FPRD research loop state

Updated after the 8 September 2026 signed six-root peripheral continuation. Other programs and the existing CK publication retain their prior scope.

Latest completed mode

ATTACK — audited native additive six-root peripheral action and explicit raw GY/CK/PP named collar dictionary.

This is a supporting research note, not a new Publication. The proof is AI-assisted and internally audited, without external specialist review or author endorsement. Novelty and first priority are unassessed.

Chen and Kricker have separately been emailed about the earlier comparison publication, with Joshua Gay CC’d. No reply, mathematical confirmation or endorsement is assumed. No new email or author contact was made in this continuation.

Active program and retained publication

Program: https://fprd-lab.joshgay.chatgpt.site/research-programs/dehn-filled-quantum-trace

Latest completed Publication: https://fprd-lab.joshgay.chatgpt.site/publications/ck-degree-zero-descent

Panitch–Romo Conjecture 7.1 remains open. The new bridge does not establish unrestricted filling nonvanishing.

Current proof packet

• FPRD-local-collar-square-2026-09-08.md: current proof and named-action formulas.
• local-collar-hostile-audit.md: current internal audit.
• local-collar-source-ledger.md: sources, credit, hashes and reading boundary.
• verify-local-collar.py, local-collar-verification-output.txt: exact local tensor audit.
• FPRD-six-root-peripheral-lift-2026-09-08.md: retained native proof, re-audited; its historical Section 5.4 open gate is superseded by the current note.
• verify-six-root-peripheral.py, verification-output.txt: retained exact algebra verifier, re-run.
• six-root-peripheral-attack.zip, SHA256SUMS.txt: complete checked evidence packet and payload hashes.

Native additive action, now audited from primary sources

Use k=Q(i,A^(1/2)). The full native GY target has left-generated tetrahedron relations f^-=z-2+(z’’)^2-1, right-generated edges H_e+A^2, and all four central vertex products iA. Opposite roots are not identified.

GY’s prequotient Appendix A calculation retains the actual bare-edge root on each parallel standard arc. Standard-off-diagonal terms vanish already in the full local target. The arbitrary extra input skein and the Type II annular factor A^-1 are retained. The geometric diagram setup, local lemmas and standard-arc spanning remain cited GY dependencies.

The two actual basis insertions give rows a,b with B6(a,b)=2 and zero pairing with every edge. Put L=W_G(a), M=W_G(b), so LM=A^2 ML. Define u_(p lambda+q mu)=pa+qb and

$$E^G_{p\lambda+q\mu}=W_G(pa+qb)=A^{-pq}L^pM^q.$$

The identity

$$K_{\lambda+\mu}= \frac{A K_\lambda K_\mu-A^{-1}K_\mu K_\lambda}{A^2-A^{-2}}$$

and the torus three-generator theorem give all named insertions. The inverse denominator is essential: no integral or A^4=1 all-slope statement is proved. Different initial-basis actions agree on the trace image; equality on every target element or Pachner-natural canonicity is not asserted.

Completed local dictionary

Let v=iA, h=iA^(3/2). PP1 Remark 3.5 gives ordinary cup = h^-1 times symmetric cup, and ordinary cap = h times symmetric cap. Both snake identities hold. The standard lantern arc retains its ribbon line under reversal of the collar normal; it acquires no uniform endpoint half-twist factor.

Use the unweighted marking reversal/state flip of PP1 Remark 3.4, s -> -s. It preserves the matching-state tensor e_+ tensor e_+ + e_- tensor e_- exactly, so the splitting square commutes term by term. On standard annular arcs the counit is delta_(mu,nu), unchanged by both state flips. The corresponding mixed-state cup/cap squares were checked before any off-diagonal quotient.

Important failed route: PP1 Proposition 3.10’s weighted self-duality is d(L_s)=v^(-sum s)L_-s. Using it on both sides of the unchanged same-state cut leaves v^(-2s), and epsilon(d(a_++))=v^-2 rather than 1. It works with the appropriate rescaled balanced actions or inverse weights on paired sides, not as a drop-in replacement here.

CK’s opposite collar reverses both stacking and the cyclic triangle form while fixing the ambient orientation and A. State degree flips at both arguments. Thus the twisted product is carried anti-multiplicatively and the corner and off-diagonal submodules are sent to their correct opposite sides.

Anchor actual edge names at a common ideal vertex and prime edge. The opposite cyclic conventions give

$$P:z_G\leftrightarrow z''_C,\quad z'_G\leftrightarrow z'_C, \quad y_G\leftrightarrow y''_C,\quad y'_G\leftrightarrow y'_C.$$

This is a same-edge coordinate dictionary, not an odd geometric move. Relabel all incidences. With D_P(W_G(u))=W_C(Pu), the complete local square and CK Theorem 4.9 yield

$$D_P\tau_G^6=\tau_{GY}^{CK}=\tau_{CK}.$$

The underlying closed framed link is unchanged. A fixed-parameter inverse-root anti-map remains impossible; the separate conjugate inverse-root anti-map is not used here.

Exact original PP named action

Retain the existing Eulerian normalization Phi(x_e)=i epsilon_e X_e, epsilon_e=epsilon_(e opposite). It relates original PP1 constants (A^-1,-i) to CK’s (-iA^-1,1). Its auxiliary face square root and shape-level descent remain as in the publication.

For the additive native row u_gamma, put

$$E^{PP}_\gamma=i^{-\sum_e(u_\gamma)_e}\epsilon^{Pu_\gamma}W_x(Pu_\gamma).$$

Then

$$\tau_{PP1}(K_\gamma\alpha) =\tau_{PP1}(\alpha)(E^{PP}_\gamma+(E^{PP}_\gamma)^{-1}).$$

The inverse summand has the reciprocal scalar. Native left multiplication becomes target right multiplication. If L_C=D_P(L), M_C=D_P(M), then M_C L_C=A^2 L_C M_C and the sum monomial is A^-1 M_C L_C. Right operators reverse composition. The central factor sigma_T^beta_T(u_gamma) is retained and beta is additive; no third sign must be guessed.

Consequences and remaining barrier

The previous generic torus-category reconstruction/excision and meridian-cokernel theorems now apply to the original named PP action with these conventions. They construct the filled target but do not ensure it is nonzero. Its nonvanishing is still equivalent to failure of surjectivity of m+A^2 in the earlier filling convention.

The full native target maps boundary-linearly onto the identified GY target, so its filled target surjects onto the identified filled target. The existing strictly positive geometric-filling theorem’s nonzero vacuum therefore transfers to the full six-root target and, through the proved comparisons, to original PP, with all of that theorem’s hypotheses retained. No claim is made beyond that chamber or for every global sign sector.

The CK comparison publication and its source-reading caveat remain unchanged: use its standard right-tetrahedron/left-edge Laurent presentation, not an alternate closure. No author-confirmed erratum is claimed. All earlier endpoint/excision, meridian-cokernel, regular-intersection, positivity and classical-specialization results retain their existing scope.

Next mode

ATTACK — intrinsic nonvanishing of the actual generic six-root meridian cokernel beyond the regular-intersection/positive-geometric chamber.

Seek a universal detecting functional, a nonvanishing deformation-theoretic construction, or a decisive structural obstruction. Do not enumerate manifolds, slopes, triangulations or skein configurations. Do not restart the generic third-sign problem or infer nonvanishing from a target comparison. Keep PR 7.1 open. Other programs remain at their latest paused or withdrawn statuses.