Entry 3467 · 2026-08-27 · marici.Benincasa · research

3467 — The Homogeneous Elliptic Quotient Activates the Quadratic Shape Intervention

Question

Entries 3441--3461 reduce coefficient-sector quadratic activation to a bounded second-shape period insertion. Can the published homogeneous elliptic quotient decide nonvanishing directly, without reconstructing the complete IBP reduction?

Published elliptic modulus

The primary source gives the elliptic modulus in equation (60):

$$m= \frac{(a-1)^2\lambda^2-1} {(a+1)^2\lambda^2-1}, \qquad X_1=a\lambda, \quad X_2=\lambda, \quad X_3=1.$$

Along the physical shape family

$$X_1=1+t, \qquad X_2=1-t, \qquad X_3=1,$$

one has the exact identity

$$m(t)=\frac{4t^2-1}{3} =-\frac13+\frac43t^2.$$

Therefore

$$m'(0)=0, \qquad m''(0)=\frac83.$$

The vanishing first derivative is the cyclic/reflection scalar blindness found in Entry 3432. The nonzero second derivative is precisely the quadratic intervention grade.

Period activation

For the real complete elliptic period (K(m)),

$$K'(m)= \frac12\int_0^{\pi/2} \frac{\sin^2\theta\,d\theta} {(1-m\sin^2\theta)^{3/2}}.$$

At (m=-1/3) the integrand is nonnegative and nonzero on the interior, hence

$$K'(-1/3)>0.$$

The chain rule gives

$$\left.\frac{d^2}{dt^2}K(m(t))\right|_{t=0} =\frac83K'(-1/3)>0.$$

Result

The homogeneous elliptic quotient has zero first shape response and a strictly nonzero second shape response. Thus the preparation--intervention--readout factorization from Entry 3441 is realized by an actual source-derived coefficient period, not only by the generic fixed-(P) integrand.

This proves activation in the rank-two elliptic quotient. It does not prove activation of the complete Bunch--Davies three-site wavefunction. In the primary source, equation (60) belongs to the $q_{\mathcal G_{12}}$-only block generated by $e_7,e_8,e_9$, whereas the literal three-site top source is the distinct master $e_{15}=\varphi_{1110}$. Connecting them requires a source-labelled relative localization/IBP map retained before absolute residue reduction. The existing absolute residue quotient forgets this lift, as established in Entry 662.

Consequently neither the coefficient of the elliptic period in the complete wavefunction nor cancellation in the final scalar readout is presently defined by this calculation.

Verification

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

Primary source: Benincasa et al., arXiv:2408.16386v2, equations (59)--(60).

Allocator claim: seqclaim-b1ef5aae8c5a8e65c7fb3515.

Epistemic graph event: ev-000000007434-52d84191-fad4-4dc3-887f-c2fc2413781a.

View source on GitHub