FPRD research loop state

Latest completed mode

ATTACK — actual-incidence normal-class translation and a two-angle filled-vacuum theorem.

The exact named longitude gate is now closed in a specified two-angle chamber. It is not closed for arbitrary triangulations or fillings. 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 remain unassessed.

Active program and retained publication

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

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

Panitch–Romo Conjecture 7.1 remains open in its unrestricted scope. No universal actual filling nonvanishing or actual full-six-root counterexample has been proved.

Chen and Kricker were separately contacted about the earlier comparison publication. No reply, mathematical confirmation, or endorsement is assumed. The present research note is not qualifying publication outreach.

Current evidence packet

The actual-incidence-attack/ packet contains:

• FPRD-actual-incidence-two-angle-vacuum-2026-09-08.md: exact native normal-class map, Euler-degree cancellation, global character consistency, two-angle actual vacuum theorem, uniform large-slope certificate, and the next pairing question.
• hostile-audit.md, source-ledger.md, source-snapshots.json: hypothesis, normalization, dependency, and source-version boundaries.
• verify-actual-incidence.py, exact-results.json: exact universal identities, fractional-lift calibration, and an optional checker for supplied native angle data. The checker is not a triangulation recognizer.
• README.md, build-release.py, SHA256SUMS.txt, actual-incidence-attack.zip, ARCHIVE-SHA256.txt, archive-validation.txt: reproduction and integrity.

The preceding longitude-difference-attack/ packet remains the exact coefficient/phase and shifted-character-family reference. Its strict-angle zero certificate is still only an abstract, incidence-infeasible witness. The global-meridian-index-attack/ packet remains the all-input convergence/descent theorem. The six-root-peripheral-attack/ packet remains the native named-action comparison. The positive-geometric and clean-excess packets retain their earlier precise scopes.

Retained exact target

Use $k=\mathbb Q(i,A^{1/2})$, $K=\mathbb Q(i)((A^{1/2}))$, $B=iA$, and $q=A^4$. Native tetrahedron relations are $S_6(z^{-2}+(z'')^2-1)$, edge relations are $(H_e+A^2)S_6$, and the four central vertex values are $iA$. These are mixed-sided relation spaces, not a two-sided algebra quotient. Opposite-root signs remain after central elimination.

Retain the exact additive native six-root peripheral action and GY/CK/original-PP collar dictionary. Its geometric and algebraic inputs, Eulerian normalization, generic parameter restriction, and CK source-reading caveat are unchanged. No author-confirmed erratum or new target-wide naturality is claimed. Do not reopen peripheral signs.

The generic intrinsic filling isomorphism sends the filled vacuum to $[1-l^2]$ in

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

The cokernel unit and this vacuum image are distinct vectors.

Retained all-input detector

In a central sector let $L=\mathbb Z^{2N}$, with local skew block $\Omega_2=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$, and keep the actual unsaturated isotropic image lattice $\Lambda$ of all edge and meridian rows. Exact character values are $\chi(h_r)=-A^2/c_r$. A character inconsistency makes that sector zero.

The seed vanishes off $2L$ and is

$$\ell_0(W(2a_1,2d_1,\ldots))=\prod_jI_\Delta(d_j,-a_j).$$

The actual average is

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

The previous theorem proves unordered summability for every algebraic input, correct-sided annihilation of all relations, and $\mathcal F(1)=1+$ positive order under consistent character and strict filling angles. It also permits nonnegative filling angles with its exact no-null-ray condition. Its failure is not target vanishing. Strict filling angles alone do not detect the filled vacuum in an abstract moment presentation.

New exact structural translation

For a genuine native raw three-coordinate row $R=(r_j,s_j,t_j)_j$, define

$$P(R)_j=(r_j-s_j,t_j-s_j),\qquad p(R)=\sum_js_j.$$

Actual nonnegative edge rows are $E_e$; native additive peripheral rows are $R_s,R_l$. Their exact operators are

$$h_e=P(E_e),\quad h_s=P(R_s),\quad b=P(R_l),\qquad c_\bullet=\epsilon_\bullet B^{p(R_\bullet)}.$$

Put $J=\bigoplus\left(\begin{smallmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{smallmatrix}\right)$. The collapse preserves the skew form. Actual edge rows are mutually orthogonal and orthogonal to peripheral rows, and $R_s^TJR_l=2$. The standard actual $Q$-normal dictionary has formal Euler values $-2,0,-1$ on $E_e,R_\gamma,\Delta_j$, and boundary values $0,2\gamma,0$.

For $\lambda\in\Lambda\cap2L$, choose any integer representation

$$\lambda=\sum_en_eh_e+n_sh_s,\qquad U=R_l+\tfrac12\left(\sum_en_eE_e+n_sR_s\right),$$ $$t_j=\max U_j,\qquad X_\lambda=\sum_jt_j\Delta_j-U.$$

This is a representation-independent injective map to integral nonnegative $Q$-normal classes, even when $U$ is half-integral. It preserves the actual image-lattice restriction; no surjectivity onto a full normal homology fiber is asserted. It has

$$\partial X_\lambda=-2l-n_ss,\qquad \chi_{\rm top}(X_\lambda)=\sum_en_e-\sum_jt_j.$$

The exact native longitude degree is

$$\boxed{D_l(\lambda)=2(\delta(X_\lambda)-\chi_{\rm top}(X_\lambda)).}$$

The filling correction $-2n_s$ cancels against the exact Weyl contribution $+2n_s$. The central scalar is essential. The leading phase remains the previous $eta_l^2\eta_\lambda^{-1}(-1)^M$; no coefficient positivity is assumed.

New two-angle theorem

An angle vector is normalized by $\theta\cdot\Delta_j=1$, $\theta\cdot E_e=2$. Assume:

1. a complete strict angular certificate $\theta^0>0$, with $\theta^0\cdot R_\gamma=0$ for every peripheral class;
2. a strict filling angular certificate $\theta^s>0$, with $\theta^s\cdot R_s=2$.

Then all-plus character consistency follows from all edge dependencies: a reduced dependency has zero meridian coefficient, its raw sum is $\sum d_j\Delta_j$, and $\sum d_j=2\sum n_e$. Its exact character product is $B^0$. Other sectors retain their phase tests; the global sum-of-edges relation necessarily imposes $\prod_j\sigma_j=1$, but this is not a full sector classification.

The filling certificate gives the retained all-input theorem. The complete certificate gives

$$D_l(\lambda)=2\delta(X_\lambda)+2\theta^0\cdot X_\lambda>0.$$

The class $X_\lambda$ cannot be zero because its longitude boundary component is nonzero, equivalently because of the meridian pairing. Consequently

$$\boxed{\mathcal F(1-l^2)=1+\text{positive }A\text{-order}\ne0.}$$

This detects the actual full-six-root filled vacuum and, using the retained named comparison, the actual named descended traces. It assumes no nonlinear shape solution, regular classical intersection, or geometric core-holonomy value. No actual chamber-separating example is asserted.

The complete certificate alone already makes the longitude series unordered-summable, by its positive linear lower bound and injectivity of $\lambda\mapsto X_\lambda$. It does not normalize or guarantee convergence of the original unit series: that degree still has the additional $-2n_s$ term. For the full theorem a nonnegative filling certificate with the prior no-null-ray condition can replace strict filling positivity.

Uniform filling-angle construction

For fixed native cusp basis rows $R_a,R_d$ with $R_a^TJR_d=2$, let $u=JR_a$, $v=JR_d$. For a primitive slope $s=pa+qd$, put

$$\theta^{p,q}=\theta^0+\frac{pv-qu}{p^2+q^2}.$$

It preserves all edge/tetrahedron equations and has exact meridian angular holonomy two. Hence every slope satisfying the coordinatewise inequalities

$$(p^2+q^2)\theta^0_j+p v_j-q u_j>0$$

is in the theorem’s chamber. A uniform sufficient bound is

$$p^2+q^2>C^2,\qquad C^2=\max_j\frac{u_j^2+v_j^2}{(\theta^0_j)^2}.$$

This proves all-sufficiently-large-primitive-slopes nonvanishing for an actual triangulation with a complete strict angular certificate. It is a basis-dependent sufficient estimate, not a classification of exceptional fillings. The real dual vector only deforms angles; the physical named integral longitude remains unchanged. Do not turn the bounded excluded region into a slope census.

Preserved failures and literature limits

The preceding abstract witness still has a nonzero cokernel unit but a zero specified vacuum image, proved by a finite correct-sided identity. It violates actual edge incidence: its second edge needs eight prime incidences but its scalar permits seven, and no change of generators in the same edge lattice repairs the failure. It also lacks a complete nonnegative angular certificate: its meridian row pairs as $1+2\theta_3$, never zero. It is not an actual triangulation counterexample or a statement about all sign sectors.

The exact coefficient stopping test, character-twisted family, and meridian-shear invariance remain available. The normal degree identity does not imply that ordinary 1-efficiency alone makes every longitude degree positive. No all-purpose residue/regularized pairing or surgery-index identification has been proved.

Primary versions freshly checked: GHHR arXiv:1604.02688v1; Futer–Guéritaud arXiv:1004.0440v2; Garoufalidis arXiv:1208.1663v2; GY author PDF dated 6 August 2025; Celoria–Hodgson–Rubinstein arXiv:2509.09886v2; Panitch–Romo arXiv:2609.02706v1. Their source roles and priority boundaries are explicit in the packet. No claimed author endorsement or external review.

Next mode and decisive question

ATTACK — exact vacuum detection beyond the two-angle chamber.

Retain the intrinsic normal-class map and all proved chamber results. Do not restart them, reopen peripheral signs, or enumerate manifolds, slopes, triangulations, normal-surface supports, or residual configurations.

When a complete strict certificate exists but the original filling certificate is absent, the prior balanced translate is available: $v=-b$ changes the meridian character valuation by $-2$, so $\theta^0$ certifies all-input convergence of

$$\mathcal G_{-b}(s')=\sum_{\lambda\in\Lambda}\chi(\lambda)^{-1} \ell_0(W(-b)W(\lambda)s').$$

It descends to the original target. The next exact test is

$$\mathcal G_{-b}(1-l^2) =\mathcal F_{\chi_{-b}}(W(-b))-c_l^2\mathcal F_{\chi_{-b}}(W(b)).$$

Determine its value or an exact structural reason it fails to detect the vacuum, or construct another correct-sided global pairing. Do not assume boundary-inversion symmetry or asymmetry. A zero or divergent pairing is not a target-vanishing theorem. Removing the complete positive angular certificate is also unresolved.

Other programs retain their latest paused or withdrawn status. In particular, Selfridge remains paused, the semicomplete-digraph target remains withdrawn, and the JT two-boundary work is not restarted by this continuation.