FPRD research loop state

Latest completed mode

ATTACK — exact balanced-pairing failure and a full integral normal-index bridge.

The balanced functional is now evaluated in the native all-plus sector: with a complete strict angular certificate, it annihilates the filled vacuum exactly. This is a failure of this detector, not a zero-vacuum theorem. It is a supporting research note, not a new Publication. AI-assisted and internally audited; no external specialist review or author endorsement. Novelty and first priority are 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 filled-vacuum nonvanishing or actual full-six-root counterexample has been proved. No new publication outreach is warranted by this note.

Chen and Kricker were separately contacted about the earlier comparison publication. No reply or endorsement is assumed. Keep the earlier publication and all established comparison hypotheses intact.

Current evidence packet

balanced-pairing-attack/:

• FPRD-balanced-pairing-vanishing-2026-09-08.md: full integral boundary-fiber bijection, native all-orders phase formula, sourced balanced vanishing, and same-complete-character one-cusp exhaustion.
• hostile-audit.md, source-ledger.md, source-snapshots.json: proof and dependency boundaries.
• verify-balanced-pairing.py, exact-results.json: exact universal algebra, parity solver, and a local seed-series check; no topological enumeration.
• README.md, build-release.py, SHA256SUMS.txt, balanced-pairing-attack.zip, ARCHIVE-SHA256.txt, archive-validation.txt: reproduction and integrity.

The preceding actual-incidence-attack/ packet remains the two-angle and uniform large-slope theorem. longitude-difference-attack/ retains the general phase formula, correct-sided shifted-character family, and incidence-infeasible abstract zero-vacuum witness. global-meridian-index-attack/ retains all-input convergence. six-root-peripheral-attack/ retains the native named-action comparison. Earlier positive-geometric and clean-excess results retain their scopes.

Retained exact presentation

Use $k=\mathbb Q(i,A^{1/2})$, $K=\mathbb Q(i)((A^{1/2}))$, $B=iA$, $q=A^4$. In a central sector,

$$C_s=S\Big/\left(\sum_jS(z_j^{-2}+w_j^2-1) +\sum_e(c_eW(h_e)+A^2)S+(c_sW(h_s)+A^2)S\right).$$

This is a mixed-sided vector-space quotient, not a two-sided algebra quotient. The four native vertex values are $iA$; opposite-root signs remain after central elimination. Retain the exact additive native peripheral action, generic parameter restriction, Eulerian normalization, and CK source-reading caveat. Do not reopen peripheral signs.

The named operators satisfy

$$m=c_sW(h_s),\quad l=c_lW(b),\quad ml=A^2lm.$$

The intrinsic filling isomorphism sends the filled vacuum to $[1-l^2]$. It is not the cokernel unit.

Keep $L=\mathbb Z^{2N}$, $\Omega(u,v)=\sum_j(u_{wj}v_{zj}-u_{zj}v_{wj})$, and the unsaturated image $\Lambda=\sum_e\mathbb Zh_e+\mathbb Zh_s$. The exact character is $\chi(h_r)=-A^2/c_r$, including phase consistency. The seed vanishes off $2L$ and equals $\prod_jI_\Delta(d_j,-a_j)$ on $W(2a_1,2d_1,\ldots)$.

Retained two-angle vacuum theorem

For an actual triangulation, raw rows satisfy $P(R)_j=(R_{j1}-R_{j2},R_{j3}-R_{j2})$, $p(R)=\sum_jR_{j2}$. All-plus native scalars are $B^{p(R)}$. Edge/peripheral/tetrahedron normal classes have Euler values $-2,0,-1$, boundary values $0,2\gamma,0$, and zero mod-2 homology.

A complete strict certificate $\theta^0$ has tetrahedron sums one, edge pairings two, and all peripheral holonomies zero. A filling certificate $\theta^s$ additionally has meridian pairing two. The earlier theorem proves

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

when both certificates exist. Its normal longitude degree is $2(\delta(X)-\chi_{\rm top}(X))>0$; the filling certificate supplies original-character all-input convergence. A nonnegative filling certificate with the exact prior no-null-ray condition can replace strict filling positivity.

For a fixed cusp basis $R_a,R_d$ with pairing two, the deformation

$$\theta^{p,q}=\theta^0+\frac{pJR_d-qJR_a}{p^2+q^2}$$

is a filling certificate whenever it is positive. A uniform sufficient condition is

$$p^2+q^2>\max_j\frac{(JR_a)_j^2+(JR_d)_j^2}{(\theta_j^0)^2}.$$

Thus the actual named filled vacuum is nonzero for all sufficiently large primitive slopes on a triangulation carrying a complete strict certificate. This is a sufficient chamber, not an exceptional-slope classification. Do not enumerate its bounded complement.

New exact normal-index bridge

Let $n(\lambda)=\Omega(\lambda,b)/2$. The balanced convergence character is

$$\chi_0(\lambda)=\chi(\lambda)A^{-2n(\lambda)} =(-1)^nB^{2E-p(R)}, \quad R=\sum_en_eE_e+nR_s,\quad E=\sum_en_e.$$

Do not discard the residual phase $(-1)^n$. Write $F_0$ for the full unsaturated average with $\chi_0$. The complete strict certificate supplies all-input convergence, and

$$\mathcal G_{-b}(a)=F_0(W(-b)a)$$

descends to the original target.

For every integer $r,n$, the map

$$\lambda\longmapsto X=\max\bigl((R+rR_l)/2\bigr)\Delta-(R+rR_l)/2$$

is a bijection from $\{\lambda\in\Lambda:n(\lambda)=n,\ \lambda+rb\in2L\}$ to the minimum-zero integral normal classes with $\partial X=-rl-ns$. The maxima are blockwise. Surjectivity follows from the integer identity

$$2X+nR_s+rR_l\in\mathbb E+\mathbb T,$$

by GHHR Theorem 7.1; it does not saturate $\Lambda$. All mod-2 homology classes in that boundary fiber are included.

Let $\mathcal Z(\gamma)=\sum_aI_{\mathcal T}^a(\gamma)$, the total GHHR boundary index. Then

$$\boxed{F_0(l^r)=\sum_{n\in\mathbb Z}(-1)^nA^{rn}\mathcal Z(-rl-ns).}$$

Every summand has exact degree $2(\delta(X)-\chi_{\rm top}(X))+rn$, and leading coefficient $(-1)^{n-\chi_{\rm top}(X)}$. The prior all-input theorem justifies grouping the individual unordered-summable terms into these fibers.

New balanced-pairing failure theorem

GHHR’s published Remark 8.3, printed p. 312, states $I^a(-\gamma)=I^a(\gamma)$. The published source and arXiv version were freshly checked. This is a formal external dependency, not a new independent inversion proof.

After the exact bridge, legitimate reindexing $n\mapsto-n$ yields $F_0(l^{-r})=F_0(l^r)$. Therefore

$$\boxed{\mathcal G_{-b}(1-l^2)=0.}$$

This is not termwise root inversion: the seed itself is not inversion-invariant. It is not zero target evidence. In fact

$$\mathcal G_{-b}(l)=c_lF_0(1) =c_l(1+\text{positive }A\text{-order})\ne0.$$

Thus the actual meridian cokernel is nonzero from the complete certificate alone; the actual vacuum remains a separate question.

For one torus boundary, every monomial left translate certified by the same complete vector also kills the vacuum. Writing $w=v+b$, orthogonality forces $w\in\Lambda\otimes\mathbb R$. If $w\in\Lambda$, the functional is $\chi(w)A^{-\Omega(-b,w)/2}\mathcal G_{-b}$. If $w\notin\Lambda$, integral normal homology forces both vacuum parity supports to be empty. The same dichotomy holds in multiple cusps when $w\in\Lambda\otimes\mathbb R$, but does not exhaust additional other-cusp directions.

The all-plus vanishing theorem has not been extended to every central sign sector. No all-purpose pairing has been excluded.

Preserved failures and limits

The preceding abstract strict-filling-angle example has a nonzero cokernel unit and an algebraically zero specified vacuum, but violates actual nonnegative edge incidence and lacks a complete nonnegative certificate. It remains an abstract obstruction only.

No target-wide nonvanishing or zero result follows from a scalar index alone. No new surgery invariant or identification with the Celoria–Hodgson–Rubinstein Dehn-filling index is asserted. Duan–Garoufalidis–Gu’s general factorization-based symmetry route is conjectural and was not used as proof. No researcher contact is authorized by this ordinary research note.

Next mode and decisive question

ATTACK — a correct-sided detector outside the exhausted balanced family.

Retain the full integral normal-index bridge and all prior chamber results. Do not repeat same-complete-character monomial translates, use lattice saturation to create artificial vacuum support, assume a descendant breaks inversion, reopen peripheral signs, or enumerate manifolds, slopes, triangulations, normal supports, or residual configurations.

For multiple cusps, there is an explicit remaining test. Let $t$ be a native curve on another cusp, with $w=P(R_t)$, $c_t=B^{p(R_t)}$. The same complete certificate already proves convergence and original-target descent of $\mathcal G_{w-b}$, and the exact normal bridge gives

$$\mathcal G_{w-b}(1-l^2)=\frac{c_l}{c_t}\sum_n(-1)^nA^n [\mathcal Z(t-l-ns)-\mathcal Z(-t-l-ns)].$$

The most explicit next gate is its exact value or a structural reason it vanishes. Do not assume independent-cusp inversion symmetry or asymmetry; the sourced simultaneous boundary inversion alone does not settle this expression. No nonzero value is currently proved.

For one cusp, seek a nonperipheral left insertion with a genuinely different certifiable moment-character valuation, or a rigorously defined second seed solution, that detects $1-l^2$ on the original target. All-input convergence and original moment descent must be proved first. A derivative of a character creates an inhomogeneous moment relation and requires an explicit correction; it is not automatically a new detector. Removing the complete positive angular certificate remains unresolved.

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