FPRD research loop state

Proposed replacement after the 8 September 2026 longitude-difference ATTACK. Other programs and the existing CK publication retain their prior scope.

Latest completed mode

ATTACK — exact longitude-shifted coefficients, a character-twisted detector family, and a strict-angle algebraic cancellation boundary.

This is supporting research, 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. No recognized open conjecture has been closed here.

The new finite mixed-sided certificate proves that strict filling angles and the extracted global-index/peripheral algebraic hypotheses alone do not force survival of the filled-vacuum image. The witness is explicitly not an actual triangulation: a necessary edge-incidence inequality fails. The actual named-longitude gate remains open.

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, actual global-index vacuum theorem in the new chamber, or actual full-six-root counterexample has been proved.

Current evidence packet

The new longitude-difference-attack/ packet contains:

• FPRD-longitude-difference-obstruction-2026-09-08.md: exact actual longitude degree/phase formulas, finite leading test, corrected character-twisted functional family, exact shear invariance, strict-angle abstract cancellation certificate, and proved failure of actual edge incidence.
• hostile-audit.md, source-ledger.md, source-snapshots.json: internal proof-boundary review, current pinned primary versions, input hashes and prior-art limits.
• verify-longitude-difference.py, verification-output.txt, exact-results.json: exact PBW verification of the all-orders certificate, scalar/lattice/angle checks, and an independent certified coefficient check through A-degree 16.
• README.md, build-release.py, SHA256SUMS.txt, longitude-difference-attack.zip, ARCHIVE-SHA256.txt, archive-validation.txt: reproduction and integrity.

The preceding global-meridian-index-attack/ packet remains the actual all-input convergence/descent and cokernel-unit theorem. The six-root-peripheral-attack/ packet remains the named-action/collar bridge. The clean-excess-meridian-attack/ packet retains the point-local obstruction and earlier weaker algebraic counterexamples. None is promoted to universal actual vacuum 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 relation spaces, not a two-sided algebra quotient. Opposite-root signs remain after central elimination.

Retain the source-native additive six-root peripheral action and its exact 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 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. Do not reopen peripheral-sign work.

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.

Retained global actual-cokernel detector

In a central sign sector use the root lattice $L=\mathbb Z^{2N}$, $w=z''$, $wz=Azw$, and local skew block

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

Keep every actual edge/filling row $(c_rW(h_r)+A^2)S$ and its isotropic image lattice $\Lambda$, without saturation. Exact values $\chi(h_r)=-A^2/c_r$, including phases, must define a character on $\Lambda$. A failure makes that sector zero.

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

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

the actual average 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 $\theta_j=(\gamma_j,\beta_j,\alpha_j)$ of sum one, with $\vartheta|_\Lambda=\nu_A\chi$, where $\vartheta(u)=\sum_j(\gamma_ju_{zj}+\alpha_ju_{wj})$. These are the actual edge and meridian affine equations.

Let $T(a,b)=(-a,0,-b)$, $\Lambda_0=\Lambda\cap2L$, $H=T(\Lambda_0/2)+\sum_j\mathbb Z\Delta_j$, $\mathcal C=(H\otimes\mathbb R)\cap\mathbb R_{\geq0}^{3N}$, and $\delta(x)=\sum_j(x_{j1}x_{j2}+x_{j2}x_{j3}+x_{j3}x_{j1})$. Require no nonzero $x\in\mathcal C$ with $\delta(x)=\theta\cdot x=0$. Strict filling angles suffice.

Under these conditions the previous theorem proves unordered summability for every algebraic root input, annihilation of all actual relations on the correct sides, and $\mathcal F(1)=1+$ positive A-order. This detects the actual generic algebraic cokernel unit without a classical shape point or regular intersection. It does not prove every actual filling has such data. Under nonnegative angles, a null ray makes this unordered sum diverge already at one; divergence is not target vanishing.

New exact longitude coefficient gate

Retain the actual named $l=c_lW(b)$, write $c_l=\eta_lA^\rho$, $\chi(\lambda)=\eta_\lambda A^{\vartheta(\lambda)}$. Only $\lambda\in\Lambda_0$ contribute to $c_l^2\mathcal F(W(2b))$. For $v=b+\lambda/2$, reduce $T(v)$ by block minima $m_j$ to $x$, and put $M=\sum_jm_j$. The exact summand degree and leading coefficient are

$$D_b(\lambda)=2\rho-\vartheta(\lambda)+\Omega_2(\lambda,b)+2\delta(x)-2M, \qquad s_b(\lambda)=\eta_l^2\eta_\lambda^{-1}(-1)^M.$$

If $d_*$ is the minimum degree and $S_*$ the finite sum of leading coefficients at that degree, then $d_*>0$, or $d_*<0,S_*\ne0$, or $d_*=0,S_*\ne1$, proves $\mathcal F(1-l^2)\ne0$. Other cases require higher coefficients. In particular $(d_*,S_*)=(0,1)$ can exhibit exact all-orders cancellation even with a unique shifted minimizer.

The corrected additional all-relations family is

$$\mathcal G_v(s)=\sum_{\lambda\in\Lambda}\chi(\lambda)^{-1}\ell_0(W(v)W(\lambda)s) =\mathcal F_{\chi_v}(W(v)s), \quad \chi_v(\lambda)=\chi(\lambda)A^{\Omega_2(\lambda,v)}.$$

It converges and descends to the original target if the shifted character has its own nonnegative-angle/no-null-ray certificate. Such feasibility is additional. The second local index recurrence cannot simply be carried through averaging without these character shifts. Translates in $\Lambda$ are scalar multiples of the first detector. Exact meridian shears $l_n=A^{-n}m^nl$ do not alter its squared longitude value under any original meridian eigenfunctional.

New algebraic failure boundary, not an actual counterexample

With $B=iA$, three tetrahedra, $x_j=z_j^2,y_j=w_j^2$, use normalized moments

$$E_1=y_1y_2^{-1}+1,\quad E_2=y_1^{-2}y_3^{-2}-B^{-5},\quad E_3=y_3-B,$$

whose exact native-form scalars are $-B^2,B^7,B$. Their image lattice is $k_1+k_2$ even in the squared-w coordinates, with character $(-1)^{k_2}B^{3(k_1+k_2)/2+k_3}$. Strict angle triples are $(1/8,1/8,3/4)$, $(1/8,1/8,3/4)$, $(1/4,1/4,1/2)$.

Take $m=By_3$ and $l=B^{-1}W(-1,0,-1,0,1,1)$. The longitude commutes with the designated edges and satisfies $ml=A^2lm$. The packet supplies literal six-root Laurent lifts and a finite identity proving $1-l^2\in\sum Sf_j+\sum E_rS$. Hence every all-relations functional on this specified sector kills its vacuum image, although the strict-angle detector makes its cokernel unit nonzero. This is stronger than a zero truncated index value. It is not a claim of zero vacuum in every sign sector of the associated full six-root Laurent lift, and it does not rule out another compatible sector as an actual detector.

It is not an actual triangulation: for a nonnegative raw edge lift $(a,b)=(r-s,t-s)$, one necessarily has $s\geq\max(0,-a,-b)$. The witness’s second edge needs at least eight total prime incidences but its exact scalar forces seven. In general, actual unscaled native edges satisfy the necessary inequality

$$\nu_A\chi(h)+\sum_j\max(0,-h_{zj},-h_{wj})\leq2.$$

The witness violates it: $-5+8=3>2$. No change of generators in its same edge lattice repairs this: every row satisfying the inequality lies in a rank-one subgroup, whereas the edge lattice has rank two. This necessary actual structure was unused by the previous global-detector proof. The inequality is not a sufficient realization test and is not itself an anti-cancellation theorem.

Literature and preserved limits

The DGG tetrahedron index, Garoufalidis’s two recurrences/triality and positivity uniqueness, GHHR’s exact degree/sign, and GY’s quantum-trace averaging remain major dependencies. Fresh current-version checks used arXiv:1208.1663v2, arXiv:1604.02688v1, and Celoria–Hodgson–Rubinstein arXiv:2509.09886v2. The latter’s surgery theorem retains its standard-cusp and efficiency scope. No comparison with its invariant or universal nonzero surgery value is established here.

The positive-geometric vacuum theorem retains its earlier hypotheses. The clean-excess result concerns disappearance of a point-local detector, not global target vanishing. Earlier Fourier/cyclic examples and the new strict-angle witness are algebraic logical-obstruction examples, not actual full-six-root triangulation counterexamples. Finite Laurent densities remain insufficient; no new residue or regularized pairing is supplied.

Next mode

ATTACK — actual-incidence anti-cancellation for the exact named longitude.

Use genuine nonnegative edge incidence, global edge dependencies and the intrinsic normal-surface/index structure to control the signed coefficients of $\mathcal F(1-l^2)$, or prove an actual correct-sided obstruction. The new witness settles only the abstract inference boundary and identifies an actual condition it fails. Excluding that witness alone does not settle the actual gate.

The shifted-character family can be pursued when its additional feasibility is proved. A surgery-index comparison could help only with the same exact longitude insertion, phases, parity/image lattice, convergence and all-input descent, plus an applicable nonvanishing result. Do not infer a nonzero trace from a nonzero target, target vanishing from a zero/divergent pairing, or actual vanishing from an abstract certificate.

Do not enumerate manifolds, slopes, triangulations or skein configurations. Do not reopen peripheral signs or promote this supporting packet to Publication. Other programs remain at their latest paused or withdrawn statuses.