Entry 3461 · 2026-08-27 · marici.Benincasa · research

3461 — The Physical Shape Insertion Closes Inside the Frozen Integral Family

Question

Entry 3455 turns the physical shape Hessian into a fixed-domain loop-momentum insertion. Does differentiating the shifted loop lengths require new denominator support or a larger Carrier?

Exact derivative structure

For a shifted loop vector (k(t)) and length (y(t)=\sqrt{k(t)^2}),

$$y'=\frac{k\cdot k'}{y},$$

and

$$y''=\frac{k'^2+k\cdot k''}{y} -\frac{(k\cdot k')^2}{y^3}.$$

Thus second differentiation introduces only the already existing propagator support (k^2=0), with inverse length powers at most (y^{-3}).

For an existing source pole,

$$\frac{d^2}{dt^2}q^{-1} =2(q')^2q^{-3}-q''q^{-2}.$$

Hence every individual source denominator is raised to power at most three. No new polynomial denominator appears.

Integral-family closure

The source three-site family already mixes integer and half-integer momentum propagator powers, as stated after equations (54)--(55). Multiplication by (y^{-1}) or (y^{-3}) preserves the half-integral exponent lattice. The second-shape insertion therefore belongs to the same IBP/Gauss--Manin family.

The finite bound is:

$$\max \tau_q=3, \qquad \max \tau_y=3.$$

Result

Physical quadratic activation requires no new Carrier cell, incidence divisor, or coefficient support. It is an existing-family reduction problem with a bounded exponent packet.

This sharpens the remaining computation: construct the explicit insertion from Entry 3455, reduce it against the source master basis using the already admitted syzygy/IBP calculus, and evaluate its component on the Bunch--Davies cycle. A nonzero period activates the quadratic intervention; zero closes it at second order.

Verification

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

Primary source: Benincasa et al., arXiv:2408.16386v2, equations (54)--(55) and the differential-equation/IBP construction in equations (22)--(26).

Allocator claim: seqclaim-ef8c0e7ce31d88a3ba711b80.

Epistemic graph event: ev-000000007417-eb5ae311-89d6-49f2-aa49-d4e15c8b3351.

View source on GitHub