FPRD research loop state

Latest ATTACK: the local move has an exact transverse cokernel

9 September 2026. This section supersedes the previous next-question summary while retaining its proof and scope.

For the generic local two-to-three tetrahedron map, with only the new interior edge and the three tetrahedron relations imposed, the even vector-space cokernel has basis e_beta for beta in Z^3 with sum beta=-1. The explicit finite normal form and source-quantum-torus action identify it with a quantum-difference representation on an affine Z^2. This is not an algebra structure on the mixed-sided quotient.

Proof: the source image spans the entire charge-zero slice, by an explicit integer lattice inverse. A correct-sided recurrence reduces positive charges to zero and charges below -1 to -1; the coefficient 1-q^(-c-1) vanishes exactly at -1. A rational dual family H_s, sum s=1, separates the surviving basis, with H_s(W^beta)=delta_(s,-beta). In particular H_(1,0,0)(W_1^-1)=1 while it kills the move image. The family extends to the primitive root edge P=-A^2, so no backwards lifting from a squared relation is used.

The local source-to-target certificate is verified symbolically. Exact rational controls cover the normal form and primitive relation sides. These are internal checks, not independent model or formal-language review. The local rational H_s is not asserted to equal the native index F or to annihilate actual exterior edge relations.

Garoufalidis-Yu’s preprint Proposition 5.5, after the all-input native normalization is checked, transports the two known actual exceptional detectors along any finite sequence of 2-3 and 0-2 moves staying 1-efficient. Their source paper is an inspected preprint, not their separate published quantum-trace paper. This proves a conditional directed-refinement extension only. The local cokernel blocks inferring reverse detection from image-wise index equality; ordinary undirected connectivity alone would not justify that inference.

Packet: /evidence/dehn-filled-quantum-trace/move-defect-attack/FPRD-local-move-defect-2026-09-09.md. Supporting files: verify-move-defect.py, exact-results.json, internal-audit.md, maturity-cards.md and move-defect-attack.zip. Public section: /research-programs/dehn-filled-quantum-trace#local-move-defect. Research note, novelty unassessed; no Publication or unrestricted conjecture resolution. Significance-based publication ordering is unchanged.

NEXT MODE: ATTACK. Impose the actual exterior edge relations on the affine-Z^2 local representation, and determine whether the native index detects a surviving charge-minus-one class whenever its charge-zero restriction vanishes. Alternatively identify that class by the intrinsic compressing-disk/product structure. Universal nonvanishing for all 1-efficient exceptional triangulations remains open. Do not assume local-to-global survival, identify H_s with F, reverse the move map, or enumerate triangulations/slopes.

Provisional maturity, discussion draft v0.2: MD-local-cokernel-v1 is M10-19 (complete local candidate proof and self-audit); NS-exceptional-all-v1 remains M0-9; NS-1efficient-v1 remains M10-19. Imported proofs and complementary subarguments are not independent reviews. Exact within-band and audit-burden scores remain unassigned. Private correspondence and operator tracking are separate from this research record.

Latest ATTACK: the closed normal-sector bridge and 1-efficient cokernels

8 September 2026. This section supersedes previous next-question summaries without removing their evidence.

The exact integer identity H=T((Lambda intersect 2L)/2)+T_Z=N_Z is now proved, where N_Z contains all closed integral Q-normal classes. The proof uses the published integer normal-homology kernel: every doubled closed class lies in E+T_Z. It never saturates Lambda. The native Gaussian phase also agrees termwise, yielding F(1)=sum_{a:boundary(a)=0} I_T^a(0), not generally only I_T^0(0).

The signed ray criterion is consequently equivalent to 1-efficiency. The preceding affine-input theorem supplies convergence on every algebraic input, including odd roots. GHHR’s published Corollary 10.19, combined with monoid decomposition and superadditivity, gives F(1)=1+positive A-order for every 1-efficient triangulation of a connected oriented torus-boundary manifold other than the solid torus and T^2 x I. Thus the retained generic native theorem extends to dim(Q^6/im tau)>=2^r-1 in this class, without strict or nonnegative angles. Every minimal compatible filling quotient is nonzero; a nonzero filled trace is not asserted. The published positivity theorem and prior Lab native comparison/image-grading proofs remain explicit dependencies.

The two published exceptional triangulations also have nonzero detectors, now on actual incidence data: solid torus cMcabbgds has F(1)=0 and F(w_1^2)=-I_Delta(0,0)!=0; product torus dLQacccbjkg has F(1)=0 and F(w_1^{-2)=I_Delta(0,0)}2!=0. Their degree-one edges force angle 2 and forbid nonnegative angles. The native rows, phases, primitive edge relations and correct-sided tetrahedron multiples were checked exactly. The resulting cokernel bounds are >=1 and >=3. These are two cited structural controls, not a triangulation census or a theorem for every triangulation of the exceptional manifolds.

Important failure avoided: direct substitution into the solid-torus generating quotient at positive difference m gives 0/0, and expanding its raw double sum gives nonpositive k-slope. Cancel the m common factors first and use the negative-parameter seed expansion. The resulting closed formula agrees with tail-certified native averages.

Packet: /evidence/dehn-filled-quantum-trace/normal-sector-attack/FPRD-normal-sector-nonvanishing-2026-09-08.md. Code/results: verify-normal-sector.py and exact-results.json; internal-audit.md; maturity-cards.md; normal-sector-attack.zip. Public section: /research-programs/dehn-filled-quantum-trace#normal-sector-nonvanishing. Supporting research note, novelty unassessed, not a Publication. Significance-based publication order is unchanged.

NEXT MODE: ATTACK. Prove nonvanishing of the native functional for arbitrary 1-efficient triangulations of the solid torus or T^2 x I. Seek correct-sided local-move transport of a nonperipheral detecting insertion with summability, or an intrinsic compressing-disk/product construction. Scalar index invariance cannot transport a detector when those indices vanish. State any move-graph connectivity hypothesis; none is assumed here. Beyond 1-efficiency the current unordered all-input seed diverges at the unit and would require a genuinely different construction. Do not enumerate triangulations, slopes or collision strata.

Maturity pilot v0.1: NS-1efficient-v1 is M10-19, a complete conditional synthesis without independent full reconstruction. NS-exceptional-all-v1 remains M0-9. The universal old question is split rather than silently promoted. Published proof inputs, exact machine calculations and the same agent’s audits are separate evidence fields, not multiple independent confirmations. No model/external review or formal proof is claimed.

Latest ATTACK: signed-angle convergence and nonunit detection

8 September 2026. This section supersedes older next-question summaries below; all earlier results and dependencies are retained.

The ordinary native index average now has a signed generalized-angle convergence criterion: on C=(H tensor R) intersect R_{>=0}^{3N}, theta dot x must be strictly positive on every nonzero x with delta(x)=0. This condition is necessary for unordered convergence at 1 and sufficient for convergence on every finite Laurent input. The proof extends the affine-input mixed-term estimate using compactness: delta(Y)+theta dot Y >= c||Y||-B. It preserves the actual unsaturated lattice, exact character and relation sides. Actual generalized-angle equations also prove all-plus character consistency without strict positivity.

For nonnegative certificates satisfying the criterion, F(1)=1+positive order and the prior cohomological bound dim(Q^6/im tau)>=2^r-1 persists. No actual triangulation outside the strict-angle class is exhibited in this run. For signed certificates, the bound follows if F has ANY nonzero monomial value: translate that degree-zero input along integral normalizing defects. Nonzero unit is unnecessary.

Exact abstract control: Lambda=Z(2,0), chi(2,0)=A^4, theta=(2,-1,0), relations Sf and (z^2-A4)S. It has all-input convergence and F(W(2a,2b))=0 for b>=0, while for b<=-1 its value is q^{a(1+b/2)}/(q;q)_{-b-1}. Thus F(1)=0, F(w^-2)=1. This is not claimed to be actual triangulation incidence. Corrected the intermediate double-sum exponent to j(j-1)/2-bj+k. The exact rational recurrence checks and tail-certified q-series controls agree; these are one internal self-audit, not independent-model review.

Packet: /evidence/dehn-filled-quantum-trace/signed-angle-attack/FPRD-signed-angle-detector-2026-09-08.md. Supporting files: verify-signed-angle.py, exact-results.json, internal-audit.md, maturity-pilot.md, signed-angle-attack.zip. The research-program section is #signed-angle-detector. Ordinary research note, not a Publication; no unrestricted conjecture or nonzero filled invariant is claimed.

NEXT MODE: ATTACK. Prove that the native averaged functional is not identically zero on actual triangulations satisfying the signed ray criterion, by finding one nonzero monomial value from genuine incidence or a second exact pairing. Do not insist on a nonzero unit. Relate the criterion to the published 1-efficient normal-surface framework only through an exact lattice identification; scalar convergence alone does not supply nonvanishing. Do not enumerate manifolds, slopes or collision strata.

Confirmation pilot (discussion rubric v0.1): the preceding strict-angle conditional theorem is provisionally M 10-19, complete candidate with internal audit but no documented independent reconstruction. General nonvanishing of F under the signed criterion is M 0-9, a precise open question without a complete proof. The six bands and full claim cards are in maturity-pilot.md. No exact within-band score, automatic legacy-status conversion or significance-based publication reorder was applied. Model agreement, formal coverage, physical evidence and independent routes remain separate evidence fields.

The specific first-contact invitation and maturity draft sent at Joshua’s request are an exception to the general no-outreach policy; they must not be resent. The private hourly instructions retain the exact email thread and feedback-check state. No new stakeholder feedback was found during this run. Private contact details and message text are not included here. The mathematics continues independently of a response.

Standing communication policy — 8 September 2026

Do not email or otherwise message researchers. All former automatic author-outreach and CC requirements are withdrawn. Publish qualifying work without waiting for contact, replies, or external review. Only a later explicit request from Joshua for a specific message permits an exception. This applies to every research run, including the Cheng manuscript, whose public release Joshua authorized on 8 September 2026. Full policy: FPRD researcher communication policy. Preserve the active mathematical target below.

Latest completed mode

ATTACK — cohomological degrees prove nonzero trace cokernels for all strict-angle triangulations, 8 September 2026.

For every connected oriented ideal triangulation with r>=1 torus cusps and a strict angle structure, the retained full six-root native trace tau over k=Q(i,A^(1/2)) satisfies dim_k(Q^6/im tau)>=2^r-1. Every minimal compatible Dehn-filling quotient D_p=Q^6/tau(ker p) is consequently nonzero, since it surjects onto Q^6/im tau. The common quotient supports a compatible zero filled trace. This verifies the literal nonzero-quotient condition in the generic strict-angle class; it does not provide a nonzero filled invariant or resolve unrestricted Panitch–Romo 7.1. Research note only.

The actual half-face map is J=L/2L -> D_Gamma, where D_Gamma is graph cohomology annihilating the all-edge cycle. Its kernel K_0 is the N-dimensional Lagrangian of dual-graph cycles modulo that all-edge cycle. Actual edge rows have zero defect, as does the entire native trace image by published PP Theorem 5.6 / Proposition 5.7. The parity lattice has exact sequence 0 -> H_1(M;F_2) -> J/B -> D_Gamma -> 0, B the actual edge image modulo 2.

The integral edge normalizer P={w:Omega(w,Lambda)=0} is primitive. Cusp incidence gives rank Lambda<=N-r, hence dim(P mod2)>=N+r and dim d(P mod2)>=r. Each such normalizing defect extends over the dual 2-cells and is a class of H^1(M;F_2). Do not replace P mod2 by the full mod-two kernel or assert every H^1 class is attained; the proved lower bound needs neither inference.

All-plus native character consistency is now derived, not separately assumed in this strict-angle class: an integral edge dependency has equal local triple counts k_t, and edge angles give sum k_t=2 sum n_e, cancelling the full (-1)^n i^(-sum k_t) phase. The unfilled edge-only index average F over the actual unsaturated Lambda converges on every algebraic input and has F(1)=1+positive A-order. For w in P, Psi_w(v)=F(W(-w)v) descends on the correct sides, is supported at d(w), kills the whole trace image when d(w)!=0, and detects W(w). Distinct nonzero defects give a diagonal evaluation matrix and at least 2^r-1 independent cokernel classes.

Next mode: ATTACK. Remove the strict-angle requirement by constructing an actual unfilled-target functional with nonzero unit, exact native character, and proved all-input convergence, or determine a sharp obstruction to such a replacement. The grading and integral rank argument are angle-independent. A nonzero filled trace is separate; the fixed figure-eight meridian kernel witness still forces every compatible map to vanish. Keep the actual image lattice and native signs, and do not enumerate manifolds or slopes.

Preceding minimal-meridian attack

ATTACK — the minimal meridian quotient is nonzero and its compatible trace is necessarily zero, 8 September 2026.

For the actual named full six-root trace T on the standard two-tetrahedron figure-eight triangulation, an explicit skein

kappa = b^(2) + A^-2 (mu^2 - 1)b + A^-4 mu^2 - 1

satisfies T(kappa)=0 and p(kappa)=A^-4, where p is the meridian-filled Kauffman bracket and b, mu have zero framing. Thus the minimal compatible quotient D_min=Q^6/T(ker p) equals Q^6/im(T), and every compatible meridian-filled trace is zero. A separate convergent parity functional proves Q^6/im(T) != 0. This settles the minimal-quotient gate, without assuming it equals the previous boundary-balanced target.

The kernel identity uses the published Panitch–Park native trace formulas, the correct two-cable framing identity b^(2)=A-8(B^(2)-1)+1, and only even-root tetrahedron multiples and a squared edge relation. The finite identity is verified in both Weyl and ordered monomials. The forward even-shape map into the full six-root target supplies native vanishing; no zero is lifted backward through a sign specialization. The retained full named meridian action on arbitrary inputs remains a proof dependency.

For the independent nonzero quotient proof, h=(1,1;-1,-1), a=(1,0;0,-1), w=(0,0;1,-1), eta(u)=u_z1+u_w0 mod 2. The actual skein basis images have eta=0. The all-plus unfilled index average F(v)=sum_n ell_0(W(nh)v) converges on every input with degree 8k^2+O(|k|). Psi(v)=F(W(-w)v) kills the full trace image by parity, while Psi(W(w))=F(1)=sum_k I_Delta(-k,-k)^2=1+positive A-order. Projection is used only in the legitimate nonzero-functional direction.

This supplies a nontrivial quotient for this meridian instance of the literal Panitch–Romo conjecture, with a necessarily zero map. It is not an unrestricted resolution or a nonzero filled invariant. The published PP article replaces its preprint’s unrestricted injectivity conjecture with Question 7.3; do not claim a current universal conjecture was refuted. This remains an ordinary research note with novelty unassessed and no external specialist review.

Former next gate, now settled in the strict-angle class by the face-defect packet above. The fixed native kernel witness still forbids restoring the meridian vacuum in any compatible quotient of this named trace.

Preceding central-meridian attack

ATTACK — full six-root meridian vacuum vanishes by a universal squared-moment certificate, 8 September 2026.

On the fixed published two-tetrahedron figure-eight datum, [1-l²]=0 in the full native six-root meridian cokernel while [1]!=0. The finite eight-tetrahedron-relation identity lifts through squared edge and meridian moments over the universal central involution ring. This is not backward inference from a specialized zero. Under the retained generic filling reconstruction, the particular boundary-balanced descended trace for meridian filling is zero, despite its nonzero target. This is a research note, not a Publication.

Former gate, now settled below. For the actual unfilled named trace T and meridian filling bracket p, the question was whether ker T ⊆ ker p. The minimal compatible linear quotient D=Q/T(ker p) has kernel p(ker T) on the filled skein module and exact sequence 0 → W/p(ker T) → D → Q/im T → 0. Do not assume the larger boundary-balanced relation space equals T(ker p). Do not seek further central-component detectors for the boundary-balanced meridian vacuum: it is now proved zero before specialization.

Panitch–Romo Conjecture 7.1 was checked literally on 8 September 2026: it requires a nontrivial quotient and commuting map, not explicitly a nonzero trace. The result is not a counterexample or unrestricted resolution. The existing native comparison/filling dependencies remain in force.

Retained preceding PUBLISH — Cheng anyon-proliferation manuscript, public edition dated 8 September 2026.

Joshua authorized public release and withdrew researcher-email outreach. The publication provides the original candidate argument as an online reading edition, PDF, and source archive at /publications/anyon-proliferation-identity. Changes are limited to distribution/review wording, a short scope paragraph, linked references, and equivalence typesetting. No new mathematical result or external review is claimed in this publication pass. The no-email policy has also been applied to all FPRD research automations, without changing their schedules or enabled states.

The preceding release is preserved. Joshua authorized routine public Site deployment as part of the continuous research loop without a new conversational permission gate. Preserve the shared significance-based publication ordering in app/lib/publication-order.ts: journal recency must not determine publication rank. The new meridian result stays outside Publications.

Preceding mathematical attack

ATTACK — exact cyclic transport covers every nonmeridional filled vacuum; a finite certificate identifies the meridian all-plus boundary.

On the fixed published two-tetrahedron figure-eight datum, the native six-root meridian cokernel is nonzero for every primitive slope. The native filled vacuum is nonzero for every nonmeridional slope q!=0, using the retained exact named-action/filling reconstruction. One exact full-root cyclic map supplies the missing convergence cone; no signs or target relations are changed. For the meridian, the transformed all-plus average has nonzero unit, and a finite quantum-algebra certificate kills the all-plus vacuum. Its formerly open central lift is now supplied by the current squared-moment packet. This remains a supporting research note.

Delivery history: the cyclic and phase-free notes were saved before this publication pass. This release preserves the complete current Site source and the earlier comparison publication; the Sites deployment record determines live delivery. Researcher outreach is disabled by the standing policy above.

Preceding phase-free result, retained

On the fixed two-tetrahedron figure-eight datum published by Dimofte–Garoufalidis (arXiv:1202.6268v2, Sections 2.6 and 4.6), the unchanged one-sided seed now proves H(1-l²)=1+strictly positive A-order for every primitive slope pμ+qλ with q!=0 and p/q>-4. There is no congruence restriction on p. The exact unit and longitude degrees have the same strictly copositive quadratic part and interchanged nonnegative linear coefficients. The only possible zero-degree longitude endpoint has nonintegral second seed coordinate (1/(2q),-1/(2q)), so is unsupported. Coefficient positivity, a new local solution, and changes to the native character are unnecessary. Through the retained native named-action and filling reconstruction, this detects the actual named filled vacuum. The proof is symbolic in the slope parameters, not a census.

The preceding opposite-sign theorem remains valid for p divisible by four, but that phase certificate is no longer a necessary gate for this datum. The new odd-numerator control (p,q)=(1,1) has genuinely negative native unit summands, while its filled vacuum starts 1+mathfrak-q+2 mathfrak-q²+4 mathfrak-q³, mathfrak-q=A⁴. This is a supporting research note, not a Publication. No external specialist review or priority claim is made. The actual odd peripheral value and unrestricted conjecture remain open.

Retained +4 application beyond the earlier ordinary-index detector: -2h+h_s=(-4,0;0,-2) has character one and every positive ray multiple has ordinary seed constant coefficient one. That average fails on the unit, while the one-sided vacuum starts 1+mathfrak-q+3 mathfrak-q²+6 mathfrak-q³. No strict filling-angle certificate exists for this datum and slope. No exclusion from every earlier regular-intersection or clean-excess criterion is claimed.

Active program and retained publication

Program: https://fprd-lab.joshgay.chatgpt.site/research-programs/dehn-filled-quantum-trace

Retained Publication: https://fprd-lab.joshgay.chatgpt.site/publications/ck-degree-zero-descent

Panitch–Romo Conjecture 7.1 remains open in its unrestricted scope. The declared full-six-root meridian vacuum is zero in a nonzero boundary-balanced target; the conjecture’s weaker nonzero-quotient condition is not contradicted. Universal vacuum nonvanishing is not claimed. No researcher outreach is permitted by the standing policy.

Chen and Kricker were separately contacted about the earlier comparison publication. No reply or endorsement is assumed. Keep the earlier publication and all established comparison hypotheses intact.

Current evidence packet

face-defect-attack/:

• FPRD-face-defect-quotient-2026-09-08.md: local half-face model; dual-graph grading; exact homology sequence; integral cusp-rank bound; native phase consistency; convergent image-annihilating functionals; nonzero minimal quotients for all fillings in the strict-angle class.
• verify-face-defect.py, exact-results.json: universal local parity/symplectic controls, native phase cancellation, primitive-lift warning, and the retained fixed figure-eight witness.
• internal-audit.md, source-ledger.md, README.md, build-release.py, hashes and face-defect-attack.zip: self-audit, sources and influence, and reproducibility.

Proof: https://fprd-lab.joshgay.chatgpt.site/evidence/dehn-filled-quantum-trace/face-defect-attack/FPRD-face-defect-quotient-2026-09-08.md

Current obligation: replace the strict-angle nonzero-unit functional while preserving the actual unfilled module and all-input descent. General H^1-surjectivity of the particular defect map is not claimed or needed.

The preceding minimal-trace-kernel-attack/:

• FPRD-minimal-meridian-quotient-2026-09-08.md: actual kernel skein, full-native squared-edge identity, cable framing, nonzero filled bracket, universal zero-map consequence, and separate nonzero quotient proof by unused parity.
• verify-minimal-quotient.py, kernel-certificate.json, exact-results.json: exact finite polynomial identity in two multiplication conventions, ribbon twist checks, image parity, and an index coefficient control with a proved tail.
• internal-audit.md, source-ledger.md, README.md, hashes and minimal-trace-kernel-attack.zip: self-audit, current published-source corrections, explicit dependencies and influence, reproducibility.

Proof: https://fprd-lab.joshgay.chatgpt.site/evidence/dehn-filled-quantum-trace/minimal-trace-kernel-attack/FPRD-minimal-meridian-quotient-2026-09-08.md

Former exact obligation: the intrinsic image-grading criterion and nonzero detector are now established for the generic strict-angle class. The fixed figure-eight minimal meridian quotient remains nonzero with zero compatible trace.

The preceding central-meridian-attack/:

• FPRD-central-meridian-vacuum-2026-09-08.md: full native meridian zero-vacuum theorem, even-moment normalization, explicit finite telescoping, formal central factorization, reversed orientation, retained nonzero unit, and minimal-compatible-quotient reduction.
• verify-central-meridian.py, central-certificate.json, exact-results.json: full finite certificate over independent formal central involutions, exact Weyl and ordered-monomial verification, and reverse transport. No sign assignments are evaluated.
• internal-audit.md, source-ledger.md, README.md, build-release.py, hashes and central-meridian-attack.zip: self-audit, explicit dependencies and influences, reproducibility and integrity.

Proof: https://fprd-lab.joshgay.chatgpt.site/evidence/dehn-filled-quantum-trace/central-meridian-attack/FPRD-central-meridian-vacuum-2026-09-08.md

The former kernel-containment obligation is closed by the current native kappa certificate and nonzero bracket.

The preceding cyclic-polarization-attack/ retains:

• FPRD-cyclic-polarization-and-meridian-boundary-2026-09-08.md: exact full-root cyclic map; all-input correct-sided transport; complementary degree gap; every nonmeridional vacuum; nonzero meridian unit; finite all-plus meridian vacuum certificate.
• verify-cyclic-polarization.py, exact-results.json: generic Laurent identities, an outside-cone original-native control, meridian series with certified tails, and full quantum-algebra verification of the certificate before and after inverse transport.
• meridian-certificate.json: eight tetrahedron left multipliers and explicit finite edge/filling right multipliers in the actual all-plus sector.
• internal-audit.md, source-ledger.md, README.md, build-release.py, manifests, archive: self-audit, source scope and influences, reproduction and integrity.

Prepared proof path: /evidence/dehn-filled-quantum-trace/cyclic-polarization-attack/FPRD-cyclic-polarization-and-meridian-boundary-2026-09-08.md. This and the preceding phase-free update are retained intact in the source used for the Cheng publication release.

The preceding phase-free-nahm-attack/ retains:

• FPRD-phase-free-Nahm-vacuum-2026-09-08.md: unchanged native setup and seed, all-input descent, general complete-square degree identity, exact fractional longitude defect, and phase-free symbolic family proof.
• internal-audit.md, source-ledger.md: self-audit, retained primary-source input and access limits, formal dependencies, and intellectual influences.
• verify-phase-free-nahm.py, exact-results.json: exact generic Laurent polynomial identities; a single odd-numerator raw Weyl-sum control with Gaussian integer coefficients, original relation sides, odd roots, and rigorous tail bounds.
• README.md, build-release.py, manifests, and phase-free-nahm-attack.zip: reproducibility and integrity.

Proof: https://fprd-lab.joshgay.chatgpt.site/evidence/dehn-filled-quantum-trace/phase-free-nahm-attack/FPRD-phase-free-Nahm-vacuum-2026-09-08.md

The preceding actual-nahm-attack/ retains:

• FPRD-actual-Nahm-vacuum-2026-09-08.md: opposite-phase lemma, actual native datum, symbolic cone and phase proof, explicit +4 Nahm formula, and old-index failure ray.
• internal-audit.md, source-ledger.md: scope, source convention checks, dependencies and intellectual influences; self-audit only.
• verify-actual-nahm.py, exact-results.json: symbolic identities, original-sided native relations, supported odd roots and exact coefficients with quadratic tail bounds.
• README.md, build-release.py, manifests, and actual-nahm-attack.zip: reproducibility and integrity.

Proof: https://fprd-lab.joshgay.chatgpt.site/evidence/dehn-filled-quantum-trace/actual-nahm-attack/FPRD-actual-Nahm-vacuum-2026-09-08.md

The preceding one-sided-seed-attack/ retains:

• FPRD-one-sided-Nahm-detector-2026-09-08.md: complete seed, exact cone theorem, original-sided descent, graph/Nahm translation, conditional vacuum theorem, and abstract example.
• internal-audit.md, source-ledger.md: self-audit, dependencies, and intellectual influences; no external or independent specialist audit is asserted.
• verify-one-sided-seed.py, exact-results.json: exact local identities and global example checks with quadratic tail certificates; full-root and unsaturated-image checks.
• README.md, build-release.py, SHA256SUMS.txt, one-sided-seed-attack.zip, ARCHIVE-SHA256.txt, archive-validation.txt: reproducibility and integrity.

Proof: https://fprd-lab.joshgay.chatgpt.site/evidence/dehn-filled-quantum-trace/one-sided-seed-attack/FPRD-one-sided-Nahm-detector-2026-09-08.md

The preceding peripheral-projection-attack/ remains the integral classification of supported same-complete-character translates, exact peripheral scalar, and odd projection. Every ordinary remaining-cusp boundary-skein insertion has vacuum value zero; the actual odd part remains unresolved.

The preceding balanced-pairing-attack/ packet remains the integral normal-fiber and exact phase bridge, sourced balanced vanishing theorem, and one-cusp stopping result.

The preceding actual-incidence-attack/ packet remains the two-angle and uniform large-slope theorem. longitude-difference-attack/ retains the general phase formula, correct-sided shifted-character family, and incidence-infeasible abstract zero-vacuum witness. global-meridian-index-attack/ retains all-input convergence. six-root-peripheral-attack/ retains the native named-action comparison. Earlier positive-geometric and clean-excess results retain their scopes.

Retained exact presentation

Use $k=\mathbb Q(i,A^{1/2})$, $K=\mathbb Q(i)((A^{1/2}))$, $B=iA$, $q=A^4$. In a central sector,

$$C_s=S\Big/\left(\sum_jS(z_j^{-2}+w_j^2-1) +\sum_e(c_eW(h_e)+A^2)S+(c_sW(h_s)+A^2)S\right).$$

This is a mixed-sided vector-space quotient, not a two-sided algebra quotient. The four native vertex values are $iA$; opposite-root signs remain after central elimination. Retain the exact additive native peripheral action, generic parameter restriction, Eulerian normalization, and CK source-reading caveat. Do not reopen peripheral signs.

The named operators satisfy

$$m=c_sW(h_s),\quad l=c_lW(b),\quad ml=A^2lm.$$

The intrinsic filling isomorphism sends the filled vacuum to $[1-l^2]$. It is not the cokernel unit.

Keep $L=\mathbb Z^{2N}$, $\Omega(u,v)=\sum_j(u_{wj}v_{zj}-u_{zj}v_{wj})$, and the unsaturated image $\Lambda=\sum_e\mathbb Zh_e+\mathbb Zh_s$. The exact character is $\chi(h_r)=-A^2/c_r$, including phase consistency. The retained ordinary-index seed vanishes off $2L$ and equals $\prod_jI_\Delta(d_j,-a_j)$ on $W(2a_1,2d_1,\ldots)$.

Current one-sided construction

Use grouped coordinates (z-vector,w-vector) for this section only. Put s(a)=sum a_j, q=A⁴. The new seed vanishes off 2L and when any a_j<0; otherwise

$$\ell_+(W(2a,2d))=\frac{(-1)^{s(a)}A^{2(\|a\|^2+s(a)-a\cdot d)}}{\prod_j(q)_{a_j}}.$$

It solves the plus recurrence required by Sf_j. Its minus-recurrence residual at e=0 is −q^(1−m)/(1−q), so the established two-recurrence rank-two classification does not restrict this seed. These local coefficients are classical Euler quantum-dilogarithm coefficients.

Define H(a)=sum_{lambda in Lambda} chi(lambda)^(-1) ell_+(W(lambda)a), keeping the actual unsaturated image. Let V=(Lambda tensor R)/2, C_+={(x,y) in V:x>=0}, Q=||x||²−x.y. All-input unordered convergence is equivalent to Q>0 on C_+ minus {0}. If the cone is zero, each input has finite support. A rational negative ray defeats the unit; a nonvertical rational null ray is defeated by a sufficiently shifted w-input, and a vertical null ray defeats one sign already at the unit. Rational null rays follow from the rational restricted Hessian kernel on the minimal face; no faces were enumerated.

For u=2(alpha,beta), lambda=2(x,y), supported only if lambda+u in 2L and x+alpha>=0, the exact summand order is

$$D_u=2Q+2s(x)-\nu_A\chi(\lambda)+4(\alpha-\beta)\cdot x +2(\|\alpha\|^2+s(\alpha)-\alpha\cdot\beta).$$

Strict cone positivity supplies quadratic coercivity on every affine support. Then H(a f_j)=0 and H(W(mu)a)=chi(mu)H(a) for every mu in Lambda, proving original-sided descent. No angle or shape point is required for this convergence theorem.

Projection to x is injective, so y=Tx on V_z; isotropy makes x.T x’ symmetric. The criterion is strict copositivity of I−T on V_z intersect the nonnegative orthant. In the full-rank one-cusp case V_z=R^N. This is not necessarily positive definiteness and not a claimed efficient decision procedure. Retain Gamma={x in Z^N:2(x,Tx) in Lambda}, not its saturation.

Sufficient vacuum certificate: the exact phase satisfies eta(2x,2Tx)=(-1)^s(x) on Gamma; the declared native longitude has b=(0,-t), nonzero integral t>=0, c_l=eta_l A^rho; rho>0. Unit summands then have positive coefficients, and the longitude-square/unit summand ratio is eta_l² A^(2rho)q(t.x). Hence H(1−l²) is nonzero. Under a native complete strict angular certificate and zero longitude holonomy, rho=p(R_l)=sum alpha_j t_j>0, so no separate filling angles are needed. The phase and pure-w form must be proved for the original data, not forced by redefinition.

For actual raw R=sum n_e E_e+n_s R_s with P(R) even, the exact phase condition is equivalently sum_j R_j1 congruent to 2(sum_e n_e+n_s) mod 4. This follows from eta(P(R))=(-1)^(sum n_e+n_s)i^(-p(R)) and 2s(x)=sum R_j1−p(R). Check it on the actual even image, not a saturation or a chosen new sign sector.

Historical applicability boundary: the original pure-w certificate had only a three-block abstract example, h=(2,2,0;0,0,0), chi(h)=A⁴, b=(0,0,0;−1,0,−2), c_l=iA, Q=2k². That example remains abstract. The new opposite-phase lemma below removes the pure-w requirement and supplies actual data; it does not retroactively make the old example a triangulation.

Retained opposite-phase actual Nahm theorem

Under all-input convergence and the native unit phase eta(2x,2y)=(-1)^sum(x), every unit summand has nonnegative coefficients. For any native longitude b, every longitude-square summand has constant phase epsilon_l=eta_l²(-1)^sum(b_z). If epsilon_l=-1, H(1-l²) is nonzero regardless of the difference of supports or scalar valuations. For a raw native longitude R_l, epsilon_l=(-1)^sum(R_l,unprimed). No signs are redefined.

Use grouped coordinates. Actual raw rows: E1=((2,0,1),(2,0,1)), E2=((0,2,1),(0,2,1)), R_mu=((0,-1,0),(1,0,0)), R_lambda=((-2,2,0),(0,0,0)). Collapsed rows are h=(2,2;1,1), -h, a=(1,1;1,0), d=(-4,0;-2,0), with scalar exponents 0,4,-1,2 respectively. Omega(a,d)=-2. These are the declared Dimofte–Garoufalidis labels and canonical longitude, not a differently twisted GY example. Do not replace the native B^(prime sum) scalar with the source’s SL2 eigenvalue lift.

For s=p mu+q lambda, h_s=(p-4q,p;p-2q,0), c_s=B^(-p+2q), Lambda=Z h+Z h_s. The two actual edge characters are exactly inverse. A complement r mu+t lambda satisfies qr-pt=1, b=(r-4t,r;r-2t,0), c_l=B^(-r+2t), Omega(h_s,b)=2.

For q!=0, y=T x with diagonal T_jj=1/2-p/(8q), off-diagonal p/(8q). The exact cone form is Q=(x1²+x2²)/2+p(x1-x2)²/(8q), strictly positive on the nonnegative cone iff p/q>-4. At equality the axes are null; below it they are negative. At q=0 a vertical ray prevents all-input convergence. These failures do not imply vacuum vanishing.

Even moment parity is e even and np even. The native phase criterion is 2e+n(p-2q-2)=0 mod4 on that full image, equivalent for primitive slopes to p in 4Z. Then q is odd, Gamma={x in 2Z²:x2-x1 in 2q Z}, and every complement has r odd, hence epsilon_l=-1. Therefore the actual filled vacuum survives for gcd(p,q)=1, q!=0, p in 4Z and p/q>-4. Sector detection detects the vacuum upstairs; no injectivity of sector projection is asserted.

For (p,q)=(4,1), choose b=a, c_l=-iA^-1. Writing x=(2u,2v), the positive vacuum formula is

H(1-l²)=sum_{u,v>=0} q^(2u²+2v²-2uv+u)/((q)(2u)(q)(2v)) + sum_{u,v>=0} q^(2u²+2v²-2uv+u+2v+1)/((q)(2u+1)(q)(2v+1)), q=A⁴.

It starts 1+q+3q²+6q³+10q⁴+17q⁵. The all-orders positivity proof, not finite coefficients, proves nonvanishing. The original-character ordinary-index ray -2h+h_s has seed I_Delta(0,2k)I_Delta(-k,0)=1+positive order for all k>0, so its unordered unit sum fails. The new seed excludes the ray. Strict filling angles would imply 2gamma1+alpha2=0. No triangulation independence, first-priority claim, or unrestricted conjecture resolution follows.

Phase-free degree-gap theorem

Keep every actual row, scalar and unsaturated image from the preceding section. Write v=(1-1/q,1+1/q); then nu_A chi(2x,2Tx)=v.x. For a general full-graph native insertion b=(alpha,beta), c_l=eta_l A^rho, put Delta_b=beta-T alpha and kappa_b=2 alpha.Delta_b+v.alpha+2rho. With h0=2*1-v and a=x+alpha, exact Weyl expansion gives

$$D_0(x)=2Q(x)+h_0\cdot x,\qquad D_l(a)=2Q(a)+(h_0-4\Delta_b)\cdot a+\kappa_b.$$

If Q is strictly copositive, both linear vectors are componentwise nonnegative, kappa_b=0, and a=0 is outside the longitude support, the unit has constant one while every longitude term has positive order. Delta_b nonintegral is sufficient to exclude that endpoint when b is integral. This is a polarization-relative algebraic criterion, not a claimed coordinate-independent topological object.

For the actual figure-eight data and every complement qr-pt=1,

$$\Delta_b=\left(\frac1{2q},-\frac1{2q}\right),\quad \kappa_b=0,$$

$$D_0(x)=2Q(x)+\left(1+\frac1q\right)x_1+\left(1-\frac1q\right)x_2, \quad D_l(a)=2Q(a)+\left(1-\frac1q\right)a_1+\left(1+\frac1q\right)a_2.$$

All nonzero supported x,a lie in the nonnegative integral orthant. For any nonzero integer q the linear coefficients are nonnegative, and Q>0 there when p/q>-4. The unit endpoint x=0 is exactly the image zero and contributes one. A longitude endpoint a=0 would require second seed coordinate Delta_b, which is not integral, and is therefore absent. Merely showing 2b not in Lambda is insufficient; nonzero pure-w seeds are allowed. Including 2rho from the native scalar is essential: deleting it leaves constant -2rho.

Thus H(1-l²)=1+strictly positive A-order for all gcd(p,q)=1, q!=0, p/q>-4, independently of phases. Positive order does not mean positive coefficients. The original opposite-sign certificate still fails outside p in 4Z; the new proof makes it unnecessary. No seed, sector, moment character or multiplier side is changed. All-input convergence comes before reindexing; this proof is not restricted to squared-root inputs.

At the single odd-numerator control (p,q)=(1,1), hs=(-3,1;-1,0), cs=iA, chi(hs)=iA, b=(1,1;1,0), cl=-iA^-1. The moment 4h+2hs=(2,10;2,4) has negative unit summand, despite positive low-order aggregated coefficients. The exact filled vacuum starts 1+mathfrak-q+2 mathfrak-q²+4 mathfrak-q³+6 mathfrak-q⁴+9 mathfrak-q⁵+14 mathfrak-q⁶. The computation uses one actual control and certified tails, not a slope census. No all-orders coefficient positivity is inferred.

Exact cyclic transport and the meridian boundary

Use the single blockwise map C(z,w)=(-w,z-w), Psi(W(u))=B^(sum u_w) W(Cu). It is an integral symplectic full-root automorphism of order three, and Psi(f_j)=-w_j^-2 f_j. The left unit is on the correct side. Raw triples cycle (R1,R2,R3)->(R2,R3,R1), so every edge/peripheral scalar transforms exactly with its new prime sum. Lambda’=C Lambda is unsaturated, and chi’(Cxi)=B^(-sum xi_w)chi(xi). Pull back the convergent transformed average by Psi to act on the original native target.

Put d=2q-p. The transformed data are h’=(-1,-1;1,1), both edge scalars -A², hs’=(d,0;-2q,p), cs’=1, b’=(-r+2t,0;-2t,r), cl’=1. Both edge characters are one; chi’(e h’+n hs’)=(-A²)^n. These scalar simplifications are outputs of the exact map, not choices of new signs.

For d!=0, e=-2x2, n=2(x1-x2)/d, and y=T’x with T’=[[-2q,p],[p,-2q]]/d. The cone form is

$$Q'(x)=(x_1+x_2)^2+\frac{2q}{d}(x_1-x_2)^2.$$

Its exact strict copositivity condition is 1+2q/d>0. At d=0 the actual moment span has a vertical ray. Under strict copositivity, all-input quadratic coercivity proves descent before reindexing. The exact native longitude defect is (-2/d,2/d), with zero complete-square constant. The unit and longitude degrees are

$$D'_0=2Q'+(2-4/d)x_1+(2+4/d)x_2,\quad D'_l=2Q'+(2+4/d)a_1+(2-4/d)a_2.$$

If |d|>2, the coefficients are positive and the only zero-coordinate longitude endpoint has nonintegral second seed coordinate. Hence the pulled-back original vacuum value is 1+positive A-order. For q!=0, use the preceding original polarization when p/q>=0; when p/q<0, q/d>0 and |d|=2|q|+|p|>=3, so the cyclic detector applies. This proves every nonmeridional native filled vacuum is nonzero on this fixed datum, without enumerating slopes or polarizations.

At the meridian (p,q)=(1,0), choose r=0,t=-1. Then T’x=(-x2,-x1), Q’=(x1+x2)^2, hs’=(-1,0;0,1), b’=(-2,0;2,0), cl’=1. The full average still converges. Its unit series is

$$H'(1)=\sum_{a,b\ge0}\frac{(-1)^{a+b}\mathfrak q^{(a+b)(a+b-1)/2+2a}}{(\mathfrak q)_a(\mathfrak q)_b}.$$

The longitude-square series interchanges a,b, so H’(1-l’^2)=0. Euler summation gives

$$H'(1)=(\mathfrak q)_\infty\sum_{a\ge1}\frac{(-1)^a\mathfrak q^{a(a-1)/2+2a}}{(\mathfrak q)_a(\mathfrak q)_{a-1}}=-\mathfrak q^2+\text{higher order}\ne0.$$

The reversed meridian orientation interchanges the degree coordinates and gives the same nonzero unit series. Thus every primitive slope has a nonzero native cokernel, even though meridian vacuum survival requires a separate argument.

Finite meridian sector certificate, stronger than detector failure: after quotienting by the left-acting moment characters W(h’)=1 and W(hs’)=-A², put e_(a,b)=[W(0,0;a,b)]. The integral full-root normal form gives relations e_(a+2,b)-e_(a,b)+A^(4-2a)e_(a,b-2)=0 and e_(a,b+2)-e_(a,b)+A^(-4-2b)e_(a-2,b)=0. In the even-even block V_(j,k)=e_(2j+2,2k-2), define

$$R_1(j,k)=V_{j+1,k}-V_{j,k}+\mathfrak q^{-j}V_{j,k-1},\quad R_2(j,k)=V_{j,k+1}-V_{j,k}+\mathfrak q^{-k}V_{j-1,k}.$$

The meridian vacuum is v=V_(-1,1)-V_(1,-1), and

$$v=\mathfrak q[R_1(1,1)-R_2(2,0)-R_1(1,0)+R_1(0,2)-R_2(1,1)+R_2(0,1)]-R_1(0,1)+R_2(1,0).$$

This has been lifted to 1-W(-4,0;4,0)=G+(W(h’)-1)U+(W(hs’)+A²)V in the full quantum Laurent algebra, with explicit finite U,V in meridian-certificate.json. The exact Weyl identity and its inverse cyclic image both check. Thus [1-l²]=0 in the actual all-plus meridian sector while [1]!=0. Exchanging tetrahedra gives the reversed-orientation statement modulo the edge relation.

The warning against lifting sector vanishing backward remains valid. The current packet instead proves the stronger finite identity 1-W(2b)=G+(W(2h)-1)U+(W(2s)-A⁴)V. Factoring squared moments with arbitrary central involutions proves zero directly in the full six-root module. Its target is still nonzero. This settles the former central-algebra gate without a sign-sector census and does not contradict the nonzero-target conjecture.

Retained two-angle vacuum theorem

For an actual triangulation, raw rows satisfy $P(R)_j=(R_{j1}-R_{j2},R_{j3}-R_{j2})$, $p(R)=\sum_jR_{j2}$. All-plus native scalars are $B^{p(R)}$. Edge/peripheral/tetrahedron normal classes have Euler values $-2,0,-1$, boundary values $0,2\gamma,0$, and zero mod-2 homology.

A complete strict certificate $\theta^0$ has tetrahedron sums one, edge pairings two, and all peripheral holonomies zero. A filling certificate $\theta^s$ additionally has meridian pairing two. The earlier theorem proves

$$\mathcal F(1-l^2)=1+\text{positive }A\text{-order}\ne0$$

when both certificates exist. Its normal longitude degree is $2(\delta(X)-\chi_{\rm top}(X))>0$; the filling certificate supplies original-character all-input convergence. A nonnegative filling certificate with the exact prior no-null-ray condition can replace strict filling positivity.

For a fixed cusp basis $R_a,R_d$ with pairing two, the deformation

$$\theta^{p,q}=\theta^0+\frac{pJR_d-qJR_a}{p^2+q^2}$$

is a filling certificate whenever it is positive. A uniform sufficient condition is

$$p^2+q^2>\max_j\frac{(JR_a)_j^2+(JR_d)_j^2}{(\theta_j^0)^2}.$$

Thus the actual named filled vacuum is nonzero for all sufficiently large primitive slopes on a triangulation carrying a complete strict certificate. This is a sufficient chamber, not an exceptional-slope classification. Do not enumerate its bounded complement.

Retained exact normal-index bridge

Let $n(\lambda)=\Omega(\lambda,b)/2$. The balanced convergence character is

$$\chi_0(\lambda)=\chi(\lambda)A^{-2n(\lambda)} =(-1)^nB^{2E-p(R)}, \quad R=\sum_en_eE_e+nR_s,\quad E=\sum_en_e.$$

Do not discard the residual phase $(-1)^n$. Write $F_0$ for the full unsaturated average with $\chi_0$. The complete strict certificate supplies all-input convergence, and

$$\mathcal G_{-b}(a)=F_0(W(-b)a)$$

descends to the original target.

For every integer $r,n$, the map

$$\lambda\longmapsto X=\max\bigl((R+rR_l)/2\bigr)\Delta-(R+rR_l)/2$$

is a bijection from $\{\lambda\in\Lambda:n(\lambda)=n,\ \lambda+rb\in2L\}$ to the minimum-zero integral normal classes with $\partial X=-rl-ns$. The maxima are blockwise. Surjectivity follows from the integer identity

$$2X+nR_s+rR_l\in\mathbb E+\mathbb T,$$

by GHHR Theorem 7.1; it does not saturate $\Lambda$. All mod-2 homology classes in that boundary fiber are included.

Let $\mathcal Z(\gamma)=\sum_aI_{\mathcal T}^a(\gamma)$, the total GHHR boundary index. Then

$$\boxed{F_0(l^r)=\sum_{n\in\mathbb Z}(-1)^nA^{rn}\mathcal Z(-rl-ns).}$$

Every summand has exact degree $2(\delta(X)-\chi_{\rm top}(X))+rn$, and leading coefficient $(-1)^{n-\chi_{\rm top}(X)}$. The prior all-input theorem justifies grouping the individual unordered-summable terms into these fibers.

Retained balanced-pairing failure theorem

GHHR’s published Remark 8.3, printed p. 312, states $I^a(-\gamma)=I^a(\gamma)$. The published source and arXiv version were freshly checked. This is a formal external dependency, not a new independent inversion proof.

After the exact bridge, legitimate reindexing $n\mapsto-n$ yields $F_0(l^{-r})=F_0(l^r)$. Therefore

$$\boxed{\mathcal G_{-b}(1-l^2)=0.}$$

This is not termwise root inversion: the seed itself is not inversion-invariant. It is not zero target evidence. In fact

$$\mathcal G_{-b}(l)=c_lF_0(1) =c_l(1+\text{positive }A\text{-order})\ne0.$$

Thus the actual meridian cokernel is nonzero from the complete certificate alone; the actual vacuum remains a separate question.

For one torus boundary, every monomial left translate certified by the same complete vector also kills the vacuum. Writing $w=v+b$, orthogonality forces $w\in\Lambda\otimes\mathbb R$. If $w\in\Lambda$, the functional is $\chi(w)A^{-\Omega(-b,w)/2}\mathcal G_{-b}$. If $w\notin\Lambda$, integral normal homology forces both vacuum parity supports to be empty. The same dichotomy holds in multiple cusps when $w\in\Lambda\otimes\mathbb R$, but does not exhaust additional other-cusp directions.

The all-plus vanishing theorem has not been extended to every central sign sector. No all-purpose pairing has been excluded.

Preserved failures and limits

The preceding abstract strict-filling-angle example has a nonzero cokernel unit and an algebraically zero specified vacuum, but violates actual nonnegative edge incidence and lacks a complete nonnegative certificate. It remains an abstract obstruction only.

No target-wide nonvanishing or zero result follows from a scalar index alone. No new surgery invariant or identification with the Celoria–Hodgson–Rubinstein Dehn-filling index is asserted. Duan–Garoufalidis–Gu’s general factorization-based symmetry route is conjectural and was not used as proof. No researcher contact is authorized by this ordinary research note.

New multiple-cusp classification and odd projection

Let $\Gamma_o=\bigoplus_{T\ne T_0}H_1(T;\mathbb Z)$. For $w=v+b\in L\cap\Lambda^\perp$, nonempty vacuum seed support is equivalent to

$$w=\lambda_0+w_\gamma,\quad\lambda_0\in\Lambda,\quad \gamma\in\Gamma_{\rm adm}:= \{\gamma:\iota_*(\gamma-l)\in\operatorname{span}_{\mathbb F_2}\{\iota_*s\}\}.$$

The representation is unique for fixed additive native rows. The integral proof chooses an integral normal lift V of w and forms the minimum-zero class X from (V−R_l+R)/2. Integrality of its boundary forces the other-cusp and meridian coefficients of ∂V to be even; the doubled normal identity forces [V]₂=0. GHHR’s integer kernel theorem then gives the decomposition. No saturation or assumed half-integral peripheral lift is used. Unsupported translates vanish on the vacuum only, not necessarily on all other inputs.

For $n_0=n(\lambda_0)$, the exact all-input scalar is

$$\mathcal G_{w-b} =\chi(\lambda_0)A^{-n_0}c_\gamma^{-1}\mathcal G[E_\gamma], \quad\mathcal G[P](a)=F_0(W(-b)Pa),\quad E_\gamma=c_\gamma W(w_\gamma).$$

On the other-cusp quantum torus define $\sigma(E_\gamma)=E_{-\gamma}$, fixing A and coefficients. This is an algebra automorphism: it preserves product order and Weyl factors. For the original-target vacuum functional

$$\mathscr B(P)=\mathcal G[P](1-l^2),$$

the exact retained normal bridge gives

$$\mathscr B(E_\gamma)=c_l\sum_n(-1)^nA^n [\mathcal Z(\gamma-l-ns)-\mathcal Z(-\gamma-l-ns)], \qquad \mathscr B\circ\sigma=-\mathscr B.$$

Consequently $\mathscr B(P)=\mathscr B((P-\sigma P)/2)$. Frohman–Gelca’s torus-skein image is generated by $E_\gamma+E_{-\gamma}$ on each cusp, hence every ordinary remaining-cusp boundary-skein insertion has vacuum value zero. This covers arbitrary noncommuting products without assuming independent-cusp symmetry of Z. It does not extend σ to the whole mixed-sided target or assert zero value on all target inputs.

The formal boundary witnesses Z₀,Z₁ have common unit normalization, simultaneous inversion, admissible Lagrangian mod-2 support, finite summability and positive nonunit order, but the oriented test is respectively 0 and A⁴. They do not satisfy any claimed actual tetrahedron presentation and are not triangulation counterexamples. They prove only that the listed boundary constraints cannot decide the odd value.

Next mode and decisive question

ATTACK — identify the intrinsic unused grading and its detecting functional.

The fixed figure-eight minimal meridian quotient is Q^6/im(T) != 0, while every compatible filled trace is zero. The witness kappa proves this is forced by the actual unfilled trace kernel, not an artifact of the larger balanced relation space. The new question is structural: for a general actual triangulation, what part of the native root grading is inaccessible to closed skeins, and when does that part survive the mixed-sided relations? Derive the image grading and a convergent nonzero pairing. Determine precisely when the literal nonzero-quotient condition can be met by a zero filled map. Do not replace this by a triangulation or slope census.

Keep all paused and withdrawn directions paused. The standing no-researcher-message policy and significance-based publication ordering remain in force.