Internal proof audit
8 September 2026. This is a self-audit of the present attack, not an independent specialist review. No external validation is claimed.
Contents
Integral classification
The strongest point needing examination is the implication from nonempty seed support to membership in the unsaturated edge-plus-meridian lattice plus a native other-cusp row. The proof does not infer this from symplectic rank. It constructs the integral class X in equation (14), obtains even boundary coefficients from the free integral cusp decomposition, obtains [V]₂=0 from equation (16), and only then applies GHHR’s integral kernel theorem. This supplies integer edge coefficients. The converse uses the image part of that same theorem and the pair long exact sequence. Both directions require actual triangulation normal data.
An integral root lift V need not initially have even peripheral boundary or zero mod-2 homology. Those properties are conclusions conditional on vacuum support. Dropping that condition would invalidate the theorem. Unsupported translates are asserted to vanish on the two vacuum monomials only, not on every algebra input.
The other-cusp class is unique: equality modulo Λ makes the difference of its doubled boundary classes zero. No real-span quotient is used to erase torsion. Γ_adm can be empty and is a parity coset condition, not necessarily a subgroup.
Scalar and sidedness
For u=wγ−b and λ₀∈Λ, Ω(λ₀,u)=−2n₀. Moving W(λ₀) to the left gives A{n₀}χ₀(λ₀)=A{−n₀}χ(λ₀). The further coefficient cγ^{-1} in equation (18) is required because the insertion is the native Eγ, not its bare W-monomial. This derivation retains χ’s phase.
Every other-cusp insertion commutes with the original moment operators and with l. The definition =F₀(W(−b)Pa) is evaluated in the ambient algebra. Correct-sided annihilation is inherited from the previously proved translated functional, or checked termwise for tetrahedron relations and by moment reindexing for left moments. No multiplication of quotient classes and no two-sided ideal is assumed.
Inversion and noncommutativity
σ fixes A and sends Eγ to E−γ. Since the Weyl form is unchanged when both arguments are negated, σ preserves product order. Thus an arbitrary word of invariant curve sums is invariant even if its factors do not commute. The exact verifier includes such a noncommuting word; treating it as a commutative polynomial would lose Weyl powers.
Only GHHR’s simultaneous boundary inversion is used. It first converts the r=−1 normal sum by n↦−n. The resulting difference is odd in γ by inspection. Independent-cusp symmetry is neither proved nor assumed. The theorem says =0 on the fixed subalgebra; it does not say G[P]=0 on the target.
The source boundary-inversion identity is an explicit external dependency. The companion code does not prove it. GY’s discussion of descendant symmetry is not used as a replacement proof, and its q↔︎q^{-1} chirality discussion is not treated as an equality of the same formal Laurent expansions.
Formal witnesses
The two boundary coefficient systems in Section 4 exactly satisfy the constraints claimed there. Their weighted sums are evaluated by solving for n in finite support, not by cutting off an infinite sum. The parity kernel is isotropic of half dimension over F₂. Their oriented values differ.
Neither witness is asserted to satisfy actual tetrahedron recurrences, to be an actual normal index, or to have a topological realization. Therefore they establish only insufficiency of the listed boundary constraints. They do not disprove independent-cusp symmetry or Panitch–Romo Conjecture 7.1.
Scope verdict
The classification and boundary-skein non-detection arguments are complete conditional on the stated retained convergence/native dictionary and cited integral normal-index inputs. The separate-cusp oriented value remains unproved. The result is a supporting research note; it does not meet the publication threshold. The prior two-angle and positive-geometric nonvanishing results retain their original hypotheses.