Generic boundary reconstruction for the Dehn-filled quantum trace

FPRD Lab research note — 8 September 2026

Status and principal result

At generic Kauffman parameter, the full boundary-category problem has a universal solution for every quantum-trace target that already carries the correct torus-skein-algebra action. The endpoint spaces, their morphisms, and the relative trace can be constructed explicitly by Morita induction.

This is an application of a decisive existing theorem: Gunningham–Jordan–Vazirani, Skeins on tori, Theorems 1.5 and 6.1, show that the generic SL₂ torus skein category has two orthogonal parity blocks. The empty object generates the even block; the one-point object generates the odd block. The relative skein functor of an unmarked three-manifold is zero on odd configurations. Combining this fact with Gunningham–Jordan–Safronov excision proves generic ordinary torus-algebra excision for arbitrary unmarked pieces, including Dehn filling.

Consequently the earlier integral empty-sector obstruction does not stop the generic construction. It locates a specialization failure. The earlier noninjectivity theorem at $A=-1$ remains valid.

The construction below is unconditional for the Garoufalidis–Yu target of arXiv:2406.04918. For the named Panitch–Park traces, it remains conditional on a boundary action matching the trace, including the stated scaling branch. Matching the identified target presentations transports a GY trace, but does not by itself identify that trace with the named PP trace. We prove an ambient normalizer lemma which removes the unknown comparison kernel as a separate obstacle for the intended peripheral multiplication operators. The remaining issue is the globally normalized boundary intertwiner before opposite roots are identified.

A further explicit reduction applies when the boundary action extends to its full peripheral quantum torus. If $m$ is the filling-meridian operator, with the conventions in Section 8, then

$$\boxed{\quad Q_s\cong\operatorname{coker}(m+A^2:Q\to Q).\quad}$$

Both maps of this isomorphism are given and proved below. Thus nonvanishing is exactly failure of surjectivity of one specified operator. A separate commuting-operator Koszul argument proves nonvanishing when the classical tetrahedron, independent edge and filling symbols form a regular sequence at a nonzero shape solution (Theorem 8.2).

Nonvanishing is proved under the explicit regular-intersection hypothesis for GY, not for all intended fillings. Panitch–Romo Conjecture 7.1 remains open. This is a research note applying and auditing existing structural theorems, not a claimed solution or a Publication.

1. Coefficients and three different parities

Work over the characteristic-zero field

$$k=\mathbb Q(i,A^{1/2}),$$

where $A$ is an indeterminate. All skein modules and quantum-gluing modules in the theorems below are extended to this field. One may instead use a characteristic-zero field and a compatible non-root-of-unity parameter covered by the cited generic theorems. No theorem below is asserted over the unlocalized Laurent ring or at $A=-1$.

The following distinctions are essential.

In particular, the phrase “even block” in this note does not restrict the trace to the homologically even skein submodule used for the 3d-index.

Let $\mathcal C=\operatorname{SkCat}_k(T^2)$, let $e$ be its empty object, and let $B=\operatorname{End}_{\mathcal C}(e)$. We use covariant module functors for the displayed construction. Reversing the collar orientation replaces $\mathcal C$ and $B$ by their opposites and interchanges left and right modules. This is relevant in Section 6, where the Panitch–Park convention uses right multiplication.

2. What the generic torus theorem supplies

We use the Temperley–Lieb presentation with fundamental-labelled endpoint configurations. Its additive idempotent completion gives the generic SL₂ version. Denote the configurations with an even or odd number of endpoints by $\mathcal C_0$ and $\mathcal C_1$.

A one-manifold has an even number of boundary points, so there are no morphisms between configurations of different parity. The cited GJV theorem implies that every object of $\mathcal C_0$ is a retract of a finite direct sum of copies of $e$. Concretely, for each even object $X$ there are finitely many collar tangles

$$a_i:e\longrightarrow X,\qquad b_i:X\longrightarrow e, \qquad \sum_i a_i b_i=\operatorname{id}_X. \tag{2.1}$$

This is the finite dual-frame identity used below. It involves only the boundary torus and does not depend on a manifold, a triangulation, a filling slope, or a link being evaluated. GJV Proposition 6.3 provides the strand reduction which constructs these factorizations recursively.

For a compact three-manifold $M$ with this torus boundary, put $F_M(X)=\operatorname{Sk}_k(M;X)$, with the collar orientation chosen to make $F_M$ covariant. Other boundary components, if present, carry no endpoints. Every relative link has an even total number of endpoints. Thus

$$F_M(X)=0\quad(X\in\mathcal C_1). \tag{2.2}$$

This argument does not require $M$ to be a handlebody. It also does not require a homological assumption on its closed links.

Proposition 2.1 — reconstruction from the empty sector

For every covariant $\mathcal C$-module $F$ which vanishes on $\mathcal C_1$, the natural map

$$\theta_X:\operatorname{Hom}_{\mathcal C}(e,X) \otimes_B F(e)\longrightarrow F(X), \qquad h\otimes m\longmapsto F(h)m \tag{2.3}$$

is an isomorphism, naturally in $X$.

Proof. Both sides are zero for odd $X$. For even $X$, use (2.1) to define

$$\psi_X(m)=\sum_i a_i\otimes F(b_i)m.$$

Then $\theta_X\psi_X(m)=F(\sum_i a_i b_i)m=m$. Conversely,

$$\psi_X\theta_X(h\otimes m) =\sum_i a_i\otimes F(b_i h)m =\sum_i a_i(b_i h)\otimes m =h\otimes m,$$

where the middle equality is precisely the balanced tensor relation. This also proves independence of the chosen dual frame. Naturality of $\theta$ follows from composition. ∎

3. An actual endpoint target and relative quantum trace

Let $Q$ be a left $B$-module, with action $\rho$, and suppose

$$\tau:\operatorname{Sk}_k(M)\longrightarrow Q$$

is $B$-linear. Define, for every boundary configuration,

$$\boxed{\quad \widetilde Q(X)=\operatorname{Hom}_{\mathcal C}(e,X) \otimes_B Q. \quad} \tag{3.1}$$

For a collar tangle $f:X\to Y$, define

$$\widetilde Q(f)(h\otimes q)=(fh)\otimes q. \tag{3.2}$$

These are specified vector spaces and linear maps. In particular, $\widetilde Q(e)=B\otimes_B Q\cong Q$, and odd values are zero. Every identity and skein relation in the collar is respected because the first factor is a relative skein module and its action is composition.

Theorem 3.1 — generic endpoint extension

The map $\tau$ extends uniquely to a natural transformation

$$\widetilde\tau:F_M\longrightarrow\widetilde Q$$

among extensions whose target vanishes in the odd block. For an even $X$, an explicit formula is

$$\boxed{\quad \widetilde\tau_X(m) =\sum_i a_i\otimes\tau\bigl(F_M(b_i)m\bigr). \quad} \tag{3.3}$$

Proof. Apply Proposition 2.1 to $F_M$ and then apply $\operatorname{Hom}(e,X)\otimes_B\tau$. The inverse in Proposition 2.1 gives (3.3), including independence of the dual frame. Naturality follows from (2.3). For uniqueness, every $m$ is $\sum_i F_M(a_i)F_M(b_i)m$, so naturality forces (3.3). ∎

Finite matrices rather than an undefined open-target symbol

Given frames (2.1), set

$$P_X=(b_i a_j)_{ij}\in\operatorname{Mat}_{r_X}(B), \qquad P_X^2=P_X.$$

The endpoint space in (3.1) is canonically the image of the corresponding idempotent matrix on $Q^{r_X}$. If $f:X\to Y$, its matrix is

$$M_f=(b_i^Y f a_j^X)_{ij}.$$

All entries are closed torus skeins, so their operators on $Q$ are already defined by $\rho$. The frame identities give

$$P_Y M_f P_X=M_f,\qquad M_gM_f=M_{gf}. \tag{3.4}$$

For example, the second equality follows by inserting $\sum_j a_j^Y b_j^Y=\operatorname{id}_Y$ between $g$ and $f$. This proves composition with the actual noncommutative order of the coefficients.

This is an intrinsic endpoint model obtained from boundary tangles and finite matrices. It is not a claim that open face-suspension pieces have already been assigned individual square-root monomials. That additional coordinate realization is unnecessary for the generic categorical extension, although it may still be useful computationally.

4. Cups, caps, transport, exchange, and ordering

The construction includes all the requested boundary operations.

For a concrete state audit, use $V=k v_+\oplus k v_-$, cup vector and cap covector

$$c=(0,A,-A^{-1},0)^T,\qquad d=(0,-A,A^{-1},0),$$

in the ordered basis $(++,+-,-+,--)$. Their coefficient matrices are mutual inverses for the two snake contractions, while

$$dc=\delta=-A^2-A^{-2}.$$

With $U=cd$ and $R=A I+A^{-1}U$, one has

$$U^2=\delta U,\qquad U_{12}U_{23}U_{12}=U_{12},\qquad R^{-1}=A^{-1}I+A U,$$

and $R$ satisfies the braid relation. The companion exact-arithmetic script checks these identities, both snakes, and the other Temperley–Lieb three-strand contraction. These are a self-contained Kauffman normalization; individual state coefficients in the cited papers use their stated orientation and rescaling conventions and must be translated accordingly.

Endpoint parity is preserved because cups and caps change the number of endpoints by two. On the odd block these maps are maps between zero spaces; there is no requirement that a category-module functor be a tensor functor with value $Q\otimes V^{\otimes n}$.

The target $Q$ retains its square-root variables and all of their relations. An entry $b\in B$ is applied through the already verified operator $\rho(b)$. When $\rho(b)$ is expressed by quantum-torus monomials, their Weyl factors and multiplication order are those of $\rho$. Equations (3.2)–(3.4) never replace a product by a commutative product, take a new square root, or identify opposite roots.

A universal four-term two-endpoint frame

The GJV calculation is especially explicit for two endpoints. Put $t=A^2$, $D=t-t^{-1}$, let $U=uv$ be cup followed by cap, and use the normalized torus braid operators $T,Z_1,Z_2,\pi$ of GJV Section 6. Their generic relations include

$$T=tI+U,\quad (T-t)(T+t^{-1})=0,\quad TZ_1T=Z_2,$$ $$\pi Z_1\pi^{-1}=t^{-1}Z_2,\qquad \pi Z_2\pi^{-1}=tZ_1.$$

Writing $r=Z_2/Z_1$, the Hecke relation gives

$$U-Z_1^{-1}UZ_2=-t+t^{-1}r, \qquad U-Z_2^{-1}UZ_1=-t^{-1}+tr^{-1}.$$

Subtract the $\pi$-conjugate of the second equality from the first to get

$$\boxed{ D\operatorname{id}_2 =-uv+Z_1^{-1}uvZ_2+\pi uv\pi^{-1} -\pi Z_2^{-1}uvZ_1\pi^{-1}.} \tag{4.1}$$

Every term factors through the empty object. Dividing by $D$ gives an explicit four-element dual frame, hence a $4\times4$ closed-skein matrix model for the entire two-endpoint target. This is the two-strand identity in GJV Proposition 6.3, rewritten without its normalized-idempotent denominators; it is not a new independent Morita theorem.

Integral limit. We have used (4.1) in the generic category. Canceling denominators in a generic formula does not, by itself, establish an integral identity in a possibly torsion relative skein module. In particular, this note does not claim that $A^2-A^{-2}$ annihilates the earlier empty-sector kernel. An integral diagrammatic lift of the torus generator relations would be needed before drawing that stronger conclusion.

5. Generic torus excision and filling

Let $M_1$ and $M_2$ be compact oriented three-manifolds glued along a torus, with no endpoints on their remaining boundaries. Choose collar orientations so that $F_1$ is a right and $F_2$ is a left $\mathcal C$-module.

Theorem 5.1 — generic ordinary-algebra torus excision

The natural gluing map is an isomorphism

$$\boxed{ \operatorname{Sk}_k(M_1)\otimes_{B}\operatorname{Sk}_k(M_2) \ \cong\ \operatorname{Sk}_k(M_1\cup_{T^2}M_2).} \tag{5.1}$$

Proof. GJS Theorem 2.5 gives the right-hand side as $F_1\otimes_{\mathcal C}F_2$. Both functors vanish on the odd block by endpoint parity. Substitute the two versions of Proposition 2.1 and apply co-Yoneda:

$$\operatorname{Hom}_{\mathcal C}(-,e) \otimes_{\mathcal C}\operatorname{Hom}_{\mathcal C}(e,-) \cong B.$$

The resulting tensor product is the left side of (5.1), and the isomorphism is the natural gluing map. ∎

For a Dehn-filling solid torus $V_s$, the filled target and descended trace are therefore

$$Q_s=\operatorname{Sk}_k(V_s)\otimes_B Q, \qquad \tau_s:\operatorname{Sk}_k(M(s))\longrightarrow Q_s. \tag{5.2}$$

The boundary collar surjects onto the skein module of the solid torus: its core powers can all be represented in the collar. Thus, with $I_s=\ker(B\to\operatorname{Sk}_k(V_s))$ in the appropriate handedness,

$$Q_s\cong Q/I_sQ. \tag{5.3}$$

This construction and the full categorical filling coend agree. At generic parameter the nonempty even filling relations are already generated by the empty-sector ones through the Morita factorizations. The odd block does not contribute.

The prior classical counterexample is compatible with (5.1). At $A=-1$ the scalar $D$ in (4.1) vanishes, and generic compact generation is not available. The kernel class in the earlier note was detected in a special fiber; it was never proved to survive localization. The present generic theorem is an independent reason that the generic gluing kernel is zero. It does not produce a specified integral torsion annihilator.

6. A proved normalizer lemma for the Panitch–Park quotient

The original face-suspension target is defined as an image in an ambient module quotient, not simply as an algebra quotient by two-sided ideals. Let $U$ be the tensor product of the face-suspension quantum tori, let $S\subset U$ be the shape-parameter quantum torus, and write its ambient relation submodule as

$$\mathcal R=V_+U+UV_-,\qquad Q_{PP}=S/(S\cap\mathcal R). \tag{6.1}$$

Here the $V_+$ relations act on the left and the $V_-$ gluing relations act on the right, as in PP Section 5.2 and CK equation (5.6).

Lemma 6.1 — ambient right normalizer

Suppose $c\in S$ commutes in $U$ with every element of $V_-$. Then right multiplication by $c$ preserves $\mathcal R$ and $S\cap\mathcal R$, and therefore induces an operator on $Q_{PP}$.

Proof. For $v_+\in V_+$, $v_-\in V_-$ and $u\in U$,

$$(v_+u)c=v_+(uc)\in V_+U, \qquad (uv_-)c=(uc)v_-\in UV_-.$$

Right multiplication also preserves $S$. Intersecting the invariant submodule with $S$ proves the second assertion. ∎

In particular, suppose a comparison map is induced by the same ambient shape torus,

$$\mu:S/I_{GY}\twoheadrightarrow S/(S\cap\mathcal R),$$

and right multiplication by $c$ acts on its source. The lemma proves

$$R_c\ker\mu\subseteq\ker\mu, \qquad \mu R_c=R_c\mu. \tag{6.2}$$

The unknown size of the comparison kernel is irrelevant to this argument. One does not need CK Conjecture 5.27. Its application does require the actual peripheral operator to be multiplication on this side by an element centralizing the gluing constraints. An ambient quantum-torus map alone would not prove (6.2).

The commutation part is compatible with retaining the six bare-edge roots. In CK (5.7), the skew form on $(z,z',z'',y,y',y'')$ is the pullback of the three-variable skew form under the map which adds exponents at opposite edges. Differences of opposite edge exponents are in its radical. Thus the peripheral–edge zero pairing from the three-variable model lifts to zero pairing for any corresponding six-variable lifts. The companion script checks this skew-form identity exactly. It does not determine the globally normalized peripheral lifts or their trace intertwiner.

7. Target audit: what is unconditional and what remains

There are two distinct opposite-root identifications in the source chain.

1. GY arXiv:2406.04918, Section 3.3 and Remark 3.1, use a target which identifies opposite roots. Theorem 5.7 is stated for that target.
2. CK Theorem 4.9 compares its construction with the extended GY target before this identification; its discussion immediately before that theorem explicitly explains the distinction. The CK surjection in Theorem 1.3 then goes to the original PP target.
3. PP arXiv:2509.09100, Definition 2.22 and footnote 11, introduce their own opposite-root-identified target and prove their Pachner theorem for that target.

These facts refine the earlier audit. It is insufficient to combine GY Theorem 5.7 and CK Theorem 1.3 without also matching the source targets.

The identified target

The GY identified target and PP Definition 2.22 have matching explicit presentations after reversing multiplication and matching parameters. In terms of the Kauffman parameter $A$, take

$$q_{GY}=A^4,\qquad q_{UV}=-A^2,\qquad c_B=1,\quad c_T=(-A^2)^{-1/2},$$

with $(-A^2)^{1/2}=iA$. In both presentations the central vertex value is $iA$, the Lagrangian equation is $z^{-2}+z''^{\,2}=1$, and the edge value is $-A^2$. The commutation forms and left/right quotient conventions are opposite. The variable-preserving anti-isomorphism carries Weyl-ordered monomials to Weyl-ordered monomials with the reversed form. This matches the target presentations and allows the GY trace and action to be transported to this identified presentation.

It does not yet establish equality with the named PP trace. PP Remark 2.21 records two scaling branches: the original PP constants correspond, after common rescaling, to the second branch, whereas the UV–IR paper fixes the first by default. The local trace maps and square-root choices must be matched as well as the equations of their targets. We therefore retain canonical PP trace matching as a condition. No endpoint extension or Pachner compatibility of the named PP trace is inferred solely from the matching presentations.

For GY itself, Theorem 5.7 supplies the boundary-linear trace, Theorem 3.1 supplies its full generic endpoint extension, and (5.2) supplies its filled descent. This named-target conclusion is unconditional under GY’s stated triangulation hypotheses, although the filled target may be zero.

The original target

GY Appendix A is a substantial positive indication for a lift: its off-diagonal elimination, height-exchange cancellation, and annulus-counit arguments are local before the final identification of shape variables. Its proof reduces the peripheral calculation to three torus generators. However, this audit has not completed the global normalization of the six-bare-edge peripheral monomials, including the sum-of-slopes generator and its possible central sign factors. We therefore do not assert the original-target boundary intertwiner as a theorem.

The remaining original-target test is exact:

Choose the six-edge lifts of the meridian, longitude, and their sum. Prove their torus-skein relations and the three peripheral intertwining identities in the extended GY target, with the actual central signs and Weyl factors. The ambient normalizer lemma then transfers the action to the original PP target, and the generic endpoint extension follows from Theorem 3.1.

This is a boundary operator problem, not an enumeration of manifolds or triangulations.

8. Explicit meridian-cokernel reduction

A stronger uniform description is available when the $B$-action extends to a full peripheral quantum torus, as it does in GY Theorem 5.7. Fix the following conventions. Let $m$ correspond to the filling meridian, let $l$ correspond to a complementary longitude, and take

$$T=k\langle m^{\pm1},l^{\pm1}\mid ml=A^2lm\rangle, \qquad B=T^{\iota},\quad\iota(m,l)=(m^{-1},l^{-1}). \tag{8.1}$$

The invariant Weyl monomial for the sum of these slopes is $A^{-1}ml+A^{-1}m^{-1}l^{-1}$. We use the right solid-torus module obtained from the left module of Frohman–Gelca, Section 5, by the opposite collar convention: their $(0,1)$ is our meridian and their $(1,0)$ is our longitude. This reverses the ordered intersection pairing. Their Theorem 3 gives the right vacuum ideal generated by

$$X=m+m^{-1}+A^2+A^{-2},$$ $$Y=A^{-1}ml+A^{-1}m^{-1}l^{-1}+A^{-3}(l+l^{-1}). \tag{8.2}$$

The $A^{-3}$ is the negative framing-twist coefficient in that source convention. Reversing the convention without also transporting this coefficient would give a different, incorrect pair of operators.

For any left $T$-module $Q$, restriction to $B$ gives

$$Q_s=Q/(XQ+YQ).$$

This follows from the right-ideal presentation: if $I_s=XB+YB$, then $I_sQ=XQ+YQ$. There is no assumption that $Q$ is cyclic or that the image of the trace spans $Q$.

Theorem 8.1 — explicit cokernel isomorphism

Put

$$r_1=-A^2,\qquad r_2=-A^{-2},\qquad d=r_1-r_2=A^{-2}-A^2.$$

Since $d\ne0$ at generic parameter, the following are mutually inverse linear isomorphisms:

$$\boxed{\quad \Phi:Q_s\longrightarrow Q/(m+A^2)Q, \qquad [q]\longmapsto[(1-l^2)q], \quad} \tag{8.3}$$ $$\boxed{\quad \Psi:Q/(m+A^2)Q\longrightarrow Q_s, \qquad [q]\longmapsto\left[\frac{m+A^{-2}}{A^{-2}-A^2}q\right]. \quad} \tag{8.4}$$

Proof. Write $F=1-l^2$ and $H=m-r_2$. All the following are identities in the noncommutative algebra $T$, with $m$ ordered before $l$:

$$H(m-r_1)=Xm,\qquad HF-d=X-AYl,\qquad FH-d=(m-r_1)(1-A^{-4}l^2). \tag{8.5}$$

Moreover,

$$FX=(m-r_1)\bigl(m^{-1}(m-r_2) -A^{-4}m^{-1}(m+A^6)l^2\bigr), \tag{8.6}$$ $$FY=(m-r_1)\bigl(-A^{-5}l^3 +A^{-1}m^{-1}(m-A^2)l+A^{-3}m^{-1}l^{-1}\bigr). \tag{8.7}$$

For example, (8.7) follows by moving $l^2$ past $m$ and $m^{-1}$ using $l^2m=A^{-4}ml^2$ and $l^2m^{-1}=A^4m^{-1}l^2$, and then collecting the $l^3,l,l^{-1}$ coefficients. These displayed factorizations prove that $\Phi$ kills $XQ+YQ$, so it is independent of representatives. The first identity in (8.5) proves the same for $\Psi$. The second proves $\Psi\Phi=\operatorname{id}_{Q_s}$; the third proves $\Phi\Psi=\operatorname{id}_{Q/(m-r_1)Q}$. This is a proof for every $T$-module, not a calculation on a chosen representation. ∎

Exact nonvanishing criterion and its remaining content

Consequently

$$Q_s\ne0\quad\Longleftrightarrow\quad m+A^2:Q\longrightarrow Q\text{ is not surjective}. \tag{8.8}$$

Equivalently, there exists a nonzero algebraic linear functional $f:Q\to k$ with

$$f(mq)=-A^2f(q)\qquad(q\in Q). \tag{8.9}$$

The corresponding functional on the filled quotient is $\ell(q)=f((1-l^2)q)$. Conversely, a filling functional $\ell$ yields $f(q)=\ell((m+A^{-2})q)/(A^{-2}-A^2)$. Equations (8.3)–(8.4) show that these correspondences preserve nonzero functionals. If survival of the vacuum is required, its exact additional test is

$$f((1-l^2)1_Q)\ne0. \tag{8.10}$$

The cokernel reduction alone does not show that such an eigenfunctional exists. Theorem 8.2 below supplies a sufficient structural condition. In particular, $m$ being invertible does not decide whether $m+A^2$ is surjective. A nonzero special fiber also does not prove (8.8) over $k$. Outside the regular-intersection class of Theorem 8.2, the unresolved nonvanishing question is the concrete operator problem (8.8).

For targets carrying only the invariant torus-skein action, the more general criterion remains a nonzero $\ell:Q\to k$ annihilating $I_sQ$ on all vectors. Nonzero factors alone do not imply a nonzero tensor product: $k[x]/(x)\otimes_{k[x]}k[x]/(x-1)=0$.

The alternative vector-space quotient by $\tau(\ker(\operatorname{Sk}(M)\to\operatorname{Sk}(M(s))))$ always gives a formal descended linear map, but its nonvanishing is another substantive obligation. It should not be substituted for the identified boundary quotient without proving they agree.

Theorem 8.2 — nonvanishing under a regular classical intersection

For the identified GY target, the cokernel in Theorem 8.1 is nonzero under an explicit local complete-intersection hypothesis. This is a conditional nonvanishing theorem, not an inference from a special fiber.

Let the triangulation have $N$ tetrahedra and $r$ torus boundary components, and fill one chosen component. After eliminating the central vertex variables, write the GY shape quantum torus as $S_A$ on $2N$ invertible variables. Let

$$f_i=z_i^{-2}+z_i''^{\,2}-1\qquad(1\le i\le N)$$

be the tetrahedron relations, let

$$g_j=e_j+A^2\qquad(1\le j\le N-r),\qquad g_{N-r+1}=m+A^2. \tag{8.11}$$

Here the independent edge relations generate the same entire right ideal as all edge relations, as stated in GY Section 3.3 immediately after (31). This exact redundancy is used, not merely independence of their classical exponent vectors. If an alternative presentation has additional relations not generated by this list, the theorem cannot be applied without including or proving redundancy of those relations.

In GY’s convention the vector-space presentation of the meridian cokernel is

$$C_s=S_A\Big/\left(\sum_i S_A f_i+\sum_j g_jS_A\right). \tag{8.12}$$

The opposite collar changes the module interpretation, not this requirement to respect the specified sides. The $g_j$ commute mutually: the edge monomials commute and the peripheral monomial commutes with every edge. The $f_i$ commute mutually because distinct tetrahedron factors commute. Left multiplication by a $g_j$ commutes with right multiplication by every $f_i$ by associativity.

Specialize the shape quantum torus at $A=1$, where these root variables commute. Assume there is a point

$$p\in(\mathbb C^*)^{2N}$$

at which all symbols $f_i^0,g_j^0$ vanish, and assume that their ordered list is a regular sequence in the formal local ring

$$R=\widehat{\mathcal O}_{(\mathbb C^*)^{2N},p}. \tag{8.13}$$

A sufficient, more concrete condition is full row rank $2N-r+1$ of their joint Jacobian at $p$. No assertion is made that every nondegenerate triangulation or every Dehn filling satisfies this condition. The ordinary phrase “nondegenerate shape” only excludes zero or forbidden shape values; it does not by itself certify (8.13).

Then $C_s\ne0$ over $k=\mathbb Q(i,A^{1/2})$. Consequently the intrinsic GY filled quotient $Q_s$ is nonzero. If additionally the complementary longitude symbol satisfies $l(p)^2\ne1$, the filled vacuum and the descended GY trace are nonzero as well.

Proof. Embed the coefficient field into $\mathbb C((\hbar))$ by

$$A\longmapsto e^{\hbar},\qquad A^{1/2}\longmapsto e^{\hbar/2}. \tag{8.14}$$

This is injective: a nonzero polynomial with constant coefficients cannot annihilate the nonconstant series $e^{\hbar/2}$. Every nonzero denominator therefore becomes an invertible Laurent series. This argument uses the commutative $A=1$ root-variable fiber; the root torus generally still has sign commutation at $A=-1$.

Choose logarithmic coordinates $x_a$ centered at $p$. If $\Omega$ is the skew commutation matrix on the $2N$ independent root variables, the formal local quantum torus is

$$\widehat S=\bigl(R[[\hbar]],\star\bigr),\qquad F\star G=\left. \exp\!\left(\frac{\hbar}{2}\sum_{a,b}\Omega_{ab} \partial_{x_a}\partial_{y_b}\right)F(x)G(y) \right|_{y=x}. \tag{8.15}$$

Weyl monomials map to their nonzero constant values at $p$ times exponentials of integral linear forms in $x$, with the central scalar factors evaluated by (8.14). Formula (8.15) reproduces their quantum-torus multiplication. It is well-defined on $R[[\hbar]]$: at each fixed power of $\hbar$ only finitely many derivative orders occur. Associativity follows because the constant-coefficient bidifferential operators commute.

On $W=R[[\hbar]]$ define the $2N-r+1$ mutually commuting operators

$$D_i(w)=w\star f_i,\qquad D_{N+j}(w)=g_j\star w.$$

Their reductions modulo $\hbar$ are multiplication by the regular sequence in (8.13). Put

$$H=W\Big/\sum_a D_a(W).$$

We claim $H$ is $\hbar$-torsion-free. Suppose $\hbar u=\sum_aD_a v_a$. Reducing modulo $\hbar$ gives a syzygy of the classical regular sequence. Every such syzygy is a sum of Koszul syzygies. For completeness, this last fact follows by induction on the sequence length: the last regular element is a non-zero-divisor modulo its predecessors, so the last syzygy coefficient lies in their ideal; subtract the corresponding pairwise Koszul syzygies and apply induction.

Lift those syzygy coefficients arbitrarily to $W$. Since the $D_a$ commute exactly, their lifted Koszul syzygies are still in the kernel of $(v_a)\mapsto\sum_aD_a v_a$. Subtract them from $v$. Every remaining coefficient is divisible by $\hbar$, say $v_a'=\hbar w_a$. The module $W$ is $\hbar$-torsion-free, and the operators are $\mathbb C[[\hbar]]$-linear, so canceling $\hbar$ gives $u=\sum_aD_aw_a$. This proves the claim. It does not require an unproved flatness theorem or an assertion that images of operators are topologically closed.

Furthermore,

$$H/\hbar H\cong R/(f_1^0,\ldots,f_N^0,g_1^0,\ldots,g_{N-r+1}^0). \tag{8.16}$$

This quotient is nonzero and its unit survives, since all listed symbols vanish at $p$. Torsion-freeness therefore implies that $1_H$ remains nonzero in $H[\hbar^{-1}]$.

The quantum-torus homomorphism above and the same complete relation list induce an actual map

$$C_s\otimes_k\mathbb C((\hbar)) \longrightarrow H[\hbar^{-1}],\qquad [1]\longmapsto[1]. \tag{8.17}$$

Its value on $[1]$ is nonzero, proving $C_s\ne0$ over $k$. Thus completion has been used to detect a vector in the original generic quotient; no identification of completed and original modules is assumed. Compose (8.17) with $\Phi$ in (8.3) to get the explicit detecting map from $Q_s$. The filled vacuum maps to $[1-l^2]$. If $l(p)^2\ne1$, this class is already nonzero in (8.16), so it survives after inverting $\hbar$. The GY trace sends the empty skein to this vacuum, proving the final assertion. ∎

The commuting-operator condition is indispensable. In the formal Weyl algebra with $xp-px=\hbar$, the same-side quotient $W/(xW+pW)$ has a nonzero classical fiber but $\hbar[1]=0$, because both $xp$ and $px$ lie in its relation subspace. By contrast, the operators used above act on opposite sides and commute exactly. This pinpoints the extra structure that repairs the earlier invalid special-fiber inference.

Theorem 8.2 proves nonvanishing on the explicit regular-intersection class for GY. It does not establish that this class covers all fillings meant by Panitch–Romo, and it does not supply the missing named-PP trace bridge.

9. Exact checks and useful failures

The archive contains this note, the updated continuation state, the exact verification script and its output, and a source ledger. The script uses integer Laurent-polynomial arithmetic and checks:

• both snake identities and the Kauffman loop value;
• both three-strand Temperley–Lieb contractions, crossing inverse, braid relation, and curl eigenvalue;
• the reduced Laurent-polynomial calculation behind (4.1), including its zero denominator at $A=-1$;
• the six-to-three-variable skew-form pullback and opposite-edge radical;
• all five quotient-map obligations in Theorem 8.1, in the noncommutative quantum torus, including independence of representatives and both inverse compositions.

These checks audit the displayed local algebra. The category-generation theorem, excision, and source comparisons are formal proof dependencies, not conclusions established by a numerical sample. Theorem 8.2 has a separate algebraic proof using classical regular-sequence syzygies and exactly commuting left/right operators; it is not inferred from the script.

Three unsuccessful shortcuts are retained because they locate real gaps.

1. Arbitrary multiplication on both sides of a quantum-torus module quotient does not descend. A fixed moment equation $X=1$ is incompatible with an invertible same-sector operator $Y$ satisfying $XY=qYX$ unless $q=1$ or the target vanishes. Charged sectors can address that local issue; the generic Morita construction does not assume these arbitrary same-sector operators.
2. Abstract coinduction from the empty object gives a formal extension once boundary linearity exists, but by itself gives no nonzero filling. The finite dual-frame model above is stronger: it reconstructs the actual even relative skein functor and proves agreement with ordinary algebra excision at generic parameter.
3. The tempting fixed annihilator $A^2-A^{-2}$ is not claimed integrally. The standard two-endpoint handle-slide presentation would make it sufficient if (4.1) were established over the integral coefficient ring, but a generic equality cannot supply that missing integral lift.

10. Sources, dependencies, and contribution

• Formal generation input: Sam Gunningham, David Jordan, Monica Vazirani, Skeins on tori, arXiv:2409.05613v1, Theorems 1.5 and 6.1, Proposition 6.3, and Section 7.1. The finite boundary generators and two-strand identity belong to their work.
• Formal excision input: Sam Gunningham, David Jordan, Pavel Safronov, The finiteness conjecture for skein modules, arXiv:1908.05233v4, Theorems 2.5 and 4.1. Its Proposition 4.3 alone only gives the earlier two-handlebody sufficient condition; GJV is the additional generic input used here.
• Formal peripheral input: Stavros Garoufalidis, Tao Yu, The 3d-index of the 3d-skein module via the quantum trace map, arXiv:2406.04918, Section 3.3, Remark 3.1, Theorem 5.7, Appendix A.
• Trace and extended-target input: Garoufalidis, Yu, A quantum trace map for 3-manifolds, arXiv:2403.12424, Section 5.1 and Remark 5.1.
• Formal PP target input: Samuel Panitch, Sunghyuk Park, 3d Quantum Trace Map, arXiv:2403.12850, Theorems 5.5–5.6 and Section 5.2.
• Formal comparison input: Qingjing Chen, Andrew Kricker, On 3d Quantum Trace Maps, arXiv:2606.13268v1, Theorems 1.3 and 4.9, equation (5.6), Definition 5.22, Remark 5.26, Conjecture 5.27.
• Modified-target input: Panitch, Park, Compatibility of quantum trace and UV–IR maps, arXiv:2509.09100v1, Definition 2.22, footnote 11, and Theorem 2.25.
• Formal solid-torus input: Charles Frohman, Răzvan Gelca, Skein modules and the noncommutative torus, arXiv:math/9806107v1, Theorems 2 and 3 and Section 5. Their two vacuum-ideal generators and framing convention are used in Theorem 8.1.
• Open objective: Samuel Panitch, Mauricio Romo, On the length conjecture for twist knots, arXiv:2609.02706v1, Conjecture 7.1.

The approach is directly influenced by the GJV finite-generation viewpoint, GJS categorical gluing, and the PP/CK ambient presentation. The formal dependencies are the particular results identified above. The contribution of this note is their explicit synthesis into the endpoint reconstruction, the generic Dehn-filling consequence, the explicit meridian-cokernel isomorphism, the regular-intersection nonvanishing criterion, the ambient normalizer proof, and the source/parameter audit. No external novelty assessment has been performed.

11. Continuing question

Next mode: ATTACK. Determine whether the intended geometric filling class satisfies the regular-sequence hypothesis in Theorem 8.2 by a universal deformation-theoretic argument. For singular or nonregular intersections, attack the surjectivity of $m+A^2$ directly. For the original PP target, complete the globally normalized six-edge peripheral intertwiner and the named-trace scaling bridge; then apply the same cokernel test. Keep the integral extension separate; do not infer it by specialization of the generic construction. All other paused programs remain paused.