A global index detector for the native meridian cokernel

FPRD Lab, 8 September 2026. ATTACK research note. AI-assisted work, internally audited; no external specialist review or author endorsement. Novelty and first priority are unassessed.

Result and scope

There is a global, explicitly defined difference-equation functional on the actual generic native six-root meridian cokernel, provided its exact moment character is consistent and it admits a nonnegative filling-angle datum with no zero-degree ray. The functional detects the original algebraic cokernel unit. Strict filling angles imply the no-ray condition, but are not necessary for this argument.

The domain is the original finite Laurent presentation, not its formal completion at a shape point. Only the coefficient field is extended to Laurent series at $A=0$. No classical shape solution, regular intersection, or positive geometric realization is required. This removes those hypotheses from this conditional nonvanishing criterion; it does not establish that every actual filling satisfies the new criterion, or exhibit an actual filling outside the earlier geometric chamber.

The construction averages the established tetrahedron index over the full actual edge-plus-meridian lattice. Its ingredients and convergence strategy substantially reuse the 3D-index literature. The contribution of this note is the explicit all-root, exact-character, filled-presentation argument and its no-ray boundary, not the invention of the tetrahedron index or of lattice averaging.

The filled vacuum is not detected by the result below. The class to test for a nonzero descended trace is $1-l^2$, not the cokernel unit. Its exact functional value is isolated in Section 8, without a nonvanishing assertion. Panitch–Romo Conjecture 7.1 remains open in this work. This is supporting research, not a new Publication.

1. The actual presentation and its exact character

Retain the preceding six-root peripheral/collar packet, including its source-reading caveat for the CK comparison. Work over

$$k=\mathbb Q(i,A^{1/2}),\qquad K=\mathbb Q(i)((A^{1/2})),\qquad q=A^4. \tag{1}$$

The embedding $k\hookrightarrow K$ is injective. It is an expansion at $A=0$, not the substitution $A=e^\hbar$ used in the clean-excess note.

Fix one central sign sector of the native full six-root presentation. Central vertex elimination gives $y=\sigma z, y'=\sigma z', y''=\sigma z''$, with the fixed value $\sigma=\pm1$ in each tetrahedron, and $W(z+z'+z'')=iA$. Set $w=z''$. The remaining quantum torus $S$ has lattice $L=\mathbb Z^{2N}$ and Weyl multiplication

$$W(u)W(v)=A^{\Omega_2(u,v)/2}W(u+v),\qquad \Omega_2=\bigoplus_{j=1}^N\begin{pmatrix}0&-1\\1&0\end{pmatrix}. \tag{2}$$

In particular $wz=Azw$ and $W(a,b)=A^{ab/2}z^aw^b$ in one block. The tetrahedron relation is

$$f_j=z_j^{-2}+w_j^2-1, \tag{3}$$

with arbitrary multipliers on its left. Every actual edge relation and the chosen native meridian relation have the form

$$(c_jW(h_j)+A^2)S, \tag{4}$$

with arbitrary multipliers on their right. The $c_j$ are the exact central-elimination scalars, including all signs and powers of $i,A$. One row is the named operator $m=c_sW(h_s)$; it is not replaced by a square or by another meridian. The prior native action supplies the isotropy

$$\Omega_2(h_j,h_k)=0 \tag{5}$$

for all actual edge and filling rows. Put $\Lambda=\sum_j\mathbb Z h_j\subset L$. This is the actual image lattice: it is neither saturated nor replaced by $2\Lambda$.

The character consistency condition is that

$$\chi(h_j)=-A^2/c_j \tag{6}$$

extends to a group homomorphism $\chi:\Lambda\to k^\times$. Equivalently, every integer relation $\sum_j n_jh_j=0$ must satisfy $\prod_j(-A^2/c_j)^{n_j}=1$. Checking only valuations is insufficient. Conversely, a failure gives a nonzero scalar among the moment relations, and that sector is zero: one can compose the commuting left monomials along the dependency. Thus this condition is necessary, not an invented normalization.

The sector of the actual cokernel is exactly

$$C=S\Big/\left(\sum_jSf_j+\sum_j(c_jW(h_j)+A^2)S\right). \tag{7}$$

All rows occur in (7); no assertion about redundant edges in every sign sector is needed. A functional on (7) pulls back through the central quotient to the full native six-root cokernel. If it detects (1), the full cokernel is nonzero as well. No converse to opposite-root identification is used.

For reference, a local six-root row $u$ reduces by

$$\begin{gathered} r=u_z+u_y,\quad s=u_{z'}+u_{y'},\quad t=u_{z''}+u_{y''},\quad b=u_y+u_{y'}+u_{y''},\\ W_6(u)=\sigma^b(iA)^sW_2(r-s,t-s). \end{gathered} \tag{8}$$

This records the scalar information needed in (6).

2. Filling angles and the zero-degree cone

For each tetrahedron choose rational numbers

$$\theta_j=(\gamma_j,\beta_j,\alpha_j),\qquad \gamma_j,\beta_j,\alpha_j\geq0,\qquad \gamma_j+\beta_j+\alpha_j=1. \tag{9}$$

For $u=(a_1,b_1,\ldots,a_N,b_N)\in L$, define

$$\vartheta(u)=\sum_j(\gamma_ja_j+\alpha_jb_j). \tag{10}$$

A nonnegative filling-angle datum means (9) and

$$\vartheta(\lambda)=\nu_A\chi(\lambda)\quad(\lambda\in\Lambda), \tag{11}$$

where $\nu_A(A^r)=r$. It is enough to impose (11) on the actual generating rows. For (8), equations (6) and (11) become, summed over tetrahedra,

$$\sum\bigl(\gamma(r-s)+\alpha(t-s)\bigr)=2-\sum s, \quad\text{equivalently}\quad \sum(\gamma r+\beta s+\alpha t)=2. \tag{12}$$

All sums in (12) run over tetrahedra. Additional scalar powers, if present in an explicitly supplied native row, must be retained through (11). Ordinary unfilled edge angles do not by themselves satisfy the extra filling equation. The value (2) is fixed by the actual $+A^2$ relation; it cannot be changed to obtain feasibility.

Introduce the linear map and tetrahedron translations

$$T(a_1,b_1,\ldots)=(-a_1,0,-b_1,\ldots),\qquad \Delta_j=(0,\ldots,1,1,1,\ldots,0). \tag{13}$$

Let

$$\Lambda_0=\Lambda\cap2L,\qquad H=T(\Lambda_0/2)+\sum_j\mathbb Z\Delta_j,\qquad \mathcal C=(H\otimes\mathbb R)\cap\mathbb R_{\geq0}^{3N}. \tag{14}$$

The intersection with $2L$ in this auxiliary convergence lattice does not change the averaging lattice $\Lambda$. Define

$$\delta(x)=\sum_j(x_{j1}x_{j2}+x_{j2}x_{j3}+x_{j3}x_{j1}). \tag{15}$$

The no-null-ray condition is

$$x\in\mathcal C,\quad \delta(x)=0,\quad\theta\cdot x=0 \quad\Longrightarrow\quad x=0. \tag{16}$$

Since coordinates are nonnegative, $\delta(x)=0$ means that at most one coordinate in each triple is positive. Condition (16) is structural; no list of such supports is generated here. Strictly positive angles imply it immediately.

The value $\theta\cdot x$ on $H\otimes\mathbb R$ is fixed by the character valuations and the sums in (9): on $T(\mu/2)+\sum d_j\Delta_j$ it is $-\nu_A\chi(\mu)/2+\sum d_j$. Thus (16) is independent of which feasible angle datum is used. Angles certify estimates; they do not enter the functional below.

3. The tetrahedron-index seed is the required adjoint solution

Use the established tetrahedron index

$$I_\Delta(m,e)=\sum_{n=\max(-e,0)}^\infty \frac{(-1)^nq^{n(n+1)/2-(n+e/2)m}} {(q;q)_n(q;q)_{n+e}},\qquad (q;q)_n=\prod_{r=1}^n(1-q^r). \tag{17}$$

It satisfies

$$q^{e/2}I_\Delta(m+1,e)+q^{-m/2}I_\Delta(m,e+1)-I_\Delta(m,e)=0. \tag{18}$$

The seed and recurrence are sourced from the DGG construction and its rigorous treatment in Garoufalidis, Definition 3.1 and Section 4.5. They are not proved from finite coefficient tests here.

Define a $k$-linear map $\ell_0:S\to K$ by

$$\ell_0(W(u))=0\quad(u\notin2L),\qquad \ell_0(W(2a_1,2b_1,\ldots))=\prod_j I_\Delta(b_j,-a_j). \tag{19}$$

The Weyl convention gives, in block $j$,

$$W(u)f_j=A^{-u_{wj}}W(u-2e_{zj}) +A^{-u_{zj}}W(u+2e_{wj})-W(u). \tag{20}$$

For even $u$, applying (19) turns (20) into (18), with $m=u_{wj}/2, e=-u_{zj}/2$. For any odd coordinate, all three seed values vanish. Therefore

$$\ell_0(Sf_j)=0\quad\text{for every actual tetrahedron relation}. \tag{21}$$

The zero extension in (19) alone does not handle odd moment shifts. The full lattice average in the next section does.

We also use the exact degree formula. Put

$$J_\Delta(a,b,c)=(-q^{1/2})^{-b}I_\Delta(b-c,a-b). \tag{22}$$

If $m=\min(a,b,c)$ and $x=(a-m,b-m,c-m)$, then

$$\deg_{q^{1/2}}J_\Delta(a,b,c)=\delta(x)-m. \tag{23}$$

In particular these series are nonzero. This is the minimum-degree formula in Garoufalidis–Hodgson–Hoffman–Rubinstein, Section 2, equation (7) of arXiv v1, also reproduced as equation (62) of GY-index. The source’s $q$-degree is multiplied by two here to obtain $q^{1/2}$-degree. The normalization gives $I_\Delta(0,0)=1+O(q)$.

4. The global average and theorem

Define the candidate on every root monomial by

$$\mathcal F(W(u))= \sum_{\lambda\in\Lambda}\chi(\lambda)^{-1} \ell_0(W(\lambda)W(u)) =\sum_{\lambda\in\Lambda} \chi(\lambda)^{-1}A^{\Omega_2(\lambda,u)/2} \ell_0(W(u+\lambda)). \tag{24}$$

A family in $K$ is summable here if, below every fixed $A$-degree, only finitely many family members are nonzero. This ensures an order-independent sum and legitimizes reindexing. Ordinary finite-box cancellation is not a substitute.

Theorem 4.1. Global actual-cokernel detection

For the actual sector presentation (7), assume exact character consistency (6), a nonnegative filling-angle datum (9)–(11), and (16). Then (24) is summable for every $u\in L$. It extends $k$-linearly to the original algebraic $S$, annihilates every tetrahedron, actual edge and filling relation, and descends to

$$\mathcal F:C\longrightarrow K, \qquad \mathcal F([1])\in1+A^{1/2}\mathbb Q(i)[[A^{1/2}]]. \tag{25}$$

In particular $C\neq0$, and the actual full six-root meridian cokernel is nonzero.

Under the nonnegative-angle hypothesis, condition (16) is also necessary for the specific unordered average (24) to converge at (1). Its failure is not a vanishing theorem for $C$ and does not rule out a different detector.

Corollary 4.2. Strict filling angles

Exact character consistency and strictly positive filling angles suffice for actual full-six-root meridian-cokernel nonvanishing. Neither a classical root point nor regularity of a classical intersection is part of this corollary.

5. Convergence on every algebraic input

We give the estimate in full because a functional defined only on the unit would not establish descent. Let

$$\Omega_3=\bigoplus_j\begin{pmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{pmatrix},\qquad D^{\rm sym}=\bigoplus_j\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}. \tag{26}$$

Then $\Omega_3(Tu,Tv)=\Omega_2(u,v)$, each $\Delta_j$ is in the radical of $\Omega_3$, and $\delta(x,y)=x^TD^{\rm sym}y/2$ polarizes (15). Entrywise nonnegativity of $D^{\rm sym}-\Omega_3$ gives

$$(2\delta-\Omega_3)(x,y)\geq0\quad(x,y\geq0). \tag{27}$$

These are the same quadratic/skew positivity ingredients used in the convergence proof of GY-index, Theorem 5.3. That theorem concerns its stated even-target presentation and hypotheses; the root-lattice and filling checks below are additional, not an application that silently drops those restrictions.

Fix $u\in L$. If $(u+\Lambda)\cap2L$ is empty, (24) is zero and there is nothing to prove. Otherwise, for each allowed $\lambda$ put $v=(u+\lambda)/2\in L$. Reduce $T(v)$ by subtracting its blockwise minima:

$$m_j=\min T(v)_j,\qquad x=T(v)-\sum_jm_j\Delta_j\in\mathbb N^{3N}. \tag{28}$$

Each block of $x$ has minimum zero. Formula (23) and $I_\Delta(b,-a)=J_\Delta(-a,0,-b)$ give the exact valuation of the corresponding nonzero summand in (24):

$$\begin{aligned} d_u(x) &= -\vartheta(\lambda)+\Omega_2(\lambda,u)/2 +2\delta(x)-2\sum_jm_j\\ &=2\delta(x)+2\theta\cdot x+\vartheta(u)+\Omega_3(x,T(u)). \end{aligned} \tag{29}$$

For the second line use $\lambda=2v-u$, $\theta\cdot T(v)=-\vartheta(v)$, $\theta\cdot\Delta_j=1$, and the radical property. This checks both the Weyl phase and the exact $A^2$ moment shift.

Choose one allowed $v_0$. Every reduced $x$ in (28) belongs to the affine lattice $T(v_0)+H$. Let

$$P_u=(T(v_0)+H)\cap\mathbb N^{3N}. \tag{30}$$

By Dickson’s lemma this set has finitely many coordinatewise minimal elements. Every $x\in P_u$ dominates one of them, say $b\leq x$; this also follows by choosing a minimal point in the finite box below $x$. No computation of these minima is needed. Dickson’s lemma itself follows by induction from the fact that every infinite sequence in $\mathbb N$ has an infinite nondecreasing subsequence, applying that fact successively to the coordinates.

For reduced $x$, the chosen $b\leq x$ is reduced too. Put $F=x-b\in H\cap\mathbb N^{3N}$. Write

$$b=T((u+\lambda_b)/2)+\Delta\text{-translation},\qquad F=T(\mu/2)+\Delta\text{-translation}, \tag{31}$$

where $\lambda_b\in\Lambda, \mu\in\Lambda_0$. Isotropy gives

$$\Omega_3(F,T(u))=-2\Omega_3(b,F). \tag{32}$$

Expanding (29) now yields the decisive identity

$$d_u(b+F)-d_u(b) =2\delta(F)+2\theta\cdot F +2(2\delta-\Omega_3)(b,F) \geq 2\delta(F)+2\theta\cdot F. \tag{33}$$

The linear phase that is uncontrolled for arbitrary input has become a nonnegative mixed term. This is why convergence at the unit alone was not enough.

Under (16), the continuous function $\delta(y)+\theta\cdot y$ is strictly positive on the compact set

$$\{y\in\mathcal C:\|y\|_1=1\}. \tag{34}$$

Both summands are nonnegative; their simultaneous vanishing is exactly the excluded case. Let its positive minimum be $\varepsilon$. For every nonzero integral $F\in H\cap\mathbb N^{3N}$, write $F=ty$ with $t=\|F\|_1\geq1$. Then

$$\delta(F)+\theta\cdot F =t^2\delta(y)+t\theta\cdot y \geq\varepsilon\|F\|_1. \tag{35}$$

There are finitely many $b$, and (33)–(35) give finitely many $x$ below any valuation bound. Finally, the map $\lambda\mapsto x$ is injective: equality of reduced representatives implies $T((\lambda-\lambda')/2)\in\sum\mathbb Z\Delta_j$; its middle coordinates are zero, so all translation coefficients and then $\lambda-\lambda'$ vanish. This proves summability in (24) for every root monomial, hence for every finite Laurent input.

6. Descent and the detected original algebraic class

For $h\in\Lambda$, isotropy and the now justified reindexing $\mu=\lambda+h$ give

$$\begin{aligned} \mathcal F(W(h)s) &=\sum_{\lambda\in\Lambda}\chi(\lambda)^{-1} \ell_0(W(\lambda+h)s)\\ &=\chi(h)\mathcal F(s). \end{aligned} \tag{36}$$

Therefore $\mathcal F((c_jW(h_j)+A^2)s)=0$ for each actual edge/filling row. Also

$$\mathcal F(sf_j)=\sum_{\lambda\in\Lambda}\chi(\lambda)^{-1} \ell_0((W(\lambda)s)f_j)=0 \tag{37}$$

by (21) and associativity. Finite linearity and convergence justify this termwise calculation. The central vertex relations were already killed by the sector projection. This proves every relation required by (7), on its required side.

Equivalently $\phi(u)=\mathcal F(W(u))$ is the global adjoint solution

$$\begin{gathered} A^{-u_{wj}}\phi(u-2e_{zj})+A^{-u_{zj}}\phi(u+2e_{wj})-\phi(u)=0,\\ \phi(u+h)=\chi(h)A^{-\Omega_2(h,u)/2}\phi(u). \end{gathered} \tag{38}$$

These equations hold on all of $L$, including the odd root cosets. Retaining only even moments would not prove the second equation.

At $u=0$, the $\lambda=0$ summand is $I_\Delta(0,0)^N=1+O(A^4)$. Every other contributing $\lambda\in\Lambda_0$ has a nonzero reduced $x\in\mathcal C$ and

$$d_0(x)=2\delta(x)+2\theta\cdot x>0 \tag{39}$$

by (16). Thus the constant coefficient of $\mathcal F(1)$ is exactly (1). This is a value on the original algebraic class, not on a newly created completed class. A zero $k$-vector space cannot admit this functional, so generic algebraic nonvanishing follows. No comparison with a shape-local completion is required.

7. The exact failure at a null ray

Suppose (9)–(11) hold but (16) fails. Choose a nonzero $x\in\mathcal C$ with $\delta(x)=\theta\cdot x=0$. Its support has at most one coordinate in each tetrahedron. On that support the equations defining $H\otimes\mathbb R$, the zero coordinates, and $\theta\cdot x=0$ are rational linear equations. A nonzero nonnegative real solution therefore supplies a nonzero rational solution. Multiplying by an integer puts it in $H$; denote the resulting vector by $F$.

Write $F=T(\mu/2)+\sum d_j\Delta_j$, $\mu\in\Lambda_0$. Since $F$ is reduced and nonzero, $\mu\neq0$. The reduced representative of $T(n\mu/2)$ is (nF) for each positive integer $n$. Its summand in (24) at $u=0$ has exact valuation

$$2n^2\delta(F)+2n\theta\cdot F=0. \tag{40}$$

Each such summand is nonzero by (23). Infinitely many family members have valuation zero, so the unordered $A$-adic sum is not defined. Oscillating signs or cancellation after a chosen regrouping do not repair this failure of summability. A regularized value would require a new definition and a new all-relations proof.

The weak-angle criterion is not merely a restatement of strict-angle feasibility at the matrix level. Consider the abstract one-block moment data $\Lambda=\mathbb Z(0,2)$, $\chi(0,2)=A^2$. Its cone (14) is $x_1=x_2\geq0, x_3\geq0$. The only nonnegative angle datum is $\theta=(0,0,1)$, so no strict datum exists. Yet $\delta(x)=x_1^2+2x_1x_3$, and $\theta\cdot x=x_3$, have no nonzero simultaneous zero in the cone. The theorem applies to this algebraic moment model. These rows are not claimed to be an actual manifold triangulation; the model proves a logical distinction between hypotheses, not a new topological example.

8. The filled-vacuum test remains separate

The retained intrinsic filling isomorphism sends the filled unit to $[(1-l^2)1]\in C_s^6$. Write the actual named longitude in the selected sector as $l=c_lW(b)$. Because $\Omega_2(b,b)=0$,

$$\mathcal F(1-l^2) =\mathcal F(1)-c_l^2\mathcal F(W(2b)). \tag{41}$$

Both series are well-defined under Theorem 4.1, but (39) only controls the first. The second uses the affine input degree $d_{2b}$, which has the extra longitude phase. There is no proof here that their difference is nonzero. One sufficient additional test would be a proved positive valuation for $c_l^2\mathcal F(W(2b))$, but no universal such bound is asserted.

Consequently this note proves target nonvanishing, not a nonzero descended trace and not detection of the filled empty skein. The earlier positive-geometric vacuum result remains available only with its earlier hypotheses. An identification with a surgery invariant or cycle pairing would have to retain the same longitude insertion, exact quantum character, parity and convergence, and is not supplied here.

9. Preserved failures and literature boundary

A finite Laurent coefficient density cannot supply the needed adjoint solution. For the nondegenerate pairing $\langle s,p\rangle=\operatorname{CT}(sp)$ on the quantum torus, annihilating $Sf_j$ forces $f_jp=0$. The quantum torus is a domain, so a finite Laurent $p$ must be zero. The same argument applies in any specified domain completion where this pairing remains separating and multiplication has no boundary terms. It is not a theorem ruling out rational residues with jumps, contours, or bilateral distributions. The functional (24) belongs to the algebraic dual; no domain-valued Laurent density is claimed.

The earlier clean-excess theorem and the global Fourier countermodel remain consistent with this result. Local formal detectors at $A=e^\hbar$ may disappear while a global finite-Laurent-domain functional with $A=0$ series values survives. Neither completion is substituted for the other.

The 3D-index and its relation to angles, normal surfaces, and skein quantum traces are substantial prior art. In addition, Celoria–Hodgson–Rubinstein, Theorem 6.2, arXiv v2 proves a Dehn-filling index formula under a standard-cusp construction and specified efficiency hypotheses, with at least two original torus boundary components. This note does not claim the first rigorous Dehn-filling index construction, infer unrestricted closed-manifold invariance, or identify (24) with that invariant. See the source ledger for exact inspected dependencies.

10. Evidence and next mode

The proof of convergence is (29)–(35), not a truncation experiment. The companion standard-library verifier checks the Weyl forms, central-elimination and angle equations, seed recurrence conventions, quadratic/skew degree identity, ray scaling, and a small exact seed-series sanity check. It does not certify angle feasibility for a manifold, enumerate topological inputs, prove the sourced index identities, or resolve the longitude cancellation in (41).

Recommended next mode: ATTACK. The sharp next question is the actual longitude difference (41) under this global detector, or a rigorously defined replacement for (24) on a null ray or outside nonnegative filling-angle feasibility. Any extension must annihilate every actual relation on every algebraic input. Do not reopen the peripheral-sign problem, infer target vanishing from series divergence, or promote this conditional target theorem to an unrestricted trace theorem.