Peripheral insertions and the balanced vacuum pairing

FPRD Lab · 8 September 2026 · Research note

Result and scope

The balanced tetrahedron-index pairing cannot be repaired by inserting ordinary boundary skeins on the remaining cusps. Under the retained complete strict angular certificate, in the native all-plus sector, every such insertion has value zero on the filled vacuum. This includes arbitrary products of boundary curves, even when curves on the same torus intersect.

There is a more precise classification. Every monomial left translate certified by the same complete angle vector that has nonempty vacuum support is an exact scalar multiple of a native peripheral insertion on the remaining cusps. This statement uses the actual integral normal-homology theorem; it does not replace the edge lattice by its saturation.

If $\mathscr T_o$ is the native quantum torus of the remaining boundary components, $\sigma(E_\gamma)=E_{-\gamma}$ is simultaneous inversion there, and $\mathscr B$ is the resulting vacuum functional, then

$$\boxed{\mathscr B\circ\sigma=-\mathscr B, \qquad \mathscr B(P)=\mathscr B\!\left(\frac{P-\sigma P}{2}\right).} \tag{1}$$

The boundary-skein image lies in the fixed subalgebra, so its value is zero. Equation (1) does not evaluate the remaining odd part. In particular, independent-cusp inversion of the actual index has not been established here, and the previously proposed single oriented insertion remains unresolved. We give explicit formal boundary coefficient systems showing that simultaneous inversion, homological support, summability, and unit normalization alone do not determine that odd part.

This is a restriction on one family of detectors. It is not vacuum vanishing, target vanishing, or a resolution of Panitch–Romo Conjecture 7.1. The earlier two-angle nonvanishing theorem remains intact. The work is AI-assisted, has received no external specialist review, and makes no novelty or first-priority claim.

1. Retained data and dependencies

Let $M$ be an oriented compact 3-manifold whose boundary consists of tori, with an actual ideal triangulation $\mathcal T$ having $N$ tetrahedra. Fix a filling cusp $T_0$, an integral basis $s,l$ of its first homology, and a complete strict angular certificate: positive tetrahedral angles with tetrahedron sums one, edge pairings two, and zero angular holonomy on every peripheral class. A separate filling-angle certificate is not assumed in this note.

Work at generic parameter over

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

The root lattice is $L=\mathbb Z^{2N}$, with

$$W(u)W(v)=A^{\Omega(u,v)/2}W(u+v),\qquad \Omega(u,v)=\sum_j(u_{wj}v_{zj}-u_{zj}v_{wj}). \tag{2}$$

For an integral raw three-coordinate vector $R$, put

$$P(R)_j=(R_{j1}-R_{j2},R_{j3}-R_{j2}),\qquad p(R)=\sum_jR_{j2}.$$

Actual edge rows $E_e$ and the fixed additive peripheral rows $R_s,R_l,R_\gamma$ give $h_e=P(E_e)$, $h_s=P(R_s)$, $b=P(R_l)$, $w_\gamma=P(R_\gamma)$. The native all-plus monomials are

$$m=c_sW(h_s),\quad l=c_lW(b),\quad E_\gamma=c_\gamma W(w_\gamma),\qquad c_\bullet=B^{p(R_\bullet)}. \tag{3}$$

The repeated symbol $l$ denotes a quantum monomial in algebra expressions and the oriented boundary class in homology expressions. No scalar in (3) is changed to force a symmetry. The additive convention gives $c_{-\gamma}=c_\gamma^{-1}$.

Keep the unsaturated image lattice

$$\Lambda=\sum_e\mathbb Zh_e+\mathbb Zh_s. \tag{4}$$

It is isotropic, $\Omega(h_s,b)=2$, and $\Omega(h_e,b)=0$. Thus $n(\lambda)=\Omega(\lambda,b)/2$ is the integer meridian coefficient. The original moment character is $\chi(h_r)=-A^2/c_r$; the complete character is

$$\chi_0(\lambda)=\chi(\lambda)A^{-2n(\lambda)}. \tag{5}$$

The original target is the mixed-sided vector-space quotient

$$C_s=S\Big/\left(\sum_j S(z_j^{-2}+w_j^2-1) +\sum_e(c_eW(h_e)+A^2)S+(c_sW(h_s)+A^2)S\right). \tag{6}$$

Its filled vacuum is represented by $1-l^2$. It is not the cokernel unit.

The preceding balanced-pairing note supplies the complete-character average $F_0$, its unordered convergence on every algebraic input, and the exact normal-index formula used below. Its seed vanishes off $2L$. The global-index note supplies the all-input convergence argument and correct-sided moment descent. These are retained proof dependencies, not new claims established by the companion computation.

For the new lattice argument we use the following integral normal-surface facts from GHHR Theorem 7.1 and Section 7.1:

$$\begin{array}{c|ccc} &E_e&R_\alpha&\Delta_j\\\hline \partial&0&2\alpha&0\\ [\ ]_2&0&0&0 \end{array},\qquad \ker(X\mapsto([X]_2,\partial X))=\mathbb E+\mathbb T. \tag{7}$$

Here $\mathbb E=\sum_e\mathbb ZE_e$, $\mathbb T=\sum_j\mathbb Z\Delta_j$; $\Delta_j$ is the tetrahedron all-one vector. The image consists of pairs $(a,\alpha)$ with $\partial_*a=\alpha\bmod2$. The matching equations identify edge-orthogonal root vectors with integral normal vectors modulo $\mathbb T$; the peripheral skew form agrees with boundary intersection. These statements concern the actual triangulation, not arbitrary isotropic matrices.

2. Classification of every supported complete-character translate

Set $\Gamma_o=\bigoplus_{T\ne T_0}H_1(T;\mathbb Z)$. Let $\iota_*$ denote inclusion into $H_1(M;\mathbb F_2)$. Define the admissible peripheral classes

$$\Gamma_{\mathrm{adm}}= \{\gamma\in\Gamma_o:\ \iota_*(\gamma-l)\in \operatorname{span}_{\mathbb F_2}\{\iota_*s\}\}. \tag{8}$$

This may be empty. It is a parity condition, not a search over normal surfaces.

A monomial translate has shifted character

$$\chi_v(\lambda)=\chi(\lambda)A^{\Omega(\lambda,v)}, \qquad \mathcal G_v(a)=F_{\chi_v}(W(v)a). \tag{9}$$

For $\chi_v$ to have the valuation certified by the same fixed complete vector, one must have

$$\Omega(\lambda,v+b)=0\quad\text{for every }\lambda\in\Lambda. \tag{10}$$

At generic parameter (10) gives the exact character $\chi_v=\chi_0$, not merely equal valuations. Put $w=v+b$. Its two vacuum terms have the same support condition

$$w-b+\lambda\in2L\quad\text{for some }\lambda\in\Lambda. \tag{11}$$

Theorem 2.1 — integral classification

Among $w\in L\cap\Lambda^\perp$, condition (11) holds exactly for

$$\boxed{w=\lambda_0+w_\gamma, \qquad\lambda_0\in\Lambda,quad\gamma\in\Gamma_{\mathrm{adm}}.} \tag{12}$$

The class $\gamma$ and the root vector $\lambda_0$ in (12) are unique for the fixed additive peripheral rows. In particular, supported translates modulo the actual image lattice are in bijection with $\Gamma_{\mathrm{adm}}$. Every translate failing (11) kills the vacuum by empty seed support; it need not be the zero functional on other inputs.

Proof. Choose an integral raw lift $V$ with $P(V)=w$. Edge orthogonality makes $V$ an integral normal class. Orthogonality to $h_s$, by the boundary intersection form, implies

$$\partial V=\kappa s+\beta, \qquad \kappa\in\mathbb Z,\quad\beta\in\Gamma_o. \tag{13}$$

There is no $l$-component. An alternative integral lift differs by an integral tetrahedron vector, so has the same boundary and mod-2 homology.

Suppose (11) holds and write $\lambda=P(R)$, where $R=\sum_en_eE_e+nR_s$. In each tetrahedron let $t_j$ be the maximum coordinate of $(V-R_l+R)/2$, and set

$$X=\sum_jt_j\Delta_j-(V-R_l+R)/2. \tag{14}$$

Condition (11) says the coordinate differences of the half-vector are integral. Therefore $X$ is a nonnegative integral normal vector with block minimum zero. Its boundary is

$$\partial X=l-(n+\kappa/2)s-\beta/2. \tag{15}$$

Since the boundary homology is free with the stated cusp decomposition, (15) forces $\kappa=2k$ and $\beta=2\gamma$ with integers $k$ and $\gamma\in\Gamma_o$. This is the essential integrality step; it cannot be replaced by real-span reasoning.

Furthermore,

$$2X+V-R_l+R=\sum_j2t_j\Delta_j\in\mathbb T. \tag{16}$$

The coefficients $2t_j$ are integral because the other terms of (16) are integral. Taking mod-2 normal homology and using (7) gives $[V]_2=0$. Thus $V-kR_s-R_\gamma$ has zero boundary and zero mod-2 homology. Equation (7) puts it in $\mathbb E+\mathbb T$. Applying $P$ yields

$$w-w_\gamma\in\Lambda,$$

with meridian coefficient $k$. The boundary of $X$ in (15) also gives (8).

Conversely, let $\gamma\in\Gamma_{\mathrm{adm}}$. By the long exact sequence of $(M,\partial M)$ over $\mathbb F_2$, choose an integer $n$ and a mod-2 relative class $a$ with $\partial_*a=l-\gamma-ns\bmod2$. The image statement of (7) supplies an integral normal class $X$ with exactly that integral boundary. Then

$$2X+R_\gamma-R_l+nR_s\in\mathbb E+\mathbb T. \tag{17}$$

Applying $P$ to an integer expression for (17) gives $w_\gamma-b+\lambda\in2L$ for some $\lambda\in\Lambda$. Adding any $\lambda_0\in\Lambda$ to $w_\gamma$ only reindexes this support condition.

Finally, if two expressions (12) hold, their difference gives $P(R_\gamma-R_{\gamma'}-R_0)=0$ for an integral edge-plus-meridian row $R_0$. The kernel of $P$ is $\mathbb T$. Taking boundary and projecting to the other cusps gives $2(\gamma-\gamma')=0$, hence $\gamma=\gamma'$ and then $\lambda_0=\lambda'_0$. ∎

Exact scalar, on every input

For the decomposition (12), write $n_0=n(\lambda_0)$. Then

$$\boxed{\mathcal G_{w-b} =\chi(\lambda_0)A^{-n_0}c_\gamma^{-1}\,mathcal G[E_\gamma], \quad \mathcal G[P](a):=F_0(W(-b)P a).} \tag{18}$$

To check the scalar directly, put $u=w_\gamma-b$. The exact Weyl product and complete moment relation give

$$F_0(W(\lambda_0+u)a) =A^{-\Omega(\lambda_0,u)/2}\chi_0(\lambda_0)F_0(W(u)a).$$

Other-cusp orthogonality gives $\Omega(\lambda_0,u)=-2n_0$. Inserting (5) gives $\chi(\lambda_0)A^{-n_0}$, and native normalization gives (18). This identity is of convergent functionals on the original algebra, not a multiplication rule for quotient classes.

3. The surviving boundary functional is odd

Let $\mathscr T_o$ be the quantum torus with native basis $E_\gamma$, $\gamma\in\Gamma_o$. Its multiplication is

$$E_\gamma E_\eta =A^{\Omega(w_\gamma,w_\eta)/2}E_{\gamma+\eta}. \tag{19}$$

Define $\sigma$ to fix the coefficient field and send $E_\gamma$ to $E_{-\gamma}$. Since $\Omega(-w_\gamma,-w_\eta)=\Omega(w_\gamma,w_\eta)$, it is an algebra automorphism, not an anti-automorphism and not a change $A\mapsto A^{-1}$.

Every $P\in\mathscr T_o$ commutes with the moment monomials and with $l$. The retained correct-sided argument therefore makes $\mathcal G[P]$ a functional on the original target (6). Equivalently it is a finite linear combination of the already convergent translates. Define

$$\mathscr B(P)=\mathcal G[P](1-l^2). \tag{20}$$

Let $\mathcal Z(\alpha)=\sum_a I_{\mathcal T}^a(\alpha)$ be the total GHHR normal boundary index, including all mod-2 homology sectors. The preceding exact integral-fiber proof extends additively to every $\gamma\in\Gamma_o$, and gives

$$F_0(E_\gamma l^r) =\sum_{n\in\mathbb Z}(-1)^n A^{rn} \mathcal Z(-\gamma-rl-ns). \tag{21}$$

For clarity, the normal-class map uses $U=(R+R_\gamma+rR_l)/2$, $X=\max(U)\Delta-U$. Its surjectivity witness is $2X+R_\gamma+rR_l+nR_s\in\mathbb E+\mathbb T$. The same native scalar cancellation leaves exactly $(-1)^n A^{rn}$. No new symmetry or saturation is used in (21).

GHHR’s published Remark 8.3 states $\mathcal Z(-\alpha)=\mathcal Z(\alpha)$. Using this simultaneous inversion and legitimate $n\mapsto-n$ reindexing in (21) gives

$$\boxed{\mathscr B(E_\gamma) =c_l\sum_n(-1)^nA^n \left[\mathcal Z(\gamma-l-ns)-\mathcal Z(-\gamma-l-ns)\right].} \tag{22}$$

All rearrangements are justified by the retained all-input unordered convergence before grouping into normal fibers. The residual sign $(-1)^n$ is retained.

Replacing $\gamma$ by $-\gamma$ in (22) changes the sign. Linearity now proves (1). The classification theorem shows that this boundary odd part contains the entire remaining vacuum-detection question for the supported, same-complete-character monomial family.

Corollary 3.1 — all remaining-cusp boundary skeins are invisible

For every element $P$ in the image of

$$\bigotimes_{T\ne T_0}\operatorname{Sk}_k(T\times I) \longrightarrow\mathscr T_o, \qquad \boxed{\mathcal G[P](1-l^2)=0.} \tag{23}$$

Proof. Frohman–Gelca’s torus-skein embedding sends a Chebyshev curve of class $\gamma$ on one cusp to $E_\gamma+E_{-\gamma}$. These elements and the unit span each torus-skein image and are fixed by $\sigma$. Because $\sigma$ is an algebra automorphism, arbitrary products remain fixed, including noncommuting products on a single cusp. Formula (1) gives (23). The retained native named-action dictionary identifies these sums with the actual boundary-skein action; GY Theorem 5.7 is the original identified-target source for that dictionary. ∎

This proof needs no independent-cusp invariance of $\mathcal Z$. It also does not extend $\sigma$ to the entire mixed-sided target. Equation (23) evaluates the vacuum after boundary insertion; it does not say that $\mathcal G[P]$ is zero on every target class. Indeed the uninserted functional detects $l$, as established previously.

4. Why the remaining oriented test is not yet decided

The two moment eigencharacters differ as in (5). Their covariance justifies (18) and descent, but supplies no extra relation between $\mathcal Z(\gamma+\alpha)$ and $\mathcal Z(-\gamma+\alpha)$ when the two classes are on different cusps. An unoriented insertion produces their sum, whereas (22) needs their difference.

A minimal formal check makes the limitation precise. Use a free boundary lattice with basis $s,l,t,u$ on two tori. Over $\mathbb F_2$ give its alternating form the pairings $\langle s,l\rangle=\langle t,u\rangle=1$, and take the Lagrangian parity kernel

$$\mathcal K=\operatorname{span}_{\mathbb F_2}\{t+l,s+u\}. \tag{24}$$

For $\varepsilon\in\{0,1\}$, define finite-support formal coefficients

$$Z_\varepsilon(0)=1,\qquad Z_\varepsilon(t-l)=Z_\varepsilon(-t+l)=\varepsilon A^4,$$

and zero elsewhere. They have simultaneous inversion, allowed parity support, unordered summability for every weighted outer sum, positive order away from zero, and

$$\sum_n(-1)^n Z_\varepsilon(-ns)=1.$$

Yet the right side of (22), divided by $c_l$, at $\gamma=t$ is respectively $0$ and $A^4$. Thus those boundary constraints do not logically determine the answer.

These are not normal indices of claimed triangulations. No tetrahedron recurrence, actual gluing presentation, or topological realization is supplied for these coefficient systems. They are not counterexamples to independent-cusp symmetry, to the named trace, or to the conjecture. They show why such a symmetry or a nonzero odd coefficient must come from additional mathematics.

5. What remains to attack

The separate-cusp oriented test (22) remains open. The useful reduction is now exact: unsupported translates vanish by parity; every supported translate reduces by (18) to a native other-cusp insertion; and every simultaneous-inversion-even insertion vanishes by (1), including the full boundary-skein image.

The next decisive object is the odd functional $\mathscr B|_{\mathscr T_o^-}$. A successful multiple-cusp continuation must either prove independent-cusp symmetry for the actual total normal index, or produce a nonzero value of (22) using the actual tetrahedron/normal data with a certified remainder. A symmetry of classical eigenvalues, unoriented boundary curves, or conjectural block factorization is insufficient. Do not repeat even boundary insertions or manufacture support by saturating $\Lambda$.

For one cusp $\Gamma_o=0$, so this route is exhausted. The retained alternative is a nonperipheral insertion with a different certifiable character, or a genuinely second seed whose all-input convergence and original mixed-sided descent are proved before evaluating the vacuum. No such construction is claimed here.

Sources and intellectual credit

• S. Garoufalidis, C. D. Hodgson, N. R. Hoffman, J. H. Rubinstein, The 3D-index and normal surfaces, Illinois Journal of Mathematics 60 (2016), 289–352. Published paper, arXiv text. Formal dependencies: Theorem 7.1, Section 7.1, Definition 8.1, Remark 8.3, and the boundary/skew dictionary. Integral normal homology is also the main structural influence on Theorem 2.1.
• C. Frohman, R. Gelca, Skein modules and the noncommutative torus. Primary preprint, author-hosted paper. Formal dependency and approach influence: the torus-skein fixed-subalgebra description and its exact curve embedding.
• S. Garoufalidis, T. Yu, The 3d-index of the 3d-skein module via the quantum trace map, author revision dated 6 August 2025. Paper. Section 5.5 and Theorem 5.7 give the identified-target peripheral action underlying the retained native dictionary. Section 5.6 discusses a stronger descendant symmetry without proving it in general; it is not used as a symmetry theorem here.
• T. Dimofte, D. Gaiotto, S. Gukov, 3-Manifolds and 3d Indices. Primary paper. Origin of the index and physical Weyl-symmetry motivation. The enhancement discussion surrounding equation (4.15) is motivation, not an independent-cusp proof dependency.
• FPRD Lab, balanced-pairing proof, global-index proof, two-angle vacuum proof, native collar dictionary, all 8 September 2026. The present attack continues their unresolved multiple-cusp gate; it does not replace their hypotheses or assert external novelty.