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Internal audit: the minimal meridian quotient

8 September 2026. This is a self-audit, not an independent agent or external specialist review.

Contents

Outcome

The finite native kernel certificate, its nonzero filled bracket, and the separate nonzero cokernel detector pass the checks described below. The theorem retains the published trace formulas and the preceding native named-action theorem as explicit mathematical dependencies. It is a result for one fixed figure-eight triangulation at generic parameter.

Vulnerable steps

  1. A projected zero is insufficient. The discovery calculation uses the identified GY formulas. The final certificate instead starts with the published PP full-root formulas and uses only even root monomials, tetrahedron left multiples, and the squared edge with a right multiplier. Panitch–Park’s forward map from the even shape module to the full root module is enough; its injectivity is not needed.
  2. Relation sides cannot be exchanged silently. The dictionary from PP to GY reverses multiplication. It takes PP right tetrahedron multiples to GY left multiples, and PP left edge multiples to GY right multiples. The final identity is checked in the unquotiented Laurent torus using both Weyl and ordered-monomial multiplication.
  3. A 2-cable framing correction is not a squared 1-cable correction. The correct identity is b(2)=A-8(B^(2)-1)+1. A two-strand Temperley–Lieb calculation verifies the twist eigenvalues A^8 and 1. The incorrect naive factor A^-6 is not used.
  4. The current source matters. The published PP p. 2310 corrects the final simplification of the preprint’s cable formula. We use the first nine-state expansion, which agrees between the two versions and matches GY equation (100) after the correct framing change. We do not use the preprint’s erroneous assertion that the result is a scalar polynomial in its one-cable representative squared.
  5. The original framed skeins matter. PP Figures 12–13 and its explicit framing change were checked against GY Figure 11. The zero-framed oval and small zero-framed meridian are split unknots after meridian filling. Their bracket gives A^-4 for kappa. This is independent evidence that kappa is not zero in the unfilled skein module.
  6. Multiplication by the meridian trace is an operator statement. The argument needs the prior named peripheral action on arbitrary skein inputs, not merely the source’s value on the empty skein. That prior theorem is explicitly retained. Its squared operator loses no residual branch character, because each such character is an involution.
  7. The minimal quotient need not equal a previously balanced construction. The equality D_min=Q/im(T) is proved anew from kappa. No equality with the boundary-balanced relation space is assumed.
  8. A nonzero unit in another target does not prove this cokernel nonzero. The new functional Psi separately annihilates im(T) and detects W(0,0;1,-1). It is defined on the unfilled target before any filling relation. The basis theorem and named action show that every actual traced skein has even eta degree.
  9. Index convergence is proved on all inputs. For a fixed input the supported terms have degree 8k^2+O(|k|), including the Weyl prefactor, by the published exact tetrahedron-index degree formula. Reindexing is justified. The exact unit degree is 8k^2+4|k|. The finite series control is not the proof of convergence.
  10. Direction of the second projection is valid. A nonzero functional on the all-plus quotient pulls back to the full target. Unlike a zero statement, nonzero evaluation on a lifted element proves nonzero in the original cokernel.
  11. Published conjecture scope is not inferred from the old preprint. The journal edition asks in Question 7.3 which triangulations have injective trace. It no longer states the preprint’s unrestricted injectivity conjecture. The note does not claim a new refutation of a current universal conjecture.

Remaining obligations

The current fixed-datum minimal quotient question is settled under the declared named-action dependency. A general theorem would need a presentation-independent interpretation of the unused parity character, an image-grading statement for arbitrary triangulations, and an all-input convergent detector outside that image. These are not established here. No claim of priority or of external confirmation is made.

Reproduction

Run python3 verify-minimal-quotient.py. kernel-certificate.json records the actual polynomial coefficients, and exact-results.json records verification and the finite index control. The verifier is exact and uses only Python’s standard library. No triangulation, slope, or central-sign census is performed.