FPRD research loop state

Latest completed mode

ATTACK — a one-sided Nahm functional with an exact all-input convergence criterion.

An Euler quantum-dilogarithm coefficient seed solves the one adjoint recurrence required by the actual tetrahedron relation, while failing the extra recurrence imposed by the ordinary tetrahedron index. Its full-image average converges on every algebraic input exactly when Q(x,y)=||x||²−x·y is strictly positive on the nonzero cone of moment directions with x≥0. The sum descends with the original relation sides, odd roots, and exact native character. A native phase and pure-w longitude criterion then detects 1−l²; a complete strict angular certificate supplies the needed positive longitude scalar. No filling-angle assumption enters that conditional implication.

No additional actual triangulation/filling has yet been certified by these new criteria. The explicit example is abstract and lacks actual edge/normal data. This is a constructed alternative detector and conditional theorem, not an established new topological chamber or Publication. The previous actual odd peripheral value and unrestricted Panitch–Romo conjecture remain open. AI-assisted; no external specialist review; novelty 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

one-sided-seed-attack/:

• FPRD-one-sided-Nahm-detector-2026-09-08.md: complete seed, exact cone theorem, original-sided descent, graph/Nahm translation, conditional vacuum theorem, and abstract example.
• internal-audit.md, source-ledger.md: self-audit, dependencies, and intellectual influences; no external or independent specialist audit is asserted.
• verify-one-sided-seed.py, exact-results.json: exact local identities and global example checks with quadratic tail certificates; full-root and unsaturated-image checks.
• README.md, build-release.py, SHA256SUMS.txt, one-sided-seed-attack.zip, ARCHIVE-SHA256.txt, archive-validation.txt: reproducibility and integrity.

Proof: https://fprd-lab.joshgay.chatgpt.site/evidence/dehn-filled-quantum-trace/one-sided-seed-attack/FPRD-one-sided-Nahm-detector-2026-09-08.md

The preceding peripheral-projection-attack/ remains the integral classification of supported same-complete-character translates, exact peripheral scalar, and odd projection. Every ordinary remaining-cusp boundary-skein insertion has vacuum value zero; the actual odd part remains unresolved.

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 retained ordinary-index seed vanishes off $2L$ and equals $\prod_jI_\Delta(d_j,-a_j)$ on $W(2a_1,2d_1,\ldots)$.

Current one-sided construction

Use grouped coordinates (z-vector,w-vector) for this section only. Put s(a)=sum a_j, q=A⁴. The new seed vanishes off 2L and when any a_j<0; otherwise

$$\ell_+(W(2a,2d))=\frac{(-1)^{s(a)}A^{2(\|a\|^2+s(a)-a\cdot d)}}{\prod_j(q)_{a_j}}.$$

It solves the plus recurrence required by Sf_j. Its minus-recurrence residual at e=0 is −q^(1−m)/(1−q), so the established two-recurrence rank-two classification does not restrict this seed. These local coefficients are classical Euler quantum-dilogarithm coefficients.

Define H(a)=sum_{lambda in Lambda} chi(lambda)^(-1) ell_+(W(lambda)a), keeping the actual unsaturated image. Let V=(Lambda tensor R)/2, C_+={(x,y) in V:x>=0}, Q=||x||²−x.y. All-input unordered convergence is equivalent to Q>0 on C_+ minus {0}. If the cone is zero, each input has finite support. A rational negative ray defeats the unit; a nonvertical rational null ray is defeated by a sufficiently shifted w-input, and a vertical null ray defeats one sign already at the unit. Rational null rays follow from the rational restricted Hessian kernel on the minimal face; no faces were enumerated.

For u=2(alpha,beta), lambda=2(x,y), supported only if lambda+u in 2L and x+alpha>=0, the exact summand order is

$$D_u=2Q+2s(x)-\nu_A\chi(\lambda)+4(\alpha-\beta)\cdot x +2(\|\alpha\|^2+s(\alpha)-\alpha\cdot\beta).$$

Strict cone positivity supplies quadratic coercivity on every affine support. Then H(a f_j)=0 and H(W(mu)a)=chi(mu)H(a) for every mu in Lambda, proving original-sided descent. No angle or shape point is required for this convergence theorem.

Projection to x is injective, so y=Tx on V_z; isotropy makes x.T x’ symmetric. The criterion is strict copositivity of I−T on V_z intersect the nonnegative orthant. In the full-rank one-cusp case V_z=R^N. This is not necessarily positive definiteness and not a claimed efficient decision procedure. Retain Gamma={x in Z^N:2(x,Tx) in Lambda}, not its saturation.

Sufficient vacuum certificate: the exact phase satisfies eta(2x,2Tx)=(-1)^s(x) on Gamma; the declared native longitude has b=(0,-t), nonzero integral t>=0, c_l=eta_l A^rho; rho>0. Unit summands then have positive coefficients, and the longitude-square/unit summand ratio is eta_l² A^(2rho)q(t.x). Hence H(1−l²) is nonzero. Under a native complete strict angular certificate and zero longitude holonomy, rho=p(R_l)=sum alpha_j t_j>0, so no separate filling angles are needed. The phase and pure-w form must be proved for the original data, not forced by redefinition.

For actual raw R=sum n_e E_e+n_s R_s with P(R) even, the exact phase condition is equivalently sum_j R_j1 congruent to 2(sum_e n_e+n_s) mod 4. This follows from eta(P(R))=(-1)^(sum n_e+n_s)i^(-p(R)) and 2s(x)=sum R_j1−p(R). Check it on the actual even image, not a saturation or a chosen new sign sector.

Unresolved applicability: no new actual triangulation/filling satisfies these joint criteria in the evidence yet. The three-block abstract example has h=(2,2,0;0,0,0), chi(h)=A⁴, b=(0,0,0;−1,0,−2), c_l=iA, Q=2k², and H(1−l²)=sum_{k>=0}q^(k²)(1+A²qk)/(q)_k². It has formal complete strict angles but no strict filling angles; it is not actual triangulation data. Do not call it an enlargement of the known topological chamber.

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 actual realizability of the new one-sided vacuum certificate, or evaluate the exact signed Nahm functional without its sufficient positivity assumptions.

The next decisive gate is whether genuine native triangulation incidence and a complete strict certificate can satisfy the joint strict-cone, native-phase, and native pure-w longitude hypotheses outside the established two-angle chamber. Work with an intrinsic construction or obstruction, preserving all edge rows, raw peripheral rows, unsaturated image and phases. The abstract three-block example does not settle this gate. If the sufficient vacuum hypotheses fail, the exact all-input cone theorem remains available and the signed Nahm sum can still be investigated.

The alternative actual odd peripheral value B restricted to T_o^- remains unresolved. Do not repeat inversion-even boundary insertions, same-complete-character translate classification, saturation constructions, or conjectural separate-cusp Weyl symmetry as if they solve it. A differentiated character has an inhomogeneous moment relation. The new half-space seed is an explicit different one-recurrence solution; do not incorrectly force it into the two-recurrence rank-two space. Reassembling the full index by the extra local sum requires a separate global interchange proof.

Retain all earlier actual two-angle and uniform large-slope nonvanishing results. Do not enumerate manifolds, slopes, triangulations, normal supports, or residual configurations. 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.