Hostile publication audit: CK comparison injectivity

8 September 2026. Verdict: PUBLISH the stated mixed-sided theorem, with the source-reading caveat and AI/internal-review labels. No mathematical gap was found in that theorem. Do not promote any original six-root peripheral-action or Dehn-filling claim.

This is a fresh internal audit, not an external referee report. The candidate, its prior internal audit and its verifier were treated as claims to test, not proof authorities. Fresh primary PDFs were downloaded; a new ordered-PBW engine independently reconstructed the finite algebraic identities. No source-author contact or configuration enumeration was used.

1. Target substitution and the exact conjecture

Attempted failure: perhaps the candidate proves an isomorphism with a newly defined target rather than the actual PP image.

Finding: CK Definition 5.22(iii), pp. 75-76, explicitly takes the image of the six-root torus in the ambient quotient (5.6). Proposition 5.25, pp. 77-80, induces the comparison by matching each of the six generators. Conjecture 5.27, p. 80, asserts injectivity of that surjective R-module homomorphism. The freshly downloaded primary PDF page was rendered and inspected. The theorem’s kernel is exactly (S intersect Rambient)/J. This is the kernel the proof computes.

2. Sidedness: an important source caveat, not a silent correction

Attempted failure: interpreting a left-generated triangle ideal and then right-extending it could introduce extra ambient relations.

Finding: this danger is real. CK Definition 5.7(i), p. 62, defines its tetrahedron ideal on the right and then calls its global extension a left ideal. Definition 5.11 explicitly recalls a right ideal, and the proof of Proposition 5.25 multiplies the trinomial by a unit on the right. PP1 independently specifies V_+ U + U V_-, with tetrahedron generators right-generated and edge generators left-generated (Section 5.2, especially Remark 5.9).

The publication states the right-tetrahedron/left-edge reading as part of its theorem. It does not assert that all possible literal readings of CK’s inconsistent sentence yield isomorphic objects, and no author-confirmed erratum is claimed.

An exact countercheck makes this more than a typographic nicety. Take the shape vertex monomial v, scalar v0, and g=G_(z,y'') in the image of the triangle algebra. Then g v g^-1 = A v, so

$$g(v-v_0)g^{-1}-A(v-v_0)=(A-1)v_0.$$

Thus left multiplication by the triangle-image algebra, followed by arbitrary right extension, can produce a nonzero scalar. Over Q(i,A^(1/2)) the scalar is invertible, collapsing that altered ambient quotient. This is checked independently by the new verifier. It does not invalidate the standard PP target; it shows why the presentation convention must remain visible.

Additional source shorthand: CK Definition 5.22(ii) lists the six generators but omits their inverse symbols. Remark 5.23, equation (5.7), explicitly gives the Laurent quantum torus. The theorem uses that explicit Laurent presentation. A polynomial-only interpretation would not even be the source comparison torus.

3. Ambient basis and lattice saturation

Attempted failure: a hidden face quotient, duplicated bare occurrence, non-saturated shape lattice, or degenerate form destroys U_0=S.

Finding: CK Definition 5.17 makes each face suspension the tensor product of two distinct face-cone quantum tori. Regrouping by tetrahedra does not identify any variables. Each bare edge uses exactly its two independent half-edge coordinates, disjoint from every other pair. The degree map is the difference of the two exponents in each pair. Its kernel is exactly the integral shape lattice and it is surjective. The twisted-group-algebra basis works over arbitrary commutative coefficient rings; a radical in the skew form does not produce linear dependence. Self-pairings of distinct face occurrences do not change this argument.

4. Is the projection used too strongly?

Attempted failure: the proof treats degree-zero projection as an algebra map, or assumes a one-sided quotient is a bimodule.

Finding: only the S-bimodule identity on P0:U -> S is used. The quotient need not be an S-bimodule. Applying P0 directly to a finite expression sum a_i u_i + sum v_j b_j is legitimate. The new verifier checks a deliberate nonmultiplicativity example: P0(h)=P0(h^-1)=0, but P0(h h^-1)=1. The proof does not invoke multiplicativity, averaging or faithful flatness of S inside U.

5. Raw local generators and right normalization

Attempted failure: the raw trinomial is not in S; a calculation modulo relations only proves one inclusion; a Weyl scalar or the side of the unit is wrong.

Finding: the raw trinomial is homogeneous of a nonzero quotient-lattice degree. Multiplication by the ordered RIGHT unit u_(z'',h3)^-1 u_(z,h3) moves all its terms to degree zero. The new verifier uses ordered PBW monomials rather than the previous Weyl multiplication engine. It checks the raw triangle images, the exact vertex image, the three-term intermediate expression, the correction identity, and explicit reverse witnesses. All identities hold before quotienting.

The key source calculation is already on CK p. 79. The manuscript credits it and identifies the contribution as the equal-right-ideal interpretation plus the full shape-degree projection.

6. Whole tetrahedron list, not one convenient relation

Attempted failure: one base-vertex calculation does not cover other source relations, or orientation-preserving symmetries change the parameter.

Finding: bare-edge labels are assigned to 01,02,03,23,13,12. The verifier checks all twelve even permutations of the four vertices, explicitly verifies preservation of the ambient 12-by-12 form, and checks both identities for every resulting pair. This fixed finite symmetry-group certificate is not a search through triangulations or skein configurations. The identity plus the three double transpositions already reaches every vertex.

The cyclic Lagrangian witness is checked in all three cyclic forms. Opposite-square elimination follows from explicit central Laurent identities; neither 2 nor A-A^-1 is divided by. Consequently all four vertex pairs, with allowed cyclic label choices, generate exactly the extension of the CK right ideal.

7. Global edge relations

Attempted failure: repeated tetrahedra or face suspensions introduce an unrecorded edge scalar, or only independent edges were retained.

Finding: every edge is retained. In the image of a face bigon, opposite face orientations cancel the skew contributions. The three bigon images within a face suspension commute pairwise, including when one edge visits that face suspension more than once. Their total exponent is exactly the sum of the shape exponents around the edge. Thus the image is the Weyl edge monomial with coefficient 1. The value is -A^2 in CK’s normalization. The exact inverse-edge redundancy is on the left. Ambient centrality of edge monomials is neither needed nor asserted.

The verifier checks the universal paired-face skew cancellation and a noncentral illustrative inverse-edge identity. The arbitrary-incidence argument is the written proof and CK’s independent source calculation, not a finite manifold example.

8. Coefficient ring and specialization

Attempted failure: the proof hides generic denominators, uses characteristic zero, or infers an intersection identity by base change.

Finding: the core proof uses only the source’s units t=A^(1/2) and v0=(-A^2)^(1/2). It is proved directly over every such R. The universal ring can be written Z[i,t^+-1] by setting i=v0/A; no numerical complex evaluation is used by either verifier. Sign idempotents and local nonvanishing are separately restricted to k=Q(i,t). The optional original-PP face scaling adjoins s^2=-i and descends the shape map faithfully; it is not needed for the core integral theorem.

9. Named-trace dependency and branch conventions

Attempted failure: a target isomorphism is mistaken for a theorem about named trace maps, or raw GY/PP conventions are silently identified.

Finding: CK Theorems 4.9 and 5.34 supply the named maps and their compatibility. The new theorem only makes the already supplied comparison invertible. CK’s Remark 4.2 and Section 4.3 make the product/orientation/label and extended-target distinctions explicit. GY’s published Remark 5.1 supplies the extended construction’s scope.

CK Lemma 5.18 uses PP2’s first normalized branch (cT,cB)=(-i/A,1). PP1 used (1/A,-i). The publication includes the Eulerian half-face scaling and its faithful descent to a shape map over Q(i,t), but claims no canonical/Pachner-natural branch choice. This auxiliary transport does not identify the raw GY-index peripheral coordinate dictionary.

The audit checks source statements and their relevant constructions, not every geometric step of the 87-page CK paper. Reliance on these primary theorems is explicit.

10. Opposite-root quotient scope

Attempted failure: the theorem discards minus sectors, inverts an identified quotient, or uses centrality in the wrong algebra.

Finding: all six roots are retained throughout the comparison. Opposite ratios are central in S, preserve J, and act on the shape image through the intersection theorem. A common quotient is therefore legitimate on both sides. No two-sided quotient of U is used.

After only vertex relations, the local algebra is a rank-two torus tensored with the group algebra of one involution. The explicit elimination map and its inverse verify the presentation. The remaining Lagrangian is a nonmonomial, hence generates a proper right ideal over the stated field by the Newton-polytope argument. The local minus sector is nonzero. This obstructs a shape-matching inverse, not arbitrary linear splittings. Some global sectors may disappear under edge relations.

11. Peripheral action and Dehn filling remain unproved

Attempted failure: the target comparison silently imports GY-index Theorem 5.7 into every six-root sector, or is announced as Panitch-Romo 7.1.

Finding: the fresh GY-index primary revision still identifies opposite roots in Section 3.3. Its Theorem 5.7 therefore cannot see a product of the central involutions. The publication supplies the exact Weyl-collapse factor and composition warning only as algebraic facts. It leaves open the additive signed lift and raw collar/state/label dictionary for the actual named peripheral skeins.

The fresh Panitch-Romo primary text still states Conjecture 7.1 as a nontrivial compatible Dehn-filled quotient problem. No filling quotient or nonvanishing proof is supplied here.

12. Priority and significance

The bounded fresh primary check found no resolution of CK 5.27. It does not establish first priority. PP1 already uses a closely related parity projection for a different image, and most of the local multiplication is in CK. Credit and the limited size of the new step are therefore explicit.

The result crosses the user’s Publications threshold because it addresses a recognized, precisely stated open conjecture rather than a small refinement of an internal note. It is a niche comparison result, not a major breakthrough. This calibration does not depend on guessed numerical benchmarks or on the conjecture’s recency.

Verdict and next action

Publish the theorem and corollary with the exact mixed-sided source-reading caveat, source credits, primary links, AI-assisted origin and absence of external review. No retraction packet is warranted for the standard target. Do not describe the internally audited paper as externally verified. Preserve earlier Dehn-filling results at their existing statuses; they were not re-proved by this audit.

The next mathematical mode is ATTACK on the signed six-root additive peripheral lift and exact named-action convention dictionary. Do not enumerate manifolds, slopes, triangulations or skein configurations, and do not contact authors without further authorization.