FPRD research loop state

Updated after the 8 September 2026 global meridian-index ATTACK. Other programs and the existing CK publication retain their prior scope.

Latest completed mode

ATTACK — a global index detector for the actual generic native six-root meridian cokernel, with a sharp weak-angle convergence boundary.

This is a supporting research note, not a new Publication. The work is AI-assisted and internally audited, without external specialist review or author endorsement. Novelty and first priority are unassessed; the construction substantially reuses established 3D-index methods.

Chen and Kricker were separately emailed about the earlier comparison publication, with Joshua Gay CC’d. No reply, mathematical confirmation or endorsement is assumed. This continuation made no new author contact.

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 in this work. No unrestricted actual filling nonvanishing, nonzero descended trace in the new chamber, or actual full-six-root counterexample has been proved.

Current evidence packet

The new global-meridian-index-attack/ packet contains:

• FPRD-global-meridian-index-detector-2026-09-08.md: full actual-presentation functional, all-input convergence, descent, detected cokernel unit, and null-ray failure theorem.
• hostile-audit.md, source-ledger.md, source-snapshots.json: proof boundaries, primary-source versions, close prior art and retained dependencies.
• verify-global-meridian-index.py, verification-output.txt, exact-results.json: 24 exact convention/algebra checks and a local seed-series sanity check.
• README.md, SHA256SUMS.txt, global-meridian-index-attack.zip, ARCHIVE-SHA256.txt: reproducibility and integrity.

The six-root-peripheral-attack/ packet remains the named-action/collar bridge. The clean-excess-meridian-attack/ packet remains the obstruction to point-local deformation detection and the source of the global Fourier countermodel. Neither is superseded by a claim of universal nonvanishing.

Retained exact target and named action

Use $k=\mathbb Q(i,A^{1/2})$. The full native GY target has tetrahedron relations $S_6(z^{-2}+(z'')^2-1)$, edge relations $(H_e+A^2)S_6$, and all four central vertex values $iA$. These are mixed-sided relations, not a two-sided algebra quotient. Opposite-root signs remain after central elimination.

Retain the preceding source-native additive six-root peripheral action and its raw named GY/CK/PP collar dictionary. It depends on GY’s local calculation, PP’s unweighted marking reversal and inverse cup/cap factors, CK’s convention-adjusted comparison, and the original-PP Eulerian normalization. The CK mixed-sided Laurent-target source-reading caveat remains; no author-confirmed erratum is claimed. No target-wide naturality across basis choices or Pachner moves is asserted.

For native additive row $u_\gamma$, the named original-PP monomial remains

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

with reciprocal coefficient on its inverse summand and all central signs retained. The generic peripheral-sign problem is not being reopened.

The intrinsic generic filled target remains isomorphic to

$$C_s^6=\operatorname{coker}(m+A^2:Q_G^6\to Q_G^6),\qquad ml=A^2lm.$$

That isomorphism sends a filled class $[v]$ to $[(1-l^2)v]$. Detecting the abstract cokernel unit does not detect the filled vacuum.

New conditional theorem: a global actual-cokernel functional

Fix one central sign sector. Reduce to the root quantum torus $S$ on $L=\mathbb Z^{2N}$, with $w=z''$, $wz=Azw$, and local skew block

$$\Omega_2=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.$$

Retain every actual edge and filling relation $(c_jW(h_j)+A^2)S$. Their rows generate the actual isotropic image lattice $\Lambda$, without saturation or replacement by an even sublattice. Require that the exact values $\chi(h_j)=-A^2/c_j$, including their phases, define a character on $\Lambda$. Failure of this scalar consistency already makes that sector zero.

Let $q=A^4$, $K=\mathbb Q(i)((A^{1/2}))$, and define the standard tetrahedron-index seed by

$$\ell_0(W(2a_1,2b_1,\ldots))=\prod_j I_\Delta(b_j,-a_j), \qquad \ell_0(W(u))=0\quad(u\notin2L).$$

The global candidate is

$$\mathcal F(W(u))=\sum_{\lambda\in\Lambda} \chi(\lambda)^{-1}A^{\Omega_2(\lambda,u)/2}\ell_0(W(u+\lambda)).$$

Require nonnegative rational triples $\theta_j=(\gamma_j,\beta_j,\alpha_j)$ summing to (1), and

$$\vartheta(\lambda)=\nu_A\chi(\lambda),\qquad \vartheta(a_1,b_1,\ldots)=\sum_j(\gamma_ja_j+\alpha_jb_j).$$

These are the actual affine edge and meridian angular equations; unfilled edge angles alone are insufficient. Let

$$T(a,b)=(-a,0,-b),\quad H=T((\Lambda\cap2L)/2)+\sum_j\mathbb Z\Delta_j,\quad \mathcal C=(H\otimes\mathbb R)\cap\mathbb R_{\geq0}^{3N},$$

where $\Delta_j$ is the all-one triple in tetrahedron $j$, and put

$$\delta(x)=\sum_j(x_{j1}x_{j2}+x_{j2}x_{j3}+x_{j3}x_{j1}).$$

The final hypothesis is the absence of a nonzero $x\in\mathcal C$ with $\delta(x)=\theta\cdot x=0$. Strict filling angles imply it. Nonnegative angles can suffice even without a strict angle solution, as an explicitly non-topological lattice model demonstrates.

Under these conditions, $\mathcal F$ converges as an order-independent Laurent-series family for every algebraic root input, kills all actual tetrahedron/edge/filling relations on the correct sides, and has

$$\mathcal F(1)=1+\text{positive }A\text{-order}\neq0.$$

This proves nonvanishing of the sector and hence of the actual full native six-root cokernel. It requires no classical shape point or regular intersection. The domain remains the original finite Laurent presentation; only the coefficient field is expanded at $A=0$. No actual topological input outside the earlier geometric chamber is exhibited, and no universal angle-feasibility theorem is asserted.

Exact convergence boundary and remaining trace gate

The all-input proof uses reduced triple representatives, finitely many coordinatewise minimal affine points, isotropy, and the nonnegative quadratic/skew cross term. Compactness supplies a linear lower degree bound on the homogeneous cone. This is a proof, not a finite truncation extrapolation.

With nonnegative filling angles, a nonzero simultaneous zero of $\delta$ and the angle weight produces an integral ray whose unit summands all have valuation zero. Thus the unordered average fails already at (1). This is a sharp obstruction to this construction, not a theorem that the actual cokernel vanishes. Regrouping or analytic continuation would require a new all-relations argument.

If the actual longitude is $l=c_lW(b)$, the filled-vacuum value is exactly

$$\mathcal F(1-l^2)=\mathcal F(1)-c_l^2\mathcal F(W(2b)).$$

Both terms converge, but their difference has not been shown nonzero under the new hypotheses. The earlier positive-geometric vacuum theorem retains all its earlier hypotheses. No equality with a Dehn-surgery index or cycle-valued pairing is claimed.

Literature and preserved limits

The DGG tetrahedron index, Garoufalidis’s angle/index work, GHHR’s exact degree formula and GY’s quantum-trace index averaging are major dependencies. Celoria–Hodgson–Rubinstein’s current arXiv v2, Theorem 6.2, already supplies rigorous Dehn-index surgery under its standard-cusp and efficiency hypotheses. The note does not claim first construction or inherit a universal surgery theorem from that literature.

The regular-intersection and positive-geometric results retain their scope. The clean-excess theorem still proves disappearance of its point-local formal detector, not disappearance of the global algebraic cokernel. The Laurent Fourier and cyclic peripheral countermodels remain countermodels to invalid inference principles, not actual triangulation counterexamples. Finite Laurent constant-term densities cannot solve the tetrahedron annihilator in a domain; bilateral or residue constructions are not thereby excluded.

Next mode

ATTACK — the actual longitude difference under the global detector, or a replacement detector at its null-ray boundary.

The most focused positive target is nonvanishing of $\mathcal F(1-l^2)$ with the exact named longitude and quantum shifts. The alternative is a rigorously defined residue/difference pairing that survives zero-degree rays or the absence of nonnegative filling angles, with descent checked on every algebraic input.

Do not infer a nonzero trace from a nonzero target, target vanishing from a divergent index, or actual vanishing from zero point-local detectors. Do not enumerate manifolds, slopes, triangulations or skein configurations. Do not restart peripheral-sign work. Other programs remain at their latest paused or withdrawn statuses.