Positive geometric fillings and the quantum-trace quotient

FPRD Lab research note — 8 September 2026

Result and scope

The regularity condition in the preceding Dehn-filling note is automatic at a strictly positively oriented shape solution. The reason is intrinsic: the logarithmic tangent of the tetrahedron equations is a definite complex Lagrangian, whereas independent edge and filling constraints span a real isotropic space. These spaces are transverse.

For a genuine hyperbolic filling realized by such tetrahedra, the geometric logarithms also provide the required signed square roots. They give the GY edge and filling values −1, not merely their squares. The complementary longitude measures the loxodromic core and has square different from 1. Consequently the identified Garoufalidis–Yu filled quotient and its descended trace are nonzero at generic parameter.

A separate construction shows that the two Panitch–Park scaling branches are conjugate even after opposite roots are identified. It uses an Eulerian orientation of the triangulation’s dual multigraph. This removes the scaling-branch issue as an existence obstruction; it does not identify the GY trace with the named PP trace.

These are supporting research results, not a solution of Panitch–Romo Conjecture 7.1. That conjecture covers arbitrary ideal triangulations and fillings. Strict geometric positivity is a real restriction. The original six-root PP boundary/trace bridge remains unproved here. External novelty is unassessed; no Publication is claimed.

1. Precise target and the previous quantum input

Let a connected oriented three-manifold have an ideal triangulation with $N$ tetrahedra and $r\geq1$ torus ends. Work over $k=\mathbb Q(i,A^{1/2})$ with $A$ indeterminate. We use the identified GY target of Garoufalidis–Yu, arXiv:2406.04918v1, Section 3.3 and Theorem 5.7, with $q_{GY}=A^4$. Opposite bare-edge roots are identified. After the central vertex variables are eliminated, its shape quantum torus $S_A$ has generators $z_j,z_j''$ for $1\leq j\leq N$. The tetrahedron relations are

$$f_j=z_j^{-2}+(z_j'')^2-1.$$

The central vertex value, before elimination, is $[z_jz_j'z_j'']=iA$. Write $e_1,\ldots,e_{N-r}$ for a complete independent edge list and $m$ for the chosen filling-meridian monomial. The right edge ideal generated by this list equals the full edge ideal, as specified by GY after (31). The meridian cokernel has vector-space presentation

$$C_s=S_A\Big/\left(\sum_{j=1}^N S_A f_j+ \sum_{a=1}^{N-r}(e_a+A^2)S_A+(m+A^2)S_A\right). \tag{1.1}$$

The preceding proof note, Theorems 8.1–8.2, proves the following two facts with the actual sidedness:

1. With $ml=A^2lm$, the generic intrinsic filled target is naturally isomorphic to $C_s$. The map from the filled quotient sends $[v]$ to $[(1-l^2)v]$ in $C_s$.
2. If the $2N-r+1$ classical symbols in (1.1), specialized at $A=1$, vanish at a nonzero root point and form a regular sequence there, then $C_s\neq0$. If $l^2\neq1$ at the point, the filled vacuum and descended GY trace are nonzero as well.

For clarity, the second statement is not special-fiber reasoning. Its proof sets $A=e^\hbar$ and uses the mutually commuting operators consisting of right tetrahedron multiplication and left edge/filling multiplication. Regular-sequence syzygies are Koszul; their exact commuting lifts prove $\hbar$-torsion-freeness of a formal local detector. A map from (1.1) to that detector after inverting $\hbar$ detects the original vector. The present note establishes the missing regularity and signed-root hypotheses on a geometric chamber; it does not replace that quantum proof.

Generic torus-category reconstruction and excision in the preceding note use the theorems of Gunningham–Jordan–Vazirani and Gunningham–Jordan–Safronov. The solid-torus relations use Frohman–Gelca. Those results remain dependencies, not new claims of this note.

2. Definite transversality: an elementary lemma

Lemma 2.1

Let $P,Q$ be real $k_0\times N$ matrices such that $(P\mid Q)$ has row rank $k_0$ and

$$PQ^t=QP^t. \tag{2.1}$$

Let $D$ be a complex diagonal matrix whose imaginary diagonal entries are all strictly positive, or all strictly negative. Then

$$\operatorname{rank}_{\mathbb C}(P+QD)=k_0. \tag{2.2}$$

Proof. If a complex row $\alpha$ annihilates $P+QD$, set $a=\alpha P$, $b=\alpha Q$, so $a=-bD$. Reality and (2.1) give $a\overline b^{,t}-b\overline a^{,t}=0$. Substitution yields

$$-2i\sum_j\operatorname{Im}(D_{jj})|b_j|^2=0.$$

Definiteness forces $b=0$, and hence $a=0$. The row rank of $(P\mid Q)$ then gives $\alpha=0$. ∎

The same argument is the promised intrinsic description. On $\mathbb C^N\oplus\mathbb C^N$ use $\Omega((u,v),(u',v'))=u\cdot v'-v\cdot u'$. The graph $L_D=\{(u,Du)\}$ is complex Lagrangian, and

$$i\Omega(\overline{(u,Du)},(u,Du)) =-2\sum_j\operatorname{Im}(D_{jj})|u_j|^2. \tag{2.3}$$

Thus it is definite when $\operatorname{Im}D$ is definite. A complexified real isotropic space has zero Hermitian form and cannot meet such a graph nontrivially. Formula (2.3), rather than an imported positivity theorem, proves the assertion used here.

3. Automatic regularity in the positive-shape chamber

Put $Z_j=z_j^2$ and $Z_j''=(z_j'')^2$ at the commutative $A=1$ fiber. The tetrahedron equation says

$$Z_j''=1-Z_j^{-1}.$$

Assume $\operatorname{Im}Z_j>0$ for every $j$. This positivity refers to ordinary shape squares, not to the individual square-root variables. Use local logarithmic coordinates $u_j=\log z_j$, $v_j=\log z_j''$. The tangent equation $df_j=0$ is

$$dv_j=\frac{1}{Z_j-1}\,du_j.$$

Consequently the tetrahedron tangent is $L_D$, where

$$D_{jj}=\frac1{Z_j-1},\qquad \operatorname{Im}D_{jj} =-\frac{\operatorname{Im}Z_j}{|Z_j-1|^2}<0. \tag{3.1}$$

After eliminating the vertex variables, write the logarithmic exponent rows of the independent edges and the filling slope as $(P\mid Q)$. There are $k_0=N-r+1$ such rows. Their coefficients are real integers; the scalar powers of $i$ introduced by elimination do not change a logarithmic derivative.

Here the topological input is precise. Neumann’s symplectic complex identifies the edge span $E$ as an isotropic space of dimension $N-r$, and $E^\perp/E$ with $H_1(\partial M;\mathbb R)$, with its symplectic pairing up to a nonzero fixed normalization. Thus a primitive slope on a torus has a nonzero class modulo $E$. Adjoining it raises the dimension by one and keeps the space isotropic. See Neumann, Theorem 4.1, and the explicit matrix formulation in Garoufalidis–Zickert, Corollaries 2.6–2.7. Equivalently, pairing the slope row with any complementary peripheral row gives a nonzero value, while every edge row pairs to zero with that row.

Theorem 3.1 — positive-shape regularity

At any root point satisfying the classical tetrahedron, edge and filling equations of (1.1), with all $\operatorname{Im}Z_j>0$, their joint Jacobian has full row rank

$$N+(N-r+1)=2N-r+1. \tag{3.2}$$

In particular, these symbols are a regular sequence in the formal local ring. Their local common zero set is smooth of dimension $r-1$.

Proof. The $N$ tetrahedron equations have independent differentials: each uses its own two variables and has nonzero $v_j$ coefficient. On their common tangent, the remaining differentials, up to nonzero row scalars, form $P+QD$. Neumann’s input gives the hypotheses of Lemma 2.1, and (3.1) gives definiteness. Thus the remaining rank is $N-r+1$. Full cotangent rank permits these functions to be completed to a formal coordinate system, proving regularity. Since $r\geq1$, $N-r+1\leq N$ and (3.2) never exceeds the $2N$ ambient dimensions. ∎

The identical rank conclusion holds if all shape squares are in the lower half-plane. This algebraic observation alone does not identify a lower- half-plane paper’s parameter, cyclic ordering, central root value, or named trace with GY’s. Those conventions must still be transported.

4. Signed-root existence from genuine geometric surgery

Theorem 3.1 assumes an actual root point. Merely solving squared equations does not certify one with the required edge and meridian signs. For a strictly positive geometric surgery solution, however, there is an explicit lift.

Use the standard positively oriented triple

$$Z_j,\quad Z_j'=(1-Z_j)^{-1},\quad Z_j''=1-Z_j^{-1}, \qquad \operatorname{Im}Z_j>0.$$

All three values lie in the upper half-plane. Their principal logarithms $U_j,U_j',U_j''$ satisfy

$$U_j+U_j'+U_j''=i\pi. \tag{4.1}$$

For example, their product is −1, and the sum of arguments is continuously $\pi$ on the connected upper half-plane (evaluate at a regular ideal tetrahedron). Define

$$z_j=e^{U_j/2},\quad z_j'=e^{U_j'/2},\quad z_j''=e^{U_j''/2}. \tag{4.2}$$

Then $z_jz_j'z_j''=i$, exactly the $A=1$ GY central vertex value, and $f_j=0$.

We use honest geometric edge equations, not just exponentiated ones: the sum of the ordinary logarithmic shape parameters around every manifold edge is $2\pi i$. Equation (4.2) then gives

$$e_a=\exp\left(\tfrac12\sum_{\text{around }a}U\right)=-1. \tag{4.3}$$

For the peripheral convention, choose the normal-curve exponent rows $C_\mu,C_\lambda$ used for the geometric logarithmic holonomies and for GY’s Weyl monomials (89). The logarithmic representatives are fixed consistently from the complete positive structure. For $s=p\mu+q\lambda$, use $C_s=pC_\mu+qC_\lambda$. Write

$$H_s=\sum C_s U.$$

The genuine hyperbolic surgery equation is $H_s=2\pi i$; reversing the slope gives $-2\pi i$. Therefore

$$m=\exp(H_s/2)=-1. \tag{4.4}$$

This matches the meridian symbol $m+1$ in (1.1). It is not a free choice between the eigenvalues $+1$ and $-1$. Normalizing an exponent row differently requires carrying its central scalar and logarithmic constant with it. In particular, an arbitrarily altered peripheral row cannot be substituted in (4.4).

Choose a complementary primitive slope $t$, so $s,t$ form a basis of the torus. Its logarithmic holonomy is the complex length of the added core, up to sign and an integral $2\pi i$ period. For a genuine hyperbolic filling the core is loxodromic, of positive real length. Hence

$$l^2=\exp(H_t)\neq1. \tag{4.5}$$

These geometric log and core conventions are the ones described in Neumann, Sections 2–3, especially the surgery equation at the beginning of Section 3 and the complementary-slope formula preceding (3.9). The sign lift (4.2)–(4.4) is an explicit calculation with those conventions. It asserts neither a new holonomy-lifting theorem nor a lift of every exponentiated PSL₂ point.

Corollary 4.1 — nonzero geometric GY filling

Suppose an incomplete hyperbolic structure on the ideally triangulated parent is realized by strictly positively oriented ideal tetrahedra, satisfies the ordinary geometric edge equations, and completes at the chosen end to a hyperbolic Dehn filling with loxodromic core. Use the compatible peripheral logarithmic convention above. Then, over $k$, the intrinsic identified GY filling target is nonzero, its vacuum is nonzero, and its descended quantum trace is nonzero.

Proof. Equations (4.2)–(4.4) give a root point for all the symbols in (1.1). Theorem 3.1 establishes their regularity. Equation (4.5) gives the additional vacuum test. Apply the quantum detector recalled in Section 1. ∎

This class is not merely formal. If a complete finite-volume cusped hyperbolic manifold is supplied with a strictly geometric ideal triangulation, the usual sufficiently long hyperbolic surgeries deform its shapes toward the complete ones. Strict positivity is open and is therefore retained for all sufficiently long slopes on one chosen end, with the others left complete. The existence/convergence here is the classical hyperbolic Dehn surgery input, not a theorem of this note; see Petronio–Porti, Theorem 0.1 and Sections 2–3. We do not assume that every manifold, or every prescribed triangulation, admits such a strictly positive realization. We have not established a numerical slope cutoff or all of the twist-knot range in Panitch–Romo.

A complete unfilled cusp has logarithmic filling holonomy zero, giving $m=+1$ in the normalization (4.2). It is not the point in (4.4). Substituting it for a genuine surgery point would lose exactly the sign which is needed here.

5. Why the positivity qualification cannot be dropped algebraically

Real isotropy and nonzero shapes alone do not force regularity. This can be seen structurally, without a triangulation census.

In a four-variable commutative torus use

$$f_1=x_1^{-2}+w_1^2-1,\qquad f_2=x_2^{-2}+w_2^2-1,$$

and the two monomial constraints

$$x_1x_2=1,\qquad x_1^{-1}w_1^{-1}w_2=i. \tag{5.1}$$

Their logarithmic rows are

$$P=\begin{pmatrix}1&1\\-1&0\end{pmatrix},\qquad Q=\begin{pmatrix}0&0\\-1&1\end{pmatrix}.$$

They have full row rank and satisfy $PQ^t=QP^t$. But (5.1) gives $x_2=x_1^{-1}$ and $w_2=i x_1w_1$, whence

$$f_2=-x_1^2 f_1. \tag{5.2}$$

Thus the four equations have a one-dimensional local solution set, not the expected zero-dimensional one. They are not a regular sequence. One may take any nondegenerate point of this curve with $\operatorname{Im}(x_1^2)>0$; then $x_2^2=x_1^{-2}$ lies in the lower half-plane. Equivalently, $D_1+D_2=-1$ and $\det(P+QD)=0$. The companion script checks the identities and an exact Gaussian-rational nondegenerate point.

This is an algebraic folded-shape example with the tetrahedron equations and an isotropic moment lattice. It is not asserted to arise from a specified manifold triangulation or filling, and it is not a counterexample to Panitch–Romo. It shows precisely why the rank proof cannot be extended to arbitrary mixed-orientation data using only the stated inputs. Nor does singular Jacobian rank alone disprove regularity: singular complete intersections can still be regular sequences. Example (5.2) proves actual excess dimension, which is the stronger failure needed here.

6. PP scaling branches: a global conjugacy

This result concerns the named PP constructions on both scaling branches, not the GY-to-PP comparison. Use PP’s parameters $q=-A^2$, $c_B^2=q c_T^2$. Their face-suspension trace sends triangle generators to $c_T[bc]$, $c_T[ca]$, $c_T[ab]$ on each of the two halves, and a bigon generator to $c_Ba_1a_2$.

PP Theorem 2.20 and Remark 2.21 already state the local branch change: multiply all generators in one half by $i$ and all generators in the other half by $-i$. Triangle images pick up $-1$, while bigon images do not change. Hence this conjugates $(c_T,c_B)$ to $(-c_T,c_B)$. The source notes that the two halves have no canonical ordering.

The remaining issue is whether simultaneous local choices preserve the opposite-root identification. Arbitrary choices need not. The following global choice does.

Proposition 6.1 — branch conjugacy after opposite-root identification

For every ideal triangulation with paired faces, there is a diagonal quantum-torus isomorphism inducing

$$\Phi:\operatorname{SQGM}_{c_T,c_B} \xrightarrow{\sim}\operatorname{SQGM}_{-c_T,c_B}, \qquad \Phi\circ\operatorname{Tr}_{c_T,c_B} =\operatorname{Tr}_{-c_T,c_B}. \tag{6.1}$$

It preserves the opposite-root identification. It is a chosen coordinate conjugacy, not a claimed canonical identification or a GY trace match.

Proof. The dual graph has a vertex for each tetrahedron and an edge for each paired face. It is a 4-regular multigraph; loops are counted twice in the degree. Each component has an Euler circuit, so it admits an orientation with two incoming and two outgoing face occurrences at every tetrahedron. This elementary graph fact is classical: orient an Euler circuit and count arrivals and departures, including loops.

For a face occurrence $f$ at tetrahedron $T$, set $s_{Tf}=i$ if its dual half-edge is outgoing and $s_{Tf}=-i$ if it is incoming. On that face half multiply all three quantum-torus generators by $s_{Tf}$. A paired face has opposite scalars, with product 1. Every face-half scalar has square −1. These diagonal scalar maps preserve all quantum commutators and all Weyl factors. On each face suspension they are exactly the local PP branch conjugacy just described; both traces send the empty skein to 1. The face trace is generated from this vector by the stated triangle and bigon actions, so equality on those actions proves local trace equality. Tensoring and imposing the same relative skein relations proves the global equality before opposite roots are identified.

A bare edge $e$ of $T$ belongs to two distinct face occurrences $f,f'$. Its shape variable is the product of the two corresponding half-face generators and is multiplied by

$$\epsilon_e=s_{Tf}s_{Tf'}\in\{1,-1\}. \tag{6.2}$$

At each tetrahedron $\prod_{f\subset T}s_{Tf}=i^2(-i)^2=1$. Opposite bare edges use complementary pairs of its four faces, so $\epsilon_e\epsilon_{e^{opp}}=1$, hence $\epsilon_e=\epsilon_{e^{opp}}$. Therefore the opposite-root relations are preserved.

For completeness, the three identified target relations can be checked directly. At a bare vertex the product of the three incident edge signs is the product of the squares of the three incident half-face scalars, namely −1. Thus the vertex monomial changes sign, exactly as required by $q^{-1}c_T^{-3}\mapsto q^{-1}(-c_T)^{-3}$. Lagrangian relations use shape squares and are unchanged. In a manifold edge monomial, the half-face occurrences pair across every glued face; each pair contributes $s_{Tf}s_{T'f}=1$. Its gluing value $q c_B^{-d}$ is unchanged. This also proves the induced target isomorphism directly from Definition 2.22. The inverse scalar automorphism gives the inverse map.

All counts use face occurrences. A self-glued face gives a dual loop with one incoming and one outgoing half-edge; its two occurrences remain distinct. Multiple dual edges remain distinct as well. No simplicial or loop-free assumption was used. ∎

Common rescaling is separate and simpler. Changing both constants by $\lambda$ is conjugated at shape level by $z_e\mapsto\lambda z_e$. At face-half level use a square root of $\lambda$, after a harmless coefficient extension if needed; the induced shape map is already defined over the field containing $\lambda$. With $q^{1/2}=iA$, the original PP constants $(A^{-1},-i)$ are taken by common factor $i$ to $(iA^{-1},1)=(-q^{-1/2},1)$; Proposition 6.1 then reaches the default $(q^{-1/2},1)$ branch. This applies to the original trace followed by opposite-root identification; it does not equate the original six-root module with its identified quotient.

Different Eulerian orientations can give different coordinate maps. Each satisfies (6.1), so their relative target automorphism fixes the named trace image. We do not conclude that it fixes the whole module or that the resulting choice is natural under Pachner moves.

7. What has and has not advanced

The former regularity obligation has been discharged for the strictly geometric hyperbolic filling class, including the signed-root and vacuum tests. The PP branch ambiguity is now an explicit conjugacy problem with a solution on every paired-face triangulation. Neither result depends on enumerating manifolds, slopes or collision configurations.

The symplectic topology is Neumann–Zagier/Neumann’s work, the quantum presentation and peripheral action are GY’s, and PP supplied the local scaling observation and trace formulas. The present note proves the displayed definite-transversality calculation, applies it with the signed geometric lift to the prior quantum-flatness detector, and supplies the global opposite-root-compatible scaling construction. These are the specific contributions claimed; novelty and priority beyond those credits have not been assessed.

Next mode: ATTACK. Match GY’s named local trace with either one PP scaling branch, including the global six-root peripheral monomials and their Weyl/central normalizations. Once one branch is matched, Section 6 transports that match to the other; it need not be rederived twice. For arbitrary mixed-shape or nonregular fillings, meridian surjectivity remains open. A new attempt there must supply more than nonzero shapes and isotropic exponent rows. All other paused programs remain paused.