JT gravity: the fixed-fan obstruction to a canonical BF–polytope bridge

6 September 2026 · ATTACK · Working research note, not a Publication

Outcome

The proposed rescue of the JT presentation bridge by permutohedral toric geometry fails. The failure is structural and already occurs at two defects: the separated BF reduction, the Losev–Manin compactification, its braid fan, the defect-fusion label, and the string/dilaton soft identities do not select a unique compactified symplectic measure.

More strongly, there is an explicit one-parameter family of generalized permutohedra with exactly the same braid fan. Their volumes obey the same two soft identities at every insertion order, but their two-defect volumes differ. Thus the toric presentation does not recover the missing collision coefficient; it gives a coordinate system for the ambiguity.

This does not prove that no fully specified BF/BFV path integral can derive the weighted Weil–Petersson contact terms. It proves that the data currently available from the local BF reduction cannot do so. One must add the relative cohomology class, equivalently the normal measure/contact prescription, at the collision locus. Wei Gu’s 2026 BF analysis explicitly identifies that global input as missing. The result below makes the obstruction exact in the permutohedral presentation and closes this attack route under the Lab’s translation criterion.

1. Set-up

Work in the same genus-zero chamber as the preceding reconstruction: two labelled geodesic boundaries, $k\geq1$ labelled defects with $q_i>0$, and

$$Q=\sum_iq_i<1.$$

Put

$$x=b_1^2/2,\qquad y=b_2^2/2,\qquad c=4\pi^2.$$

For a real parameter $\eta\geq0$, define positive pair weights

$$w_{ij}^{(\eta)} =cq_iq_j+\eta q_i^2q_j^2(1-q_i-q_j)^2.$$

Let $\Delta=\operatorname{conv}(e_1,\ldots,e_k)$, and define

$$\mathcal P_k^{(\eta)} =x\Delta+y(-\Delta) +\sum_{i<j}w_{ij}^{(\eta)}[e_i,e_j]$$

in an affine translate of the type-$A$ root space. Volumes are normalized by the lattice

$$A_{k-1}=\{z\in\mathbb Z^k:\sum_i z_i=0\}.$$

At $\eta=0$, this is the polytope whose scalar volume was identified with the known weighted Weil–Petersson volume in the preceding note. Here the question is whether its toric geometry makes that choice canonical.

2. The normal fan does not see the deformation

Theorem 1 (fixed braid fan). For every interior charge vector and every $\eta\geq0$, the normal fan of $\mathcal P_k^{(\eta)}$ is exactly the braid fan. Consequently all these polytopes define toric symplectic forms on the same smooth permutohedral toric variety, the Losev–Manin space.

Proof. Choose a covector $u$ with distinct coordinates and write its decreasing order as

$$u_{\pi_1}>u_{\pi_2}>\cdots>u_{\pi_k}.$$

The exposed vertex is

$$v_\pi =xe_{\pi_1}-ye_{\pi_k} +\sum_{a<b}w_{\pi_a\pi_b}^{(\eta)}e_{\pi_a}.$$

Indeed, the positive simplex chooses the largest coordinate, the negative simplex chooses the smallest, and every segment chooses its endpoint with larger $u$-coordinate.

Now swap the adjacent labels $a=\pi_r$ and $b=\pi_{r+1}$. Only their mutual segment and, at the two ends, a simplex vertex can change. Hence

$$v_\pi-v_{\pi\circ(r\ r+1)} =\left(w_{ab}^{(\eta)} +x\mathbf1_{r=1}+y\mathbf1_{r=k-1}\right)(e_a-e_b).$$

The coefficient is strictly positive. No adjacent wall of a braid chamber is therefore collapsed. All $k!$ chambers expose distinct vertices, so the normal fan is the full braid fan. That fan is smooth in the root lattice and is the fan of the permutohedral/Losev–Manin toric variety. The Delzant correspondence gives the asserted family of toric symplectic forms. ∎

The underlying point is classical: the complete graphical zonotope is a permutohedron, and positive changes of its segment lengths do not change its normal fan. Postnikov’s work on generalized permutohedra is the relevant polyhedral antecedent. The theorem records the consequence for this specific JT candidate rather than claiming the toric fact as new.

3. The same soft identities survive at every order

Let

$$W_k^{(\eta)} =\operatorname{Vol}_{A_{k-1}}(\mathcal P_k^{(\eta)}), \qquad P_m(x,y)=S^m1,$$

where

$$Sf=\int_0^x f(t,y)\,dt+\int_0^y f(x,t)\,dt.$$

The same root-lattice mixed-volume argument as in the first packet, now with arbitrary positive edge weights, gives

$$W_k^{(\eta)} =\sum_{F\in\mathcal F_k} P_{k-1-|E(F)|}(x,y) \prod_{ij\in E(F)}w_{ij}^{(\eta)}, \tag{1}$$

where $\mathcal F_k$ is the set of labelled forests on $[k]$.

Theorem 2 (soft-indistinguishable toric family). Polynomial continuation to a soft charge satisfies, for every $k$,

$$W_k^{(\eta)}(q_1,\ldots,q_{k-1},0)=S W_{k-1}^{(\eta)}, \tag{2}$$

and

$$\left.\partial_{q_k}W_k^{(\eta)}\right|_{q_k=0} =c\left(\sum_{i<k}q_i\right)W_{k-1}^{(\eta)}. \tag{3}$$

Proof. When $q_k=0$, every weight incident with $k$ is zero. Only forests in which $k$ is isolated survive in (1). Deleting that isolated vertex and using $SP_m=P_{m+1}$ proves (2).

At first order in $q_k$, only forests in which $k$ is a leaf survive. For each possible neighbor $i$,

$$\left.\partial_{q_k}w_{ik}^{(\eta)}\right|_{q_k=0}=cq_i;$$

the deformation has a double soft zero. Deleting the leaf recovers an arbitrary forest on the old labels. Summing over its possible neighbor gives the factor $c\sum_{i<k}q_i$, proving (3). ∎

The family is not constant. At two defects,

$$W_2^{(\eta)} =x+y+cqp+\eta q^2p^2(1-q-p)^2. \tag{4}$$

The last term is nonzero in the chamber. Yet it and its first normal derivative vanish both at either soft face and, in this two-defect benchmark, at the collision wall $q+p=1$. The deformation violates the geometric deficit-degree bound; the point is precisely that neither the braid fan nor the BF soft data imply that bound.

Equations (1)–(4) prove a stronger obstruction than matching one scalar volume: even the same toric manifold, the same complete normal fan, and the full all-orders soft recursion leave an infinite measure ambiguity.

For $k>2$, this statement concerns the toric/fan and soft-recursion data: it does not assert that the deformed toric form agrees pointwise with the Goldman form everywhere on the separated locus. The combined obstruction including the separated BF class is already complete at $k=2$, as the next section shows. No Hamiltonian-reduction origin is being assumed for the deformed collision extension; deriving such an origin is exactly the missing input.

4. Where the missing datum lives

The obstruction has a coordinate-free formulation. Let $X$ be the weighted compactification and $U\subset X$ the separated-defect locus on which the local BF reduction produces the Goldman/Weil–Petersson form. If $D\subset X$ is a collision divisor, the canonical section of $\mathcal O_X(D)$ is nowhere zero on $U$. Therefore

$$i^*[D] =i^*c_1(\mathcal O_X(D))=0 \qquad(i:U\hookrightarrow X). \tag{5}$$

Thus collision-divisor classes lie in the kernel of

$$H^2(X;\mathbb R)\longrightarrow H^2(U;\mathbb R).$$

Because the Kähler cone is open, sufficiently small changes of a compactified Kähler class in one of these directions remain Kähler. A Thom representative can be supported in an arbitrarily small collision neighborhood, so the forms can be chosen to agree away from that neighborhood. Their compactified volumes nevertheless differ.

For two defects this is completely visible: $X\cong\mathbb P^1$, the separated open curve has $H^2(U)=0$, and the collision divisor is a point. The shift in (4) is the class

$$\eta q^2p^2(1-q-p)^2[D].$$

It changes the interval length and total area while leaving the restricted cohomology class, fusion label, fan, and both soft jets unchanged. At the level of symplectic representatives, a small Thom form places the difference in an arbitrarily small collision neighborhood. Thus the combined data fail to select the compactified class already in the first stable example.

The missing datum is therefore not another choice of gauge representative. It is the relative extension class across the collision locus—equivalently, the normal measure or contact coefficient. A canonical BF derivation would have to calculate this class. Naming the Losev–Manin space or its moment fan does not calculate it.

5. Comparison with the primary sources

Eberhardt and Turiaci identify the smooth Weil–Petersson form with the gauge theory form

$$\omega_{\mathrm{WP}}(\delta A,\delta A') =\frac1{4\pi}\int_\Sigma\operatorname{tr}(\delta A\wedge\delta A')$$

and determine the compactified conical class by geometric consistency and weighted reduction. Their paper explicitly asks for a first-principles derivation of that measure.

Gu derives the reduced Atiyah–Bott–Goldman measure on the smooth separated locus from BF gauge fixing, and fixes the one-defect normalization through the disk and sewing convention. He also states that extending the form to the weighted compactification—including collision coefficients—requires the global Hassett and conical Weil–Petersson geometry; local sourced flatness does not determine collision codimension, incidence, multiplicity, or normal geometry. The relative-class calculation above is an exact mathematical reason for that separation.

Blommaert, Mertens, and Verschelde already emphasize that local BF equivalence does not choose the global gravitational integration space; Teichmüller space and its mapping-class quotient yield different quantum integrals. The present obstruction is the collision analogue inside the selected genus-zero chamber.

6. What this does and does not establish

Established here:

1. The preceding root-segment polytopes are genuine moment polytopes for the same Losev–Manin toric variety, not merely unrelated convex bodies.
2. Their moment-polytope presentation is radically nonunique: an explicit one-parameter fixed-fan deformation survives both soft identities at every order.
3. The missing collision coefficient is relative compactification data that the separated reduced form cannot determine.
4. Therefore the proposed route from local BF reduction plus toric geometry to a canonical weighted measure is closed.

Not established:

1. No inconsistency in JT gravity or in the cited papers is claimed.
2. No new propagating mode, black-hole observable, or quantum-gravity dynamics has been found.
3. No theorem says a fully specified global BF/BFV path integral, with an independently justified contact prescription, cannot reproduce the known class.
4. The application-specific formulation has not been externally reviewed; priority is unassessed. The core toric and relative-cohomology mechanisms are classical.

This is consequently a useful negative research result, not a Publication. It answers the attack gate: the alternate presentation does not compress the missing physics into a canonical map. It preserves a family of measures and therefore flattens rather than reveals the decisive collision datum.

7. Verification

jt-fixed-fan-ambiguity-check.py uses exact rational arithmetic. For $2\leq k\leq6$, it constructs every chamber vertex, verifies every oriented adjacent-swap edge formula, checks all $k!$ vertices are distinct, and checks the string and dilaton identities from the forest expansion. It also verifies the nonzero two-defect shift and its zero value/first derivative at the collision wall. These are finite controls; the theorems above are analytic and do not rely on the tested cutoff.

8. Dependency and influence ledger

• Wei Gu, Conical Defects in JT Gravity from BF Theory: Quantization, Fusion, and Weighted Moduli Spaces, arXiv:2609.03347v1: selected BF target, local reduction, normalization, and the explicit compactification boundary.
• Lorenz Eberhardt and Gustavo J. Turiaci, 2D dilaton gravity and the Weil–Petersson volumes with conical defects, arXiv:2304.14948v2: established conical Weil–Petersson class and contact coefficients.
• Alexander Postnikov, Permutohedra, associahedra, and beyond, arXiv:math/0507163: generalized-permutohedron and graphical-zonotope machinery; direct intellectual influence on the fixed-fan test.
• Valery Alexeev, Weighted Grassmannians and stable hyperplane arrangements, arXiv:math/0607683: the Losev–Manin space as the toric variety of the permutohedron.
• Andreas Blommaert, Thomas G. Mertens, and Henri Verschelde, Fine Structure of Jackiw–Teitelboim Quantum Gravity, arXiv:1812.00918v3: established distinction between local BF data and the global gravitational integration space.
• Delzant correspondence, divisor-line-bundle restriction, Thom classes, and openness of the Kähler cone: classical mathematical ingredients.

9. Stopping decision

Recommended next mode: HUNT. Do not continue the JT program by selecting a contact convention and then reconstructing more amplitudes. Reopen it only if a genuinely BF-native calculation produces the relative collision class or a scheme-independent observable sensitive to it. The present alternative presentation has now supplied a decisive obstruction rather than a translation theorem.