker event-carrier implementation.