FPRD research loop state

Latest completed mode

ATTACK — integral classification of complete-character translates and boundary-skein non-detection.

Every supported monomial translate certified by the same complete strict angle vector now reduces, with an exact native scalar, to a peripheral insertion on the remaining cusps. The resulting vacuum functional is odd under simultaneous inversion of those peripheral classes. Consequently every ordinary remaining-cusp boundary-skein insertion, including arbitrary noncommuting products of boundary curves, annihilates the vacuum. The separate-cusp oriented odd value remains unresolved. This is a supporting research note, not a Publication. AI-assisted; no external specialist review or novelty claim.

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

peripheral-projection-attack/:

• FPRD-peripheral-projection-2026-09-08.md: full integral multiple-cusp classification, exact scalar, inversion-odd vacuum projection, and boundary-skein non-detection.
• internal-audit.md, source-ledger.md: self-audit and exact dependency boundaries; no independent specialist audit is asserted.
• verify-peripheral-projection.py, exact-results.json: noncommutative quantum-torus certificates and exact finite-support formal boundary witnesses. These witnesses are not actual normal indices.
• README.md, build-release.py, SHA256SUMS.txt, peripheral-projection-attack.zip, ARCHIVE-SHA256.txt, archive-validation.txt: reproduction and integrity.

Proof: https://fprd-lab.joshgay.chatgpt.site/evidence/dehn-filled-quantum-trace/peripheral-projection-attack/FPRD-peripheral-projection-2026-09-08.md

The preceding balanced-pairing-attack/ packet remains the integral normal-fiber and exact phase bridge, sourced balanced vanishing theorem, and one-cusp stopping result.

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.

Retained 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.

Retained 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.

New multiple-cusp classification and odd projection

Let $\Gamma_o=\bigoplus_{T\ne T_0}H_1(T;\mathbb Z)$. For $w=v+b\in L\cap\Lambda^\perp$, nonempty vacuum seed support is equivalent to

$$w=\lambda_0+w_\gamma,\quad\lambda_0\in\Lambda,\quad \gamma\in\Gamma_{\rm adm}:= \{\gamma:\iota_*(\gamma-l)\in\operatorname{span}_{\mathbb F_2}\{\iota_*s\}\}.$$

The representation is unique for fixed additive native rows. The integral proof chooses an integral normal lift V of w and forms the minimum-zero class X from (V−R_l+R)/2. Integrality of its boundary forces the other-cusp and meridian coefficients of ∂V to be even; the doubled normal identity forces [V]₂=0. GHHR’s integer kernel theorem then gives the decomposition. No saturation or assumed half-integral peripheral lift is used. Unsupported translates vanish on the vacuum only, not necessarily on all other inputs.

For $n_0=n(\lambda_0)$, the exact all-input scalar is

$$\mathcal G_{w-b} =\chi(\lambda_0)A^{-n_0}c_\gamma^{-1}\mathcal G[E_\gamma], \quad\mathcal G[P](a)=F_0(W(-b)Pa),\quad E_\gamma=c_\gamma W(w_\gamma).$$

On the other-cusp quantum torus define $\sigma(E_\gamma)=E_{-\gamma}$, fixing A and coefficients. This is an algebra automorphism: it preserves product order and Weyl factors. For the original-target vacuum functional

$$\mathscr B(P)=\mathcal G[P](1-l^2),$$

the exact retained normal bridge gives

$$\mathscr B(E_\gamma)=c_l\sum_n(-1)^nA^n [\mathcal Z(\gamma-l-ns)-\mathcal Z(-\gamma-l-ns)], \qquad \mathscr B\circ\sigma=-\mathscr B.$$

Consequently $\mathscr B(P)=\mathscr B((P-\sigma P)/2)$. Frohman–Gelca’s torus-skein image is generated by $E_\gamma+E_{-\gamma}$ on each cusp, hence every ordinary remaining-cusp boundary-skein insertion has vacuum value zero. This covers arbitrary noncommuting products without assuming independent-cusp symmetry of Z. It does not extend σ to the whole mixed-sided target or assert zero value on all target inputs.

The formal boundary witnesses Z₀,Z₁ have common unit normalization, simultaneous inversion, admissible Lagrangian mod-2 support, finite summability and positive nonunit order, but the oriented test is respectively 0 and A⁴. They do not satisfy any claimed actual tetrahedron presentation and are not triangulation counterexamples. They prove only that the listed boundary constraints cannot decide the odd value.

Next mode and decisive question

ATTACK — determine the actual odd peripheral functional, or construct a different correct-sided seed.

The precise multiple-cusp gate is now $\mathscr B|_{\mathscr T_o^-}$. A successful continuation must either prove independent-cusp inversion for the actual total GHHR normal index, or produce a nonzero oriented value of the displayed difference from actual tetrahedron/normal data with a certified remainder. Its all-input convergence and original-target descent are already established. Its value is still open. The new classification is a reduction, not its evaluation.

Do not repeat inversion-even boundary insertions, polynomial words of ordinary boundary curves, saturation-based support constructions, or same-complete-character translations as if they were unclassified. Do not treat classical Weyl symmetry, unoriented curve invariance, or conjectural block factorization as a proof of the missing separate-cusp symmetry. The normal-index formal witnesses do not settle the actual topological question.

For one cusp, $\Gamma_o=0$ and the same-complete-character route remains exhausted. Seek a nonperipheral insertion with a different certifiable moment character, or a rigorously constructed second seed, proving all-input convergence and original mixed-sided descent first. A differentiated character has an inhomogeneous moment relation and needs an explicit correction.

Retain the two-angle and uniform sufficiently-large-slope nonvanishing theorems. Do not enumerate manifolds, slopes, triangulations, normal supports, or residual configurations. Do not reopen peripheral signs. Panitch–Romo Conjecture 7.1 remains open; no new publication or author outreach is warranted.

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.