Internal hostile audit: the local collar square

8 September 2026. An internal adversarial audit, not an external referee report or independent expert review.

Verdict

The completed local dictionary and named six-root action survive the checks below, with the geometric source inputs retained explicitly. The appropriate status is a supporting ATTACK result. Do not make a new Publication merely because an internal gate has been closed. Unrestricted Dehn-filling nonvanishing is still open.

1. Preserved native theorem

The preserved script was re-run without altering its arithmetic. Its 20 exact identities pass. Its output was not treated as proof of the geometry.

The whole GY-index Appendix A was re-read. Lemma A.3 replaces a substandard arc by a parallel arc at the same bare edge. Standard-off-diagonal annihilation already holds in GY (60), before opposite-root identification. GY Lemma 4.22 supplies prequotient standard-arc spanning. Appendix A carries an arbitrary extra skein, not just the vacuum. The Type II counit factor remains A^-1. The six-root form is exactly the pullback of the collapsed form.

The two-generator extension uses A^2-A-2 in the denominator. Its generic limitation is unchanged. GY’s admissible-diagram construction and local geometric lemmas remain substantive dependencies, not independently re-proved here. The native theorem is a source-proof extension.

2. Did a cup ratio get mistaken for a functor?

No. PP1 Remark 3.5 supplies the actual geometric conversion of vector-framed pictures to symmetric ribbon pictures. Both operations were visually inspected on printed p. 2265. The cup conversion is h^-1 and the cap conversion is h, with h=i A^(3/2). Both snake identities are exact. Using the same power on both operations leaves -A^3 and fails the audit.

Crossing and loop compatibility is imported from the primary geometric construction, not inferred from a matrix. The standard lantern arc retains its framing line under n -> -n; the cup-picture factor must not be attached as a uniform endpoint scalar to every arc.

3. Wrong self-duality: a genuine failed route

PP1 Proposition 3.10 gives d([L,s])=v^(-sum s)[L,-s] on unrescaled generators. Using it on both sides of a same-state cut creates v^(-2s); the standard annular counit on d(a_++) is v^-2 instead of 1. It also changes the fixed corner character.

The proof uses the unweighted marking-reversal symmetry of PP1 Remark 3.4. The weighted map remains appropriate for PP’s own rescaled balanced actions; the failure here does not refute that theorem. Reciprocal weights on paired sides cancel. The checker verifies this with a formal Laurent gauge variable.

4. Cut contraction versus cup/cap

The cut tensor is e_+ tensor e_+ + e_- tensor e_-; the cup/cap matrices are off-diagonal. The unweighted flip preserves the cut tensor exactly. The paired picture changes have inverse duality factors because the same framed ribbon is cut and rejoined. The proof applies this to arbitrary tensor contractions, not a list of normal-curve configurations.

5. Was the counit redefined?

No. GY Corollary 4.19 and its standard parallel-arc spanning description give the source counit delta_(mu,nu). The target is the same untwisted standard ribbon with both states flipped, whose usual matrix-element counit is delta_(-mu,-nu). Equality holds on the spanning module. Counit uniqueness is an alternative check after splitting compatibility.

The nonstandard Type II closing diagram is not assigned counit 1. Its height exchange remains part of the calculation and produces A^-1.

6. Hidden mirror or parameter inversion?

The interval direction and induced surface orientation both reverse; the ambient oriented neighborhood is fixed. This is CK’s explicit opposite collar. The ribbon does not move to a mirror manifold. The state flip compares two differently marked/cooriented collars, not one fixed surface algebra whose closed curves are purportedly pointwise fixed by an anti-automorphism.

The cyclic triangle form changes sign. Changing both state degrees alone does not change a bilinear pairing. Combining cyclic reversal and product reversal gives exactly the cocycle identity (19) in the note. All these distinctions are necessary.

7. Is the odd bare-edge dictionary arbitrary?

It is anchored at a common embedded ideal vertex and prime edge in the two opposite cyclic label conventions. After permitted cyclic choices it is z_G -> z’’_C, z’_G -> z’_C, z’’_G -> z_C, and the same on opposite partners. This names the same actual edges in two conventions; it is not an odd geometric symmetry of a tetrahedron. Every face occurrence and edge row must be carried through the dictionary.

The oriented labeling and collar diagrams were visually inspected. P^t B6 P=-B6 is a universal matrix identity. The source Lagrangian, four vertex monomials and all actual edge rows are carried with their correct sides. Independently stored label anchors require their actual cyclic relabelings; matching printed letters is not sufficient.

8. Is diagonal agreement enough?

Not alone. Marking reversal is defined on every state, and both mixed-state dualities are checked before the off-diagonal quotient. Source corner values –:v^-1 and ++:v become target ++:v^-1 and –:v. The good mixed corner remains 1 and the bad corner remains 0 under simultaneous marking/viewpoint reversal and state flip. Product reversal sends each corner relation and arbitrary left multiplier to the correct right-sided relation. Standard-off-diagonal submodules also change side.

Only then do diagonal standard-arc values and the spanning theorem determine the local target map. This is the substantive improvement over the earlier purely algebraic dictionary.

9. Actual named closed link

There are no states on the closed input to flip. The picture conversion keeps the framed ribbon; reversing the normal vector preserves its ribbon line. Splitting uses the same physical circles and heights, with state assignments reindexed by s -> -s. The two maps agree before descending from the dual-surface skein to the three-manifold skein. CK Theorem 4.9 is used after this raw local comparison.

The operator identity retains every input skein. It is not vacuum-only and is not inferred from the identified quotient.

10. Sides, branches and central signs

The shape dictionary is anti and induces a module bijection. Native left multiplication becomes CK right multiplication. The target sum monomial is A^-1 M_C L_C. Right operators reverse composition and recover the source operator sign.

Original PP1 requires the prior Eulerian scaling x_e -> i epsilon_e X_e, with equal opposite-pair signs. The new formula contains i^(-sum u) epsilon^(Pu); its inverse term has the reciprocal coefficient. That branch theorem and its auxiliary-extension descent are inherited dependencies, not consequences of the endpoint matrices.

The existing CK comparison uses the disclosed right-tetrahedron/left-edge Laurent presentation. No author-confirmed correction to its contrary isolated word is claimed. All central sigma_T factors are retained; no opposite-root quotient is inverted.

11. Nonvanishing remains separate

The generic boundary theorem gives a filled target, which may be zero. The meridian-cokernel theorem reduces nonvanishing to nonsurjectivity of m+A^2. Neither unrestricted nonsurjectivity nor survival of every global sign sector is proved here.

The positive-geometric consequence retains all hypotheses of the earlier theorem. A boundary-linear surjection from the full six-root target to the identified target induces a surjection after filling; a nonzero identified filled vacuum detects the full one. This does not reverse the quotient.

12. Credit and status

The newly used marking and picture conversions belong to PP1. CK already supplies its adjusted comparison, and GY performs nearly all native diagrammatics. This note composes those facts into the raw dictionary and signed action. The bounded search certifies neither novelty nor priority or publication-level magnitude.

Chen and Kricker were separately contacted about the previous comparison publication, with Joshua Gay copied. This continuation made no new contact and assumes no reply or endorsement. No external specialist review has occurred.

The next attack is an intrinsic nonvanishing detector for the actual generic meridian cokernel beyond the regular-intersection/positive chamber, or a decisive structural obstruction. Keep Panitch–Romo Conjecture 7.1 open. Do not restart the generic sum-slope sign calculation or replace the structural question by a census.