Hostile internal audit — CK comparison injectivity
8 September 2026. This is the sole mathematical researcher’s internal adversarial audit, not independent expert review. All mathematics in this attack was performed by the selected GPT-6 Astra; no fallback model or mathematical sub-agent was used.
Contents
Verdict
The proof of CK Conjecture 5.27 survives the checks below. It warrants PUBLISH-mode candidate-proof preparation, not automatic treatment as an externally reviewed Publication. The raw six-root peripheral named-action extension remains unproved and must be withheld.
1. Did we change the target to make the theorem true?
No. CK Definition 5.22(iii) defines the PP target as the image of the six-root torus in the ambient reduced tensor product of face tori. The proof begins with that definition. The equality (SR=J) is the conclusion, not an input. CK’s actual map is the one induced by generator matching, so the computed kernel is precisely its kernel.
2. Are the half-edge shape generators algebraically independent?
Yes as quantum-torus monomials. Each bare edge contributes the sum of its two distinct half-edge basis vectors, and different bare edges use disjoint coordinate pairs. The six-root exponent lattice embeds in the full half-edge lattice. The degree map is the difference in each pair, whose kernel is exactly that embedded lattice. Opposite tetrahedral edges are different pairs and are not identified by the degree map.
The skew form may have a radical; that does not make distinct Laurent monomials linearly dependent. This distinction is essential.
3. Is projection onto degree zero legitimate for a mixed-sided quotient?
Yes. (P_0) is an (S)-bimodule map because (S=U_0), although it is not an algebra map. Applying it term by term to (x=r_i u_i+v_j e_jS) produces (x=r_iP_0(u_i)+P_0(v_j)e_j). No ideal is treated as two-sided. No cancellation among nonzero grading components can create an extra degree-zero relation.
4. Were nonhomogeneous tetrahedron relations wrongly put in (S)?
The raw trinomials are not in (S). The proof uses an explicit invertible half-edge monomial on the RIGHT. The exact identity is checked in a twelve-variable quantum torus before any quotient:
Since (h) is invertible, (rU=(rh)U), and the vertex relation is already present. This proves equality of generated right ideals in both directions. Normalizing on the left instead would not establish the claimed equality and is not done.
5. Are all tetrahedron and edge relations present?
CK Definition 5.7 gives four vertex/trinomial pairs per tetrahedron. The single calculation is transported by oriented tetrahedral symmetries to all four vertices; cyclic Lagrangians are in the same right ideal, with an exact witness also checked by the script. Opposite squares agree by the central vertex-ratio argument. Thus one does not merely show that the desired relations happen to hold in the PP image.
Every edge is retained; no independent-edge reduction is used. The bigon images around an edge multiply to the exact Weyl edge monomial, by cancellation of the opposite face commutation factors. Inverse edge generators are redundant by an explicit left-ideal formula. The proof does not assume edge monomials are central in the full face torus.
6. Is a claimed centrality used in the wrong ambient algebra?
No. Vertex monomials and opposite-root ratios are central in (S), not generally in (U). The proof never moves them through arbitrary half-edge multipliers. The only ambient operation needed is right multiplication by the specified unit and the (S)-bimodule projection.
7. Does the source have a sidedness defect?
Yes, one word: CK Definition 5.7(i) says “left ideal” when extending an ideal it has just defined on the right. The candidate uses the actual right-generated object consistently appearing in CK equation (5.6), Definition 5.11, its Proposition 5.25 proof, and the original PP ambient presentation. Reading that isolated word as a new two-sided closure would change the target and is outside the theorem. This is disclosed in the note and ledger, not silently corrected.
8. Did we infer a named-trace equality from an algebra presentation?
No. CK Theorems 4.9 and 5.34 supply the named equality through the common face-cone construction. The new theorem proves that their existing map is invertible. CK’s face trace is explicitly in the PP2 first normalized branch, with (c_T=(-A2){-1/2},c_B=1). It is not the raw PP1 scaling. The original-PP conjugacy is separately checked on triangle and bigon trace generators and then transported by the relative tensor construction.
9. Was an opposite-root quotient inverted?
No. The six-root comparison is proved first, then the same central quotient can be taken on both sides. Locally the discarded minus sector is nonzero: after central elimination the tetrahedron Lagrangian remains a nonmonomial in a rank-two quantum torus and cannot have a right inverse. Thus the vector (z-y) does not vanish in that sector. A shape-matching inverse to the opposite-root quotient is genuinely obstructed.
This does not imply all minus sectors survive every global edge quotient. The proof makes no such assertion. The quotient is taken on the shape image module, using its central shape action, never as a two-sided opposite-root quotient of the full half-edge torus, where those ratios need not be central.
10. Were meridian, longitude, or sum-of-slopes signs hidden?
No. The note gives the exact central reduction
and the exact original scaling (i{-a_e}a). It also gives (W(a+b)=A^{-}W(a)W(b)), with pairing (2), and explicitly notes reversal of products when right multiplications are composed.
However, we have not proved that chosen six-root lifts of the named peripheral curves have these operators. GY-index Theorem 5.7 is a theorem on the identified target, and CK’s GY conventions need their own collar/state/label translation. A central factor (_T^{_T}) in a lifted sum-of-slopes row is invisible to the identified theorem. The note labels all such unverified lift formulas as candidates, not as an established boundary module structure.
11. Were characteristic or parameter denominators suppressed?
The comparison proof uses only (A^{/2}) and (v_0^{}), already required by CK. It neither divides by (A2-A{-2}) nor requires a generic parameter. The sign-sector decomposition uses (1/2) only after passing to the stated characteristic-zero field. The Eulerian face scaling uses (s^2=-i) over an auxiliary extension, then descends its shape-level map to (k). Assuming (sk) would be wrong and is explicitly avoided.
12. Did symbolic checks substitute for a general proof?
No. The general proof is the lattice-graded retraction and equal-sided- ideal argument. The script validates its exact local coefficient identities, skew form and normalization. It performs no manifold, triangulation, slope, or skein configuration search. It does not claim to prove the source trace theorems or the unproved six-root peripheral lift.
Stopping boundary
Publishable candidate target: the explicit comparison injectivity conjecture CK 5.27. Still-open research target: the intrinsic signed six-root peripheral lift and the exact raw GY-index coordinate dictionary. The larger Panitch–Romo Dehn-filling conjecture remains open. No broader nonvanishing or filling theorem is asserted by this packet.