The balanced pairing: an exact normal-index bridge and a vanishing theorem

FPRD Lab, 8 September 2026. ATTACK research note. AI-assisted work, internally audited, without external specialist review or author endorsement. Novelty and first priority are unassessed. This is not a Publication.

Outcome

The balanced pairing does not detect the filled vacuum. In the native all-plus central sector, on an actual ideal triangulation with a complete strict angular certificate,

$$\boxed{\mathcal G_{-b}(1-l^2)=0.} \tag{1}$$

This is an all-orders equality, not a numerical cancellation. The proof first identifies the exact, unsaturated root-lattice sum with a sum over all integral normal classes in specified boundary fibers. It then applies the published boundary-inversion identity of Garoufalidis–Hodgson–Hoffman–Rubinstein (GHHR). That identity is a formal source dependency, not a new symmetry theorem proved here, and it is applied only after the lattice and scalar translation is established.

The pairing itself is nonzero: it detects the native longitude class. Moreover, for a triangulation with one torus boundary component, every monomial translate whose shifted character is certified by the same complete angle vector also kills the filled vacuum. Such a translate is either a scalar multiple of the balanced pairing or has empty parity support on the vacuum.

None of this says that the actual vacuum is zero. Indeed the preceding two-angle theorem proves it is nonzero in its chamber. The present result explains why one natural convergent family cannot extend that theorem beyond its chamber. Panitch–Romo Conjecture 7.1 remains open.

1. The retained native presentation

Let

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

The reduced root torus has lattice $L=\mathbb Z^{2N}$ and multiplication

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

We retain the vector-space quotient

$$C_s=S\Big/\left(\sum_j S(z_j^{-2}+w_j^2-1) +\sum_e(c_eW(h_e)+A^2)S+(c_sW(h_s)+A^2)S\right). \tag{3}$$

These are mixed-sided relation spaces, not a two-sided ideal. The native peripheral operators are

$$m=c_sW(h_s),\qquad l=c_lW(b),\qquad ml=A^2lm. \tag{4}$$

Under the retained filling reconstruction the filled vacuum maps to $[1-l^2]$, not to the cokernel unit. The native six-root comparison, generic-parameter conditions, Eulerian normalization, and CK source-reading caveat are unchanged.

For a three-coordinate raw row $R=(r_j,s_j,t_j)_j$, put

$$P(R)_j=(r_j-s_j,t_j-s_j),\qquad p(R)=\sum_js_j. \tag{5}$$

Actual edge rows $E_e$ and the fixed additive native peripheral rows $R_s,R_l$ give

$$h_e=P(E_e),\quad h_s=P(R_s),\quad b=P(R_l).$$

In the all-plus sector the unchanged native scalars are

$$c_e=B^{p(E_e)},\qquad c_s=B^{p(R_s)},\qquad c_l=B^\rho, \quad \rho=p(R_l). \tag{6}$$

No longitude rescaling is chosen to force (1). Throughout the proof, $s,l$ inside a boundary-homology expression denote the underlying oriented curves; inside a quantum-torus expression they denote (4).

Keep the actual image lattice

$$\Lambda=\sum_e\mathbb Zh_e+\mathbb Zh_s, \tag{7}$$

without saturation. It is isotropic, and $\Omega(h_e,b)=0$, $\Omega(h_s,b)=2$. Consequently each $\lambda\in\Lambda$ has a well-defined integer

$$n(\lambda)=\Omega(\lambda,b)/2, \tag{8}$$

its meridian coefficient in every integer row representation.

2. The complete character and correct-sided pairing

A complete strict angular certificate $\theta^0$ satisfies

$$\theta^0>0,\quad \theta^0\cdot\Delta_j=1,\quad \theta^0\cdot E_e=2,\quad\theta^0\cdot R_\gamma=0 \tag{9}$$

for every peripheral class. Here $\Delta_j$ is the all-one vector in block $j$.

The original exact moment character is $\chi(h_r)=-A^2/c_r$. Its all-plus consistency follows from all raw edge dependencies, as proved in the preceding actual-incidence packet. Define

$$\chi_0(\lambda)=\chi(\lambda)A^{-\Omega(\lambda,b)}. \tag{10}$$

If

$$\lambda=P(R),\qquad R=\sum_en_eE_e+nR_s, \quad E=\sum_en_e,\quad S_R=p(R),$$

then, including the phase,

$$\boxed{\chi_0(\lambda)=(-1)^nB^{2E-S_R}.} \tag{11}$$

In particular the $(-1)^n$ does not disappear when the meridian valuation becomes zero. Equation (9) certifies precisely the valuation of (10).

Let the index seed vanish off $2L$, with

$$\ell_0(W(2a_1,2d_1,\ldots))=\prod_j I_\Delta(d_j,-a_j).$$

Write

$$F_0(W(u))=\sum_{\lambda\in\Lambda}\chi_0(\lambda)^{-1} A^{\Omega(\lambda,u)/2}\ell_0(W(u+\lambda)). \tag{12}$$

The retained all-input convergence theorem applies to (12). The retained left-seed-translate theorem then gives

$$\mathcal G_{-b}(a)=F_0(W(-b)a) =\sum_{\lambda\in\Lambda}\chi(\lambda)^{-1} \ell_0(W(-b)W(\lambda)a). \tag{13}$$

It converges unordered on every algebraic input and annihilates exactly the original relation spaces (3). This passage does not replace the original meridian relation by the complete-character relation: $F_0$ and $\mathcal G_{-b}$ have different left moment eigencharacters.

Although (1) vanishes,

$$\mathcal G_{-b}(l)=c_lF_0(1) =c_l\bigl(1+\text{positive }A\text{-order}\bigr)\ne0. \tag{14}$$

Thus a complete strict certificate alone implies that the actual meridian cokernel is nonzero. This is a short consequence of the already-established all-input theorem; it does not prove that the filled vacuum survives.

3. The integral normal-homology input

We use the actual normal-surface dictionary, not merely a matrix satisfying isotropy. In $Q(\mathcal T;\mathbb Z)$, the edge, peripheral, and tetrahedron rows satisfy

$$\begin{array}{c|ccc} &E_e&R_\gamma&\Delta_j\\\hline \partial&0&2\gamma&0\\ \chi_{\rm top}&-2&0&-1\\ [\ ]_2&0&0&0 \end{array} \tag{15}$$

The formal Euler characteristic is integral on integral normal classes. Put $\mathbb E=\sum_e\mathbb ZE_e$, $\mathbb T=\sum_j\mathbb Z\Delta_j$.

The crucial integral theorem of Neumann in the normal formulation of GHHR is

$$\ker\bigl(X\mapsto([X]_2,\partial X)\bigr)=\mathbb E+\mathbb T, \tag{16}$$

and its image consists exactly of pairs $(a,\gamma)$ with $\partial_*a=\gamma\bmod2$. These are GHHR Theorem 7.1 and Section 7.1, with the peripheral-row property in the paragraph before Remark 7.5. This theorem is a substantive formal dependency.

For an integral normal class define

$$\mathcal I(X)=(-q^{1/2})^{-\chi_{\rm top}(X)} \prod_jJ_\Delta(X_{j1},X_{j2},X_{j3}), \tag{17}$$

which is unchanged by tetrahedron translations. Define the total boundary fiber

$$\mathcal Z(\gamma)=\sum_a I_{\mathcal T}^a(\gamma) =\sum_{[X]\in Q(\mathcal T;\mathbb Z)/\mathbb T\ :\ \partial X=\gamma} \mathcal I(X). \tag{18}$$

An empty boundary fiber has value zero. This is the total of the already-defined GHHR extended indices, not a new index. Only boundary vectors supported on the chosen cusp are used below. Complete strict angles imply spun 1-efficiency and the relevant convergence; the stronger all-input convergence of (12) will justify the weighted outer sums directly.

4. An exact bijection, without saturating the edge lattice

Fix integers $r,n$. Let

$$\Lambda_{r,n}=\{\lambda\in\Lambda:n(\lambda)=n, \quad\lambda+rb\in2L\}. \tag{19}$$

For $\lambda=P(R)$ with the integer representation in Section 2, put

$$U=(R+rR_l)/2,\quad t_j=\max(U_{j1},U_{j2},U_{j3}), \quad X_\lambda^r=\sum_jt_j\Delta_j-U. \tag{20}$$

Theorem 4.1 — full integral boundary fiber

Equation (20) is a bijection from $\Lambda_{r,n}$ onto the minimum-zero representatives of

$$\{[X]\in Q(\mathcal T;\mathbb Z)/\mathbb T: \partial X=-rl-ns\}. \tag{21}$$

It has

$$P(X_\lambda^r)=-(\lambda+rb)/2, \qquad \chi_{\rm top}(X_\lambda^r)=E-T, \quad T=\sum_jt_j. \tag{22}$$

Proof. The parity condition in (19) makes the coordinate differences of $U$ integral. Subtracting $U$ from its block maxima therefore gives a nonnegative integral vector with block minimum zero, even when $U$ itself is half-integral. The actual matching equations, (15), and (20) give (21)–(22). A change of row representation adds only a blockwise constant to $U$, leaving $X$ unchanged. Equation (22) proves injectivity.

For surjectivity, take a minimum-zero integral class $X$ with the boundary in (21). The integral normal class

$$2X+nR_s+rR_l$$

has zero boundary and zero mod-2 homology. By (16), there are integers $k_e,d_j$ with

$$2X+nR_s+rR_l=\sum_ek_eE_e+\sum_jd_j\Delta_j. \tag{23}$$

Set $R=-\sum_ek_eE_e+nR_s$ and $\lambda=P(R)$. Then

$$\lambda=-2P(X)-rb\in\Lambda, \qquad \lambda+rb\in2L,$$

and pairing with $b$ shows that its meridian coefficient is exactly $n$. Furthermore $U=-X+\sum_j(d_j/2)\Delta_j$. Since $X$ has minimum zero, the block maximum of $U$ is $d_j/2$, so (20) returns $X$. Every coefficient used to produce $\lambda$ is integral. No element of the saturation of $\Lambda$ was added. ∎

This strengthens the earlier injective normal-class translation: it proves surjectivity onto the total homology fiber, not onto one selected mod-2 class. Suppressing the sum over $a$ in (18) is generally incorrect.

An immediate exact support test is

$$\Lambda_{r,n}\ne\varnothing \quad\Longleftrightarrow\quad \iota_*(rl+ns)=0\text{ in }H_1(M;\mathbb Z/2). \tag{24}$$

The reverse implication uses the image assertion in GHHR Theorem 7.1 and the long exact sequence of the pair. This is a homological criterion, not a search through surfaces. For the balanced vacuum both root terms use the same parity coset $\lambda\equiv b\pmod{2L}$. If it is empty, they vanish separately. For example, for one torus boundary, if $s$ is zero in $H_1(M;\mathbb Z/2)$, an independent longitude cannot be zero there too; the boundary kernel has dimension one, so that parity coset is empty.

5. Exact scalar and phase translation

Define the normalized complete-character peripheral value

$$H_r=F_0(l^r)=c_l^rF_0(W(rb)),\qquad r\in\mathbb Z. \tag{25}$$

Theorem 5.1 — the complete normal-index bridge

With every phase and native scalar retained,

$$\boxed{H_r=\sum_{n\in\mathbb Z}(-1)^nA^{rn} \mathcal Z(-rl-ns).} \tag{26}$$

The outer sum is unordered-summable. The exact degree of the individual normal-class term is

$$D_r(X)=2\bigl(\delta(X)-\chi_{\rm top}(X)\bigr)+rn, \tag{27}$$

and its leading coefficient is $(-1)^{n-\chi_{\rm top}(X)}$.

Proof. Take $\lambda\in\Lambda_{r,n}$, with $E,S_R,T$ as above. The seed triple in each tetrahedron is

$$(-P(U)_z,0,-P(U)_w)=X+(U_{j2}-t_j)\Delta_j.$$

Its integer minimum has total

$$M=(S_R+r\rho)/2-T\in\mathbb Z. \tag{28}$$

Since $-q^{1/2}=B^2$, the exact tetrahedron translation identity gives

$$\ell_0(W(\lambda+rb))=B^{-2M}\prod_jJ_\Delta(X_j).$$

Thus the complete $\lambda$-summand in $H_r$ is

$$\begin{aligned} B^{r\rho}\,(-1)^nB^{-2E+S_R}\,A^{rn}\,B^{-2M} \prod_jJ_\Delta(X_j) &=(-1)^nA^{rn}B^{2T-2E}\prod_jJ_\Delta(X_j)\\ &=(-1)^nA^{rn}\mathcal I(X). \end{aligned} \tag{29}$$

The exponent $rn$ comes from the exact Weyl factor $\Omega(\lambda,rb)/2$. The sign $(-1)^n$ comes from (11); replacing $B$ by $A$ would lose it. Theorem 4.1 now identifies the full fixed-$n$ sum with (18), proving (26). The sourced tetrahedron leading term and (17) give (27).

Every term in (29) is exactly a term of the all-input-summable family (12), up to the fixed scalar $c_l^r$. Grouping by $n$ is therefore legitimate: below any fixed degree only finitely many individual terms occur, hence only finitely many nonzero grouped terms can contribute there. No unproved interchange of conditionally convergent sums is used. ∎

The positive complete angle vector does not make the term $rn$ positive. Convergence comes from the all-input theorem, not from discarding that term or asserting coefficient positivity.

6. The balanced vacuum vanishes

GHHR published Remark 8.3, printed page 312, states

$$I_{\mathcal T}^a(-\gamma)=I_{\mathcal T}^a(\gamma). \tag{30}$$

The statement was freshly checked in the published author-hosted PDF, and also occurs in arXiv:1604.02688v1, Remark 8.3. We use precisely this established boundary-index equality. We do not assume a symmetry of our root seed, of individual summands, or of an unspecified surgery index.

Theorem 6.1 — balanced non-detection

Under the hypotheses above, $H_{-r}=H_r$ for every integer $r$, and equation (1) follows.

Proof. By (26), the legitimate reindexing $n\mapsto-n$, and (30),

$$\begin{aligned} H_{-r} &=\sum_n(-1)^nA^{-rn}\mathcal Z(rl-ns)\\ &=\sum_n(-1)^nA^{rn}\mathcal Z(rl+ns)\\ &=\sum_n(-1)^nA^{rn}\mathcal Z(-rl-ns)=H_r. \end{aligned}$$

Finally, (13) and $\Omega(-b,b)=0$ give

$$\mathcal G_{-b}(1-l^2) =F_0(W(-b))-c_l^2F_0(W(b)) =c_l(H_{-1}-H_1)=0.$$

All equalities concern convergent functions, not products of quotient classes. ∎

The naive termwise proof is false. Already the local seed values $I_\Delta(0,-1)$ and $I_\Delta(0,1)$ have respective leading terms $-A^4$ and $1$. Therefore root inversion does not preserve the seed. The proof uses the full normal boundary-fiber sum and the separately credited identity (30).

7. No monomial rescue at the same complete character

Let

$$\chi_v(\lambda)=\chi(\lambda)A^{\Omega(\lambda,v)},\qquad \mathcal G_v(a)=F_{\chi_v}(W(v)a). \tag{31}$$

Every $v$ whose shifted character is certified by the same complete angle vector satisfies

$$\Omega(\lambda,v+b)=0\quad(\lambda\in\Lambda). \tag{32}$$

Theorem 7.1 — exhaustion for one cusp

Suppose $\partial M$ is a single torus. Every monomial translate (31) satisfying (32) kills $1-l^2$. More precisely, write $w=v+b$. Then:

• if $w\in\Lambda$, $\mathcal G_v$ is an explicit nonzero scalar multiple of $\mathcal G_{-b}$;
• if $w\notin\Lambda$, both vacuum terms have empty parity support.

The same dichotomy holds with several cusps whenever additionally $w\in\Lambda\otimes\mathbb R$. It does not exhaust translates carrying other-cusp boundary data.

Proof. For one cusp, the standard actual rank statement gives $\dim\operatorname{span}_{\mathbb R}\{h_e\}=N-1$, and adjoining $h_s$ gives an $N$-dimensional isotropic subspace of the symplectic $2N$-space. It is Lagrangian. Thus (32) gives $w\in\Lambda\otimes\mathbb R$.

If $w\in\Lambda$, reindexing the original-character average gives, with its exact Weyl scalar,

$$\boxed{\mathcal G_{-b+w} =\chi(w)A^{-\Omega(-b,w)/2}\mathcal G_{-b}.} \tag{33}$$

Now suppose $w\in(\Lambda\otimes\mathbb R)\cap L$ and there is a contributing $\lambda$ for one vacuum term. Since their exponents differ by $2b$, they have the same parity-support condition,

$$w-b+\lambda\in2L. \tag{34}$$

Choose any integral lift $V$ of $w$ under $P$. It is an integral normal class because $w$ is in the real span of the actual edge and meridian rows, and the kernel of $P$ consists of tetrahedron vectors. Its boundary has the form $\partial V=\kappa s$, with $\kappa\in\mathbb Z$.

Write $\lambda=P(R)$, with meridian coefficient $n$. By (34), the minimum-zero reduction of $-(V-R_l+R)/2$ is an integral normal class $X$. Its boundary is

$$\partial X=l-(n+\kappa/2)s. \tag{35}$$

Since $s,l$ are an integral basis of this torus and $\partial X$ is integral, $\kappa=2t$ with $t\in\mathbb Z$. Moreover $2X+V-R_l+R$ is an integral tetrahedron combination. Taking mod-2 homology and using (15) gives $[V]_2=0$. Therefore $V-tR_s$ has zero boundary and zero mod-2 homology; (16) puts it in $\mathbb E+\mathbb T$. Applying $P$ proves $w-th_s\in\sum_e\mathbb Zh_e$, hence $w\in\Lambda$.

Contrapositively, if $w\notin\Lambda$, (34) is impossible and both vacuum terms vanish separately. The other case follows from (33) and Theorem 6.1. ∎

This is a restriction on a particular family of linear detectors, not a restriction on all linear functionals on $C_s$. In particular, the proof never saturates the moment lattice to manufacture extra functionals on the vacuum component.

7.2. An exact remaining test in a second cusp

There is a concrete expression for the other-cusp direction not exhausted above. Let $t$ be a fixed native peripheral curve on another cusp, with raw row $R_t$, root row $w=P(R_t)$, and native all-plus scalar $c_t=B^{p(R_t)}$. Then $w$ is orthogonal to $\Lambda$ and to $b$. Hence $v=w-b$ still has $\chi_v=\chi_0$, so its functional is convergent on all algebraic inputs and descends to the original target.

The exact same proof as Theorems 4.1 and 5.1, now with $U=(R+R_t+rR_l)/2$, gives

$$c_t c_l^rF_0(W(w+rb)) =\sum_n(-1)^nA^{rn}\mathcal Z(-t-rl-ns). \tag{36}$$

Here the doubled surjectivity witness is $2X+R_t+nR_s+rR_l\in\mathbb E+\mathbb T$; the peripheral row $R_t$ again has zero mod-2 homology and zero formal Euler characteristic. No new lattice assumption is introduced.

Thus, using the published simultaneous boundary inversion only,

$$\boxed{\mathcal G_{w-b}(1-l^2) =\frac{c_l}{c_t}\sum_n(-1)^nA^n \bigl[\mathcal Z(t-l-ns)-\mathcal Z(-t-l-ns)\bigr].} \tag{37}$$

This is an exact candidate test, not a nonvanishing theorem. It reduces the new question to the weighted part of the total normal index that changes under reversing the other-cusp class while the chosen filling-cusp class is held fixed. The simultaneous inversion identity (30) alone does not answer that question. Neither separate-cusp inversion symmetry nor its failure is assumed here.

8. Scope, preserved failures, and the next decisive question

What is established here:

• an exact unsaturated-lattice bijection with the full integral normal boundary fibers;
• the all-orders native scalar/phase formula (26);
• balanced non-detection in the all-plus sector, using the explicitly identified published inversion dependency;
• exhaustion of same-complete-certificate monomial translates in the one-cusp case;
• the explicit other-cusp vacuum test (37), with convergence and original moment descent already established;
• nonzero meridian cokernel from the nonzero value (14), independently of an original filling certificate.

What is not established:

• $[1-l^2]=0$, or a counterexample to an actual named filled trace;
• an extension of the vanishing theorem to every opposite-root sign sector;
• a proof or disproof of the unrestricted Panitch–Romo conjecture;
• a universal nondegenerate pairing, a closed-manifold index, or a new surgery formula;
• exhaustion of different convergence characters, non-monomial seeds, or other-cusp boundary insertions.

The earlier two-angle and all-sufficiently-large-slope vacuum nonvanishing theorems remain intact. Their detector differs from the balanced one; the zero in (1) cannot contradict a nonzero value of another functional.

The next ATTACK should seek a correct-sided pairing outside the family exhausted by Theorem 7.1. For multiple cusps, the most explicit remaining gate is the exact value or a structural vanishing mechanism of (37); its convergence and original moment descent are already established. For one cusp, the gate is whether a nonperipheral left insertion with a different certifiable moment-character valuation, or a rigorously constructed second seed solution, detects $1-l^2$. It must establish all-input convergence and the original moment eigencharacter before evaluating the vacuum. Merely translating within the one-cusp same-complete-character fiber, selecting another element of its saturation, or assuming that a descendant breaks inversion cannot succeed as a proof.

A derivative of the moment-character weight is also not automatically a detector: differentiating an eigencharacter equation produces an inhomogeneous term, so original moment descent must be repaired explicitly. No such repair is claimed here.

References and dependencies

1. 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; final form received 17 December 2016. Published author-hosted PDF; arXiv:1604.02688v1. Formal dependencies: Theorem 7.1, Section 7.1, Definition 8.1, Remark 8.3, normal Euler and tetrahedral-index identities. The boundary-inversion statement is theirs; the present work proves the precise bridge that makes it applicable.
2. S. Garoufalidis, The 3D index of an ideal triangulation and angle structures, arXiv:1208.1663v2. Retained tetrahedron recurrence, degree, and formal-solution theory. The seed and its recurrence are prior mathematics.
3. S. Garoufalidis and T. Yu, The 3d-index of the 3d-skein module via the quantum trace map, author revision dated 6 August 2025. Prior mixed-sided index descent and quadratic/skew convergence ideas, and retained native-comparison dependencies. Its even/collapsed target is not substituted for the original six-root target.
4. Z. Duan, S. Garoufalidis, J. Gu, The descendants of the 3d-index, arXiv:2301.00098v1, Section 3.2. The factorization route to rotated-index symmetries was inspected but is not a proof dependency: its general factorization is presented as a conjecture. Its discussion of descendants informs possible future approaches only.
5. D. Celoria, C. Hodgson, J. H. Rubinstein, The 3D index and Dehn filling, arXiv:2509.09886v2. Scope comparator for Dehn-filling index theory. The present functional is not identified with its surgery invariant.
6. S. Panitch and M. Romo, On the length conjecture for twist knots, arXiv:2609.02706v1, 2 September 2026. Conjecture 7.1 is the broader external question, not a formal dependency in the proof above.
7. FPRD Lab, Actual incidence, normal Euler degree, and a two-angle vacuum theorem, The longitude difference, and A global tetrahedron-index detector, 8 September 2026. Retained actual incidence, native scalars, correct-sided translate construction, and all-input convergence. The present work was directly motivated by their unresolved balanced-pairing gate.