Closed normal sectors and nonzero quantum-trace cokernels

FPRD Lab · 8 September 2026 · Supporting research note

Result and status

Let M be a connected oriented compact 3-manifold with r>=1 torus boundary components and let T be a 1-efficient ideal triangulation. Retain the native convention dictionary and generic field k=Q(i,A^(1/2)) of the preceding six-root and face-defect notes. If M is neither a solid torus nor T^2 x I, then

$$F(1)=1+\text{positive }A\text{-order},\qquad \dim_k(Q^6_T/\operatorname{im}\tau)\ge 2^r-1. \tag{1}$$

Thus every minimal compatible Dehn-filling quotient is nonzero in this class. Strict or nonnegative angles are not required. The compatible map to the common cokernel is zero; (1) is not a nonzero filled invariant.

The proof combines an exact integer normal-sector identification with Garoufalidis–Hodgson–Hoffman–Rubinstein’s published positivity theorem, not a new proof of that theorem. We also prove F!=0 for the particular published solid-torus and T^2 x I triangulations below, using explicit values away from the unit. We do not prove this for every triangulation of those exceptional manifolds or for non-1-efficient triangulations.

This is an internally checked conditional synthesis. Novelty is unassessed; no independent full-proof review or formal verification is claimed. It remains a research note, not a resolution of the unrestricted Panitch–Romo conjecture.

1. The native data and the external inputs

Write L=Z^(2N), P(R)=(R_1-R_2,R_3-R_2) blockwise and

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

For actual opposite-pair edge incidences R_e, put h_e=P(R_e), c_e=(iA)^(sum_t R_{e,t2}), Lambda=sum Z h_e and chi(h_e)=-A^2/c_e. The all-plus target is

$$Q_+=S/(\sum_t S(z_t^{-2}+w_t^2-1)+\sum_e(c_eW(h_e)+A^2)S). \tag{2}$$

It is a mixed-sided vector-space quotient, not a two-sided algebra quotient. Projection Q^6 -> Q_+ is used only to pull back detecting functionals. Lambda is never replaced by its saturation.

Set q=A^4, T(a,b)=(-a,0,-b), and let Delta_t=(1,1,1) in block t. The seed is ell(W(2a,2b))=product I_Delta(b_t,-a_t), zero otherwise. Our functional is the existing average

$$F(W(u))=\sum_{\lambda\in\Lambda}\chi(\lambda)^{-1} A^{\Omega(\lambda,u)/2}\ell(W(u+\lambda)). \tag{3}$$

Formal external inputs from GHHR [1]: Theorem 7.1 identifies the integer kernel of normal boundary and mod-two homology as E+T; Proposition A.2 supplies generalized angles; Corollary 10.19 gives positive degree for nonzero irreducible closed normal classes on 1-efficient triangulations, excluding the two manifolds above. Their index identities and Examples 11.1/11.3 supply the local controls. The prior signed-angle theorem [2] supplies all-input convergence from its ray criterion, and the native comparison and face-defect theorem [3] supply the trace grading and integral normalizers. These are proof dependencies, not independent confirmations of (1).

2. The exact integer bridge

Let N_Z be the lattice of all closed integral Q-normal classes, with arbitrary mod-two homology, and let T_Z=sum Z Delta_t. Define

$$H=T((\Lambda\cap2L)/2)+T_Z.$$

Lemma. H=N_Z. If lambda=P(R) is even, for an integer edge combination R, then Y=T(lambda/2)=-R/2+sum_t(R_{t2}/2)Delta_t. Consequently 2Y is in E+T_Z. The Q-matching and integral boundary maps vanish on 2Y, hence on Y. This proves H subset N_Z.

Conversely, X in N_Z has 2X in E+T_Z by GHHR Theorem 7.1: doubling kills mod-two homology, and its boundary remains zero. Write -2X=R+D with R an integer edge combination and D in T_Z. Then lambda=P(R)=-2P(X) belongs to the actual Lambda intersect 2L, and

$$T(\lambda/2)=T(-P(X))=X-\sum_tX_{t2}\Delta_t.$$

Thus X belongs to H. No half-integral edge coefficient or saturated lattice has been introduced.

Reduction by the minimum in each triple gives a bijection

$$\Lambda\cap2L\ \longleftrightarrow\ N_Z/T_Z, \quad\lambda\mapsto [T(\lambda/2)]. \tag{4}$$

Injectivity follows from the fixed zero middle coordinate of T(lambda/2). Surjectivity is the preceding doubling argument. Nonnegative representatives on the right are chosen with minimum zero in every block.

3. The phase, not just the degree

Choose R=sum n_e R_e for lambda, and set E=sum n_e, p=sum_t R_{t2}. Put Y=T(lambda/2), m_t=min(Y_t), m=sum m_t and X=Y-sum m_t Delta_t. The formal Euler characteristic, with chi_top(R_e)=-2 and chi_top(Delta_t)=-1, gives

$$\chi_{\rm top}(Y)=E-p/2,\qquad \chi_{\rm top}(X)=E-p/2+m.$$

The Euler characteristic of an integral closed class is integral; therefore p is even. The native character and the tetrahedron-index translation identity give

$$\chi(\lambda)^{-1}=(-1)^Ei^p A^{-2E+p},\qquad \ell(W(\lambda))=J(Y)=(-A^2)^{-m}J(X).$$

Since E+p/2-m and -chi_top(X) differ by 2E, their signs agree. Hence the entire summand, not only its valuation, satisfies

$$\boxed{\chi(\lambda)^{-1}\ell(W(\lambda)) =(-A^2)^{-\chi_{\rm top}(X)}J(X)=I([X]).} \tag{5}$$

In particular, when summable,

$$\boxed{F(1)=\sum_{a:\partial_*a=0} I_T^a(0).} \tag{6}$$

It would generally be wrong to replace the right side by I_T^0(0). The distinction is invisible in some knot-exterior examples but matters for the general proof. Equations (4)-(5) establish the needed integer and phase alignment.

4. Convergence is exactly 1-efficiency

For any generalized angle vector theta satisfying the actual equations, theta.X=-chi_top(X) on N_Z. By Section 2, the cone in the signed-angle criterion is N_R intersect R_{>=0}^{3N}. Its nonzero delta-zero points, where

$$\delta(X)=\sum_t(X_{t1}X_{t2}+X_{t2}X_{t3}+X_{t3}X_{t1}),$$

have at most one nonzero quad coordinate per block. Rational points scale to admissible closed embedded normal surfaces. Vertex-linking tori have zero quad vector and are absent from these nonzero rays. Thus theta.X>0 on every such ray is precisely absence of a nonzero admissible closed normal class with nonnegative Euler characteristic: 1-efficiency.

A failure on a real ray also fails on a rational ray, since its support face and the Euler inequality are rational. Conversely any forbidden embedded surface supplies such a ray. The preceding all-input theorem [2] now proves that (3) is summable on every Laurent input for every 1-efficient T. If T is not 1-efficient, the unordered family at 1 fails to converge. This is an exact boundary for this construction, not a zero-module theorem.

5. Nonvanishing outside the two exceptional manifolds

Let X be a nonzero reduced nonnegative closed class. Decompose X as a sum of irreducible elements in the nonnegative closed-normal monoid. Such a decomposition terminates because the sum of its integer coordinates decreases at each proper split. Each summand is coordinatewise bounded by X, so has a zero in every block and cannot be a tetrahedral solution.

GHHR Corollary 10.19 applies to each nonzero summand X_j when M is not a solid torus or T^2 x I. With d(X)=delta(X)-chi_top(X), it gives d(X_j)>=1. Since delta has nonnegative mixed terms and chi_top is additive,

$$d(X)\ge\sum_jd(X_j)\ge1. \tag{7}$$

All nonzero classes in (6), including nontrivial mod-two sectors, therefore have strictly positive A-order. The zero class contributes J(0)^N with constant coefficient one. This proves F(1)=1+positive order.

The integral normalizers from [3] have at least r independent nonzero face-defect directions. Their translated functionals descend through (2), vanish on the entire native trace image and have a diagonal nonzero evaluation matrix. This proves (1). The quotient Q^6/tau(ker p) surjects onto Q^6/im tau for every filling p, so it too is nonzero. All of the original comparison and grading hypotheses are retained.

6. A genuine solid-torus detector away from the unit

For GHHR’s fixed solid-torus triangulation cMcabbgds, the two edge rows are

$$R_1=(2,1,2;2,2,2),\qquad R_2=(0,1,0;0,0,0).$$

Thus h_2=(-1,-1;0,0), h_1=-h_2, c_2=iA, c_1=(iA)^3, and chi(h_2)=iA. These include the degree-one edge exactly. A generalized certificate is (-1/2,2,-1/2;1/3,1/3,1/3); the forced angle 2 proves there is no nonnegative angle structure on this datum.

For even input u=(2a,2b;2c,2d), put m=b-a. The second block factors as I_Delta(d,-c). Direct triality in the first block yields

$$F(W(u))=(-1)^a q^{a(m-1)/2} \sum_{e\in\mathbb Z}q^{e(1-m/2)}I_\Delta(m,e) I_\Delta(d,-c). \tag{8}$$

The sum in (8) is zero for m<=0, and for m>=1 is

$$\frac{(-1)^m q^{m(m-1)/2}}{(q;q)_{m-1}}. \tag{9}$$

Here is a summability-safe derivation of the second case. In the generating function

$$\sum_e I_\Delta(m,e)z^e =\frac{(q^{1-m/2}z^{-1};q)_\infty} {(q^{-m/2}z;q)_\infty},$$

cancel the m common factors before evaluation, giving

$$(-1)^m q^{m/2}z^{-m} \frac{(q^{1+m/2}z^{-1};q)_\infty} {(q^{m/2}z;q)_\infty}. \tag{10}$$

Its Laurent coefficients are (-1)^m q^{m/2}I_Delta(-m,e+m). At z=q^{1-m/2}, the denominator expansion has argument q, and the numerator expansion is quadratically summable. Evaluation therefore gives (9). For m<=0 the original expansion is summable at that z; its numerator has the factor 1-1 and its denominator has positive q-order, giving zero.

Substituting directly into the uncancelled ratio for m>=1 would incorrectly produce 0/0. Interchanging its raw seed double sum also fails: its exponent includes k(1-m), which does not tend to positive infinity. Neither operation is used.

Taking a=c=d=0, b=1 gives the actual native values

$$\boxed{F(1)=0,\qquad F(w_1^2)=-I_\Delta(0,0)\ne0.} \tag{11}$$

The unaveraged seed has I_Delta(0,0)=1+positive order. The detecting input is even and hence has face-defect zero; it may be translated by the integral normalizing defects as in [2]. The cokernel bound >=1 holds for this fixed triangulation. This is no longer merely an abstract incidence control. The vanishing of the boundary index is already GHHR’s result; the use of the nonperipheral root input is the point of this calculation.

7. The published product-torus datum

For GHHR’s T^2 x I triangulation dLQacccbjkg, the edge rows are

$$(1,2,2;1,1,1;1,1,1),\quad (1,0,0;0,0,0;0,0,0),\quad (0,0,0;1,1,1;1,1,1).$$

Their projections are -h,h,0 with h=(1,0;0,0;0,0), and their characters are -A^(-2),-A2,1. Thus Lambda=Z h with chi(h)=-A^2; no row is dropped without checking its scalar. The generalized certificate (2,-1/2,-1/2;1/3,1/3,1/3;1/3,1/3,1/3) satisfies the equations and again excludes nonnegative angles.

For even inputs the first-block even edge sum is exactly the one-block control from [2]. The other two blocks factor. Consequently

$$\boxed{F(1)=0,\qquad F(w_1^{-2})=I_\Delta(0,0)^2\ne0.} \tag{12}$$

It follows that the fixed native cokernel has dimension at least 2^2-1=3. This does not rely on its scalar boundary index, which GHHR already showed to vanish.

8. What is settled and what is next

The signed criterion has an exact intrinsic identification: 1-efficiency. Under it, the general nonvanishing question is settled here outside the two exceptional manifolds using a published theorem, and the two published exceptional triangulations admit explicit nonunit detectors.

The remaining precise question is F!=0 for arbitrary 1-efficient triangulations of the solid torus or T^2 x I. Scalar index invariance is insufficient, since those scalar indices can be zero. A useful next route is a correct-sided transport identity for a nonperipheral detecting insertion under a local triangulation move, with all-input summability and the actual lattice retained. Any move argument must state its connectivity hypotheses; no general connectedness of the 1-efficient move graph is assumed. Alternatively, derive the insertion from an intrinsic compressing-disk or product structure. Do not enumerate triangulations or slopes.

Beyond 1-efficiency this unordered seed cannot be used on all inputs. A different completion or seed would require a fresh construction and descent proof.

References and credit

1. S. Garoufalidis, C. D. Hodgson, N. R. Hoffman and J. H. Rubinstein, The 3D-index and normal surfaces, Illinois Journal of Mathematics 60 (2016), 289–352; printed copyright 2017. Published author-hosted PDF, arXiv:1604.02688v1 HTML. The published Theorem 7.1, Corollary 10.19 (pp. 326–327), and examples on pp. 328–330 were inspected. Their integer normal-homology theory, positivity theorem and seed identities are formal dependencies. Their degree-one-edge examples directly influenced the nonperipheral test, separately from supplying its incidence data. No novelty is claimed for the known scalar index or its vanishing.
2. Signed-angle convergence and detection away from the unit. Formal dependency for all-input convergence and detection away from 1.
3. Cohomological degrees outside the quantum-trace image, with its cited native comparison and Panitch–Park inputs. Formal dependency for the entire image grading, integral lift rank and pullback to the full six-root module.

The exact controls and internal audit are accompanying files. They check phase and recurrence vulnerabilities; finite calculations do not certify the general topology or replace an independent reconstruction.