Cohomological degrees outside the quantum-trace image

FPRD Lab · 8 September 2026 · Research note

AI-assisted work with an internal self-audit. No external specialist review or priority claim.

Result

Let M be a connected oriented compact 3-manifold whose boundary is a nonempty union of r tori. Let T be an ideal triangulation with N tetrahedra, and suppose T admits a strict angle structure. Work at generic parameter over k=Q(i,A^(1/2)), with the retained native convention dictionary and full six-root quantum trace

$$\tau:\operatorname{Sk}(M)\longrightarrow Q^6_{\mathcal T}.$$

Then

$$\boxed{\dim_k\bigl(Q^6_{\mathcal T}/\operatorname{im}\tau\bigr) \ \geq\ 2^r-1>0.} \tag{1}$$

The surviving classes have nonzero degrees defined from the triangulation’s dual graph. For the monomials used in the proof, these degrees are cohomology classes in H^1(M;F_2). The proof establishes a subspace of such degrees of dimension at least r; it does not assert that every class of H^1(M;F_2) has the required integral lift.

Consequently, for every Dehn filling p:Sk(M)→Sk(M_fill), the minimal compatible linear quotient

$$D_p=Q^6_{\mathcal T}/\tau(\ker p)$$

is nonzero. Indeed, it surjects onto the common quotient Q^6_T/im(tau). The latter supports the compatible zero filled trace. This proves the literal nonzero-quotient condition in this generic, strict-angle class. It supplies no nonzero filled invariant and does not settle Panitch–Romo Conjecture 7.1 without these restrictions.

No filling-angle condition, complete peripheral angular holonomy, classical shape point, or slope bound is imposed. All slopes use the same quotient. The result remains a supporting research note: the distinction between a nonzero quotient and a nonzero trace is essential to its significance.

1. Native sector and relation sides

Use the all-plus central projection of the retained full six-root presentation. In every tetrahedron opposite roots are identified, the Weyl-ordered vertex value is iA, and z’=iA W(-e_z-e_w), where w=z’’. The resulting lattice is L=Z^(2N), written in blocks (z_t,w_t), and

$$W(u)W(v)=A^{\Omega(u,v)/2}W(u+v),\qquad \Omega(u,v)=\sum_t(u_{wt}v_{zt}-u_{zt}v_{wt}). \tag{2}$$

The projected unfilled target is the vector-space quotient

$$Q_+=S\Big/\left(\sum_t S f_t+ \sum_e(c_eW(h_e)+A^2)S\right), \qquad f_t=z_t^{-2}+w_t^2-1. \tag{3}$$

Tetrahedron relations have multipliers on the left; edge relations have multipliers on the right. This is not a two-sided algebra quotient.

For an actual edge e, write its three opposite-pair incidence counts in tetrahedron t as (R_et,S_et,T_et). Then

$$h_e=(R_{et}-S_{et},\ T_{et}-S_{et})_t, \qquad c_e=(iA)^{\sum_t S_{et}}. \tag{4}$$

All counts include repeated occurrences. Formula (4) follows by substituting the native vertex value into the Weyl-ordered bare-edge row; it retains its phase. Set

$$\Lambda=\sum_e\mathbb Z h_e\subset L.$$

This is the actual image lattice, not its saturation. The edge rows are isotropic for Omega. These conventions and the projection Q^6→Q_+ are retained from the preceding native comparison and index packets. Projection will only be used to pull a nonzero functional back to Q^6.

2. Local parity as half-face data

Let Gamma be the dual graph: one vertex per tetrahedron and one edge per glued face. Thus it has N vertices and 2N edges. Loops and parallel edges are allowed, with their two half-edges counted separately.

For a tetrahedron t, label its four half-faces by the opposite vertices 0,1,2,3. Define

$$J_t=\{\alpha\in\mathbb F_2^4:\sum_i\alpha_i=0\} /\langle(1,1,1,1)\rangle. \tag{5}$$

A bare edge ij represents the indicator of the two faces containing ij, namely the two faces opposite the other vertices. Opposite bare edges differ by (1,1,1,1). At a vertex the three incident edges sum to zero. Consequently (5) identifies with the local root lattice modulo 2, after vertex elimination. For example, one may take

$$z=01\mapsto(0,0,1,1),\qquad w=03\mapsto(0,1,1,0),\qquad z'=02\mapsto z+w.$$

The dot product on even half-face vectors descends to J_t and equals Omega modulo 2. It is alternating and nondegenerate. Thus

$$J=\bigoplus_tJ_t\cong L/2L$$

is a 2N-dimensional symplectic vector space over F_2.

3. The face-defect map

Choose even half-face representatives alpha_t for u in J. On a glued face f, add the two half-face entries to obtain c_f. This defines a graph cochain c. Changing alpha_t by (1,1,1,1) changes c by a graph coboundary. Therefore

$$d:J\longrightarrow H^1(\Gamma;\mathbb F_2),\qquad d(u)=[c] \tag{6}$$

is well-defined.

Write zeta for the sum of every dual edge, with coefficient one. Since every dual vertex has degree four, zeta is a graph cycle, including when there are loops. The sum of all entries of c is zero. Hence d takes values in

$$D_\Gamma=\{[c]\in H^1(\Gamma;\mathbb F_2):c(\zeta)=0\}. \tag{7}$$

The dimension of D_Gamma is N: connectedness gives dim H^1(Gamma)=N+1, and the nonzero graph cycle zeta imposes one independent equation.

The map d is onto (7). To see this, distribute a representative c arbitrarily between the two half-faces of each face. The resulting vertex parity defects have total zero. Flipping both ends of a dual edge preserves c and changes the parities at its endpoints. Connectedness implies that such flips realize every even-total vertex parity vector. Make every local representative even. This also proves surjectivity with self-gluings; loops are unnecessary for the parity correction.

4. The kernel is Lagrangian and contains the trace image

Put K_0=ker d. If d(u)=0, change the local representatives by all-ones vectors until their sums across each glued face vanish. They now agree across each face and define a graph cycle gamma. Conversely every graph cycle gives such an element of K_0. The only graph cycles mapping to zero in J are 0 and zeta: at each vertex a zero local class is either all zero or all one, and agreement along edges forces the choice to be constant on the connected graph. Therefore

$$K_0\cong Z_1(\Gamma;\mathbb F_2)/\langle\zeta\rangle, \qquad \dim K_0=N. \tag{8}$$

For two graph cycles gamma and gamma’, their local dot products count each glued face twice. Thus Omega vanishes on K_0. Since its dimension is half that of J, K_0 is Lagrangian.

Each actual edge row h_e is in K_0: its half-face data describe the boundary of the dual 2-cell around e. In particular every relation in (3) is homogeneous for d. The tetrahedron exponents differ by even vectors, and the two terms of each edge relation have the same degree. Hence Q_+ is D_Gamma-graded.

The projected entire quantum-trace image has degree zero:

$$\pi_+\tau(\operatorname{Sk}(M))\subset(Q_+)_0. \tag{9}$$

Here is the native source of (9). The published Panitch–Park face-suspension trace has even total degree at each face; this is Theorem 5.6 and the input to Proposition 5.7. A bare shape edge uses its two incident half-faces, exactly as in (5). Thus its face parity maps to (6). Vertex elimination preserves this parity class; opposite-root identification changes a representative by a local graph coboundary. The all-plus projection of every native traced skein therefore has d=0. The retained side reversal and branch scalars do not alter parity. This argument applies to the whole state sum and requires no manifold-specific skein basis.

Garoufalidis–Yu already note that the quantum trace preserves mod-two homology in their published Remark 4.5. The extension below makes explicit the additional degrees of the target that are unused by closed traced skeins; the familiar homology grading itself is not a new claim.

5. The topological part of the grading

Let B be the image of Lambda in J. Under (8), its generators are boundaries of dual 2-cells. Moreover, zeta is a boundary: add all primal-edge dual 2-cells modulo 2. Every glued triangular face has three edge occurrences, so its dual edge appears three times and has coefficient one. Thus

$$0\longrightarrow H_1(M;\mathbb F_2) \longrightarrow J/B\xrightarrow{d}D_\Gamma\longrightarrow0. \tag{10}$$

The dual 2-skeleton is a spine of the ideal triangulation. This proves that the degree-zero part of the parity lattice recovers the usual homological grades.

For an integral edge-normalizing exponent w, meaning

$$P=\{w\in L:\Omega(w,h_e)=0\text{ for every }e\}, \tag{11}$$

d(w) has a further intrinsic interpretation. If gamma is any graph cycle and u_gamma its image in K_0, direct half-face summation gives

$$d(w)(\gamma)=\Omega(\bar w,u_\gamma)\pmod2. \tag{12}$$

In particular it vanishes on every dual 2-cell boundary, by (11). Thus

$$d(\bar P)\subset H^1(M;\mathbb F_2)\subset H^1(\Gamma;\mathbb F_2), \tag{13}$$

where bar P is the reduction of the integral lattice P. Formula (12), rather than a coordinate label in one example, defines the cohomology class. General graph defects in D_Gamma need not extend over M; the ones used by the detector do.

6. Enough defects have integral normalizing lifts

It is not legitimate to replace bar P by the whole mod-two kernel of the edge matrix. We prove the needed integral lower bound directly.

There are N edges because the boundary components are tori. Define the unsigned vertex-edge incidence map alpha:Q^r→QN by putting x_v+x_v’ on an edge with endpoints v,v’; a loop has value 2x_v. In each tetrahedron the contribution of one corner is the sum of its three bare edges, which is zero after vertex elimination. Hence

$$\beta\alpha=0,\qquad \beta(e)=h_e. \tag{14}$$

The map alpha is injective over Q. If all endpoint sums are zero, then for any three corners of a tetrahedral face the equations x_i+x_j=x_j+x_k=x_k+x_i=0 imply x_i=x_j=x_k=0. This argument permits identified corners and loops. Every cusp occurs in a tetrahedron. Consequently

$$\operatorname{rank}_{\mathbb Z}\Lambda\leq N-r, \qquad \operatorname{rank}_{\mathbb Z}P =2N-\operatorname{rank}_{\mathbb Z}\Lambda\geq N+r. \tag{15}$$

The lattice P is primitive: if an integer multiple of w pairs to zero with all h_e, so does w. Therefore P∩2L=2P, and its reduction in J has dimension equal to its integral rank. Combining this fact with dim K_0=N yields

$$\boxed{\dim_{\mathbb F_2}d(\bar P) \geq \dim\bar P-N\geq r.} \tag{16}$$

Thus at least r independent nonzero cohomological defects have integral lifts commuting with every actual edge. This argument uses no saturation of the averaging lattice and no assertion that an arbitrary mod-two normal vector lifts. Neumann’s integral chain complex and GHHR’s normal-homology formulation motivated the question; their full classification is not needed for (16).

7. Strict angles force the exact native character to be consistent

Write the strict angles divided by pi as theta_t=(gamma_t,beta_t,alpha_t), with all entries positive and sum one. The actual edge equations are

$$\sum_t(\gamma_t R_{et}+\beta_t S_{et}+\alpha_t T_{et})=2. \tag{17}$$

A rational strict solution may be chosen because these are rational affine equations with strict inequalities.

Set chi(h_e)=-A^2/c_e using the actual c_e in (4). Suppose sum_e n_e h_e=0. Then in each tetrahedron the combined three incidence counts are equal, say to k_t. Evaluating the same combination of (17) gives

$$\sum_t k_t=2\sum_e n_e.$$

Therefore, including the complex phase,

$$\prod_e(-A^2/c_e)^{n_e} =(-1)^{\sum n_e}A^{2\sum n_e}(iA)^{-\sum k_t}=1. \tag{18}$$

Thus the native values extend to a character chi:Lambda→k*. This is a proof of consistency on every integral dependency, not an assumption based only on valuations. It does not select a new sign sector: the all-plus sector and the native value iA were fixed in Section 1.

Define vartheta(u)=sum_t(gamma_t u_zt+alpha_t u_wt). Equation (17) also gives vartheta(lambda)=nu_A chi(lambda). The ordinary unfilled strict angles are exactly the valuation certificate needed for the edge-only average below.

8. A convergent functional and its support

Put q=A^4, K=Q(i)((A^(1/2))), and use the established tetrahedron index

$$I_\Delta(m,e)=\sum_{n\geq\max(0,-e)} \frac{(-1)^n q^{n(n+1)/2-(n+e/2)m}} {(q;q)_n(q;q)_{n+e}}.$$

Define the k-linear seed ell on the algebraic quantum torus by

$$\ell(W(2a,2b))=\prod_t I_\Delta(b_t,-a_t), \qquad\ell(W(u))=0\quad(u\notin2L). \tag{19}$$

The index recurrence gives ell(Sf_t)=0 on the required side. Average over the actual edge image:

$$F(v)=\sum_{\lambda\in\Lambda}\chi(\lambda)^{-1} \ell(W(\lambda)v). \tag{20}$$

The all-input convergence theorem in the preceding global-index packet applies verbatim with only edge rows. For clarity, its decisive estimate is summarized here. For a supported summand on W(u), put v=(u+lambda)/2, T(v)=(-v_z,0,-v_w), and subtract each triple’s minimum m_t to obtain a nonnegative reduced vector x. With

$$\delta(x)=\sum_t(x_{t1}x_{t2}+x_{t2}x_{t3}+x_{t3}x_{t1}),$$

the exact order is

$$d_u(x)=2\delta(x)+2\theta\cdot x+\vartheta(u)+\Omega_3(x,T(u)), \tag{21}$$

where Omega_3 has block matrix [[0,1,-1],[-1,0,1],[1,-1,0]]. The allowed reduced vectors lie in one affine integral lattice. Dickson’s lemma supplies finitely many coordinatewise minimal vectors b. For x=b+Y, isotropy of the edge rows gives

$$d_u(b+Y)-d_u(b) =2\delta(Y)+2\theta\cdot Y+2(2\delta-\Omega_3)(b,Y) \geq2\epsilon\|Y\|_1, \tag{22}$$

where epsilon is the smallest angle, positive. Here delta(b,Y)=b^TD_sym Y/2 is the polar form, and D_sym-Omega_3 is entrywise nonnegative. Distinct allowed lambda have distinct reduced x. Hence below every order bound only finitely many nonzero summands occur. This proves order-independent convergence on every finite Laurent input, including all translated odd-root inputs.

Associativity and the seed recurrence give F(Sf_t)=0. Reindexing the convergent sum gives F(W(h_e)v)=chi(h_e)F(v), hence F((c_eW(h_e)+A^2)v)=0. It follows that F descends to Q_+.

At u=0 the order is 2delta(x)+2theta·x. Every nonzero supported lambda has x≠0 and positive order. The lambda=0 summand is I_Delta(0,0)^N with constant coefficient one. Therefore

$$F(1)=1+\text{strictly positive A-order}\neq0. \tag{23}$$

Finally the support of (20) is contained in 2L+Lambda. All its degrees d are zero.

The tetrahedron index and its recurrence are established literature, originating with Dimofte–Gaiotto–Gukov and rigorously developed by Garoufalidis and subsequent 3D-index work. The convergence estimate reuses the earlier Lab extension of the Garoufalidis–Yu/GHHR method; neither the seed nor lattice averaging is introduced here.

9. Detecting classes outside the entire image

Choose an integral w in P. Define

$$\Psi_w(v)=F(W(-w)v). \tag{24}$$

It annihilates every tetrahedron left multiple by associativity. It also annihilates every edge right multiple, because Omega(w,h_e)=0 lets W(-w) commute across W(h_e) without changing chi. This proves correct-sided descent to Q_+ on all inputs, not just on 1.

Its support lies in w+2L+Lambda, so it is zero on every degree other than d(w). If d(w)≠0, (9) gives

$$\Psi_w(\pi_+\tau(x))=0\quad(x\in\operatorname{Sk}(M)), \qquad \Psi_w(W(w))=F(1)\neq0. \tag{25}$$

Let s=dim d(bar P), at least r. Choose a lift w_f for each nonzero degree f in this subspace. There are 2^s-1 of them, and

$$\Psi_{w_f}(W(w_g))=\begin{cases}F(1),&f=g,\\0,&f\neq g.\end{cases} \tag{26}$$

Thus these classes are linearly independent in Q_+/im(pi_+tau). Pulling the functionals back through Q^6→Q_+ proves (1) over k: a k-linear dependence would also be killed by each K-valued functional. No conclusion is inferred backward from a specialized zero.

10. Every filling and the exact limit of the conclusion

For any filling surjection p, tau(ker p) is contained in im tau. Consequently

$$D_p\twoheadrightarrow Q^6/\operatorname{im}\tau\neq0. \tag{27}$$

The induced map Sk(M_fill)→D_p is well-defined by construction. Alternatively, the quotient map q:Q^6→Q6/im tau satisfies q tau=0, so the zero map on Sk(M_fill) makes the desired square commute. The quotient in this second construction is independent of the filling.

Panitch–Romo’s literal Conjecture 7.1 specifies a nontrivial quotient and a commuting trace map; it does not explicitly require the map to be nonzero. Equations (1) and (27) verify this condition after passage to the generic field for the strict-angle class, using the retained identification with their named square-root target. They do not deliver the nonzero insertion that motivates the conjecture, establish its unrestricted coefficient-ring version, or show that every triangulation admits strict angles.

In the fixed figure-eight meridian case, the preceding native kernel witness additionally proves D_p=Q^6/im tau and forces every compatible trace to be zero. That stronger assertion is specific to the preceding kernel calculation and is not generalized here.

11. Checks, useful failure, and next question

The accompanying exact verifier checks the universal local half-face identities, the symplectic form, the primitive-lattice distinction in a controlled example, the native character cancellation including fourth-root phases, and the previously fixed figure-eight normalizer/parity witness. These finite checks support the algebra; the graph, rank, and convergence arguments above prove the all-triangulation statement.

A tempting stronger route was to identify d(bar P) with all of H^1(M;F_2) directly from the normal-homology classification. Merely quoting an abstract group isomorphism does not identify the particular half-face map. That extra identification was unnecessary and is not asserted. The primitive integral kernel and cusp-incidence rank bound suffice for (16). Likewise, replacing Lambda by its saturation in (20) would change the actual relations and was not done.

Next decisive question. Can an index functional with nonzero unit be constructed for the actual unfilled target when strict angles are unavailable, while retaining every edge character and all-input convergence? The topological part of this proof is angle-independent. A replacement for Section 8 would extend the same nonzero-quotient theorem. Recovering a nonzero filled trace is a separate problem, and the fixed meridian kernel witness forbids that for this named figure-eight trace.

Sources and attribution

• Panitch–Park, 3d quantum trace map, AGT 26 (2026), 2255–2314, Theorem 5.6 and Proposition 5.7: even face-suspension image. Formal input for the whole-image assertion.
• Garoufalidis–Yu, A quantum trace map for 3-manifolds, Advances in Mathematics 486 (2026), 110735, Remark 4.5 and the native trace construction: established mod-two homology grading and comparison context.
• Garoufalidis–Hodgson–Hoffman–Rubinstein, The 3D-index and normal surfaces, Sections 2 and 7: exact tetrahedron-index degree and integral normal-homology perspective. The integral classification motivated the search; the stronger surjectivity identification is not used.
• Garoufalidis, The 3D index of an ideal triangulation and angle structures: tetrahedron-index recurrence and convergence background.
• Neumann, Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic 3-manifolds, Topology ’90 (1992), 243–271: source of the integral chain complex, consulted through GHHR’s explicit formulation. No claim to have freshly inspected the original article.
• Panitch–Romo, On the length conjecture for twist knots, Conjecture 7.1: exact external nonzero-quotient condition.
• Prior global-index packet, Sections 3–6: retained all-root, correct-sided, all-input convergence theorem.
• Prior minimal-meridian packet: fixed-datum parity detector and the kernel witness that motivated this generalization.