Internal self-audit: face-defect quotient
8 September 2026. This is a self-audit, not independent or external specialist review.
Contents
Finding
The written argument proves a lower bound of 2^r-1 for the dimension of the full native trace cokernel over the generic field on a connected ideal triangulation with r torus cusps and a strict angle structure. Consequently every minimal compatible filling quotient is nonzero. A common quotient with zero filled trace satisfies the weak commuting-square condition. The unrestricted conjecture and existence of a nonzero filled trace remain outside this result.
Checks of the actual proof obligations
| Obligation | Resolution |
|---|---|
| Native full-root target rather than a replacement presentation | Use the retained forward all-plus central projection. Nonzero functionals pull back; no specialized zero is lifted backward. |
| Entire trace image, not a few sample skeins | Published PP Theorem 5.6 / Proposition 5.7 give even parity for every face-suspension trace. The explicit half-face map transports that condition to degree zero after native vertex elimination. |
| Is the defect well-defined after opposite identification? | Opposite bare-edge indicators differ by the all-ones vector at one tetrahedron; its glued-face change is a graph coboundary. |
| Repeated faces, loops, and repeated edge incidences | Half-faces are separate occurrences; both halves of a loop are counted. Every face contributes three incidences to the sum of dual 2-cell boundaries, even if labels repeat. |
| Dimension of the degree-zero space | Its exact presentation is graph cycles modulo the nonzero all-edge cycle, giving dimension N; its local pairing counts every face twice, so it is Lagrangian. |
| Intrinsic meaning | An integral normalizer has zero pairing with each edge row; its graph defect therefore vanishes on the dual 2-cell boundaries and is a class of H^1(M;F_2). General graph defects need not extend over M. |
| Integral lift, including 2-torsion issues | P is the primitive integral kernel of the edge-pairing matrix. Its reduction has its integral rank. The rational cusp-incidence injection supplies rank P >= N+r, hence at least r independent defect degrees. No arbitrary mod-two lift is assumed. |
| Native character phases | A dependency gives equal local triple counts k_t. Strict edge angles imply sum k_t=2 sum n_e. The exact scalar (-1)^(sum n_e) i^(-sum k_t) is then one. |
| Changed lattice or signs? | The average uses the original edge image Lambda and the original all-plus native character. Saturating Lambda or modifying a meridian is not part of the argument. There are no filling relations in this detector. |
| All-input convergence | The Dickson-minimal-vector estimate bounds each translate below by 2 epsilon times its nonnegative displacement. It applies to all finite Laurent inputs, not only the unit. |
| Relation sides | Left multiplication by W(-w) preserves tetrahedron left multiples; exact integral commutation with all edge monomials preserves the edge right multiples and their characters. |
| Nonzero classes and independence | The translated functionals form a diagonal evaluation matrix with nonzero entry F(1), separated by distinct defect degrees. |
| Generic versus coefficient-ring assertion | The theorem is over k=Q(i,A^(1/2)). It does not claim the unrestricted coefficient-ring formulation of Panitch–Romo. |
| General versus fixed zero-trace claim | A zero map can always be chosen on the common quotient. Only the prior fixed figure-eight meridian kernel witness forces every compatible trace to be zero. |
Unused stronger inference
The GHHR/Neumann integral normal-homology isomorphism suggests a possible description of all attainable cohomological degrees, but an abstract isomorphism alone does not identify this particular half-face map. The note does not claim d(bar P)=H^1(M;F_2). Its weaker lower bound is proved without that identification.
Validation limit
The exact program checks universal local vector identities and the previously fixed figure-eight witness. It does not prove convergence by truncation, sample triangulations to infer a theorem, or claim a computer proof of the general topological argument. No manifold or slope census was performed.
Remaining mathematical gate
Replace the strict-angle nonzero-unit detector by an equally rigorous construction when strict angles are unavailable. The grading and integral-rank argument require no angles. A nonzero filled invariant remains a separate question.