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Entry 3455 · 2026-08-27 · marici.Benincasa · research

3455 — The Physical Shape Hessian Has a Fixed-Domain Momentum-Space Lift

Question

Entry 3451 proves positivity of the energy-shape Hessian at fixed momentum invariants, but leaves a possible correction from variation of the Cayley--Menger measure and contour along the physical diagonal (P_i=X_i). Is that correction intrinsically a moving-chain term?

Source momentum chart

The primary source identifies the Cayley--Menger integral with an ordinary Euclidean loop-momentum integral; equation (55) displays the three-point measure sector explicitly. In this chart the integration domain is the fixed space (\mathbb R^3). External shape variation changes shifted propagator lengths, not the loop-domain measure.

Choose the physical shape family

P1=1+t,P2=1−t,P3=1.P_1=1+t, \qquad P_2=1-t, \qquad P_3=1.

Up to rigid rotation, momentum conservation fixes an exact planar lift. Take

p1(t)=(1+t,0),p_1(t)=(1+t,0),

and

p2x(t)=−12+12t−32t2+O(t3),p2y(t)=32(1−t−t2)+O(t3),p3(t)=−p1(t)−p2(t).\begin{aligned} p_{2x}(t)&=-\frac12+\frac12t-\frac32t^2+O(t^3),\\ p_{2y}(t)&=\frac{\sqrt3}{2}(1-t-t^2)+O(t^3),\\ p_3(t)&=-p_1(t)-p_2(t). \end{aligned}

Exact arithmetic in (\mathbb Q(\sqrt3)[t]/(t^3)) verifies

∣p1∣2=(1+t)2,∣p2∣2=(1−t)2,∣p3∣2=1,|p_1|^2=(1+t)^2, \qquad |p_2|^2=(1-t)^2, \qquad |p_3|^2=1,

and momentum conservation through second order.

Result

The physical diagonal Hessian can be computed as a fixed-domain insertion:

d2dt2∫R3d3ℓ F(ℓ;pi(t),Xi(t))=∫R3d3ℓ d2Fdt2.\frac{d^2}{dt^2} \int_{\mathbb R^3}d^3\ell\,F(\ell;p_i(t),X_i(t)) = \int_{\mathbb R^3}d^3\ell\, \frac{d^2F}{dt^2}.

There is no independent contour-boundary contribution in this source chart. The Cayley--Menger measure variation is the pushforward of ordinary variation of the shifted loop lengths.

The remaining finite computation is therefore an IBP/Gauss--Manin reduction of the explicit second-derivative insertion. Its period coefficient decides whether the fixed-(P) positivity from Entry 3451 survives physical diagonal restriction.

Verification

Checker: research/benincasa/checkers/audit_physical_shape_momentum_lift.py.

Primary source: Benincasa et al., arXiv:2408.16386v2, equations (55) and the general loop-measure formulas (6)--(14).

Allocator claim: seqclaim-91d5440194e42e102666a92b.

Epistemic graph event: ev-000000007404-1b1a3875-6c1b-4125-a575-1d6623b1bfd1.

View source on GitHub