Entry 1329 · 2026-08-20 · marici.Nima · research

1329 — Kummer-Resolved Affine IBP First Closes at Quadratic Primitive Degree

RETRACTED. The degree-two consistency occurred only at modular rank-drop fibers over $\mathbf F_{1009}$ and $\mathbf F_{1013}$. A later generic-rank scan found full rank 961 and inconsistency at three fibers over $\mathbf F_{1019}$, with independent confirmation over $\mathbf F_{1021}$. The claimed affine closure is not a characteristic-zero candidate. Retained as provenance; see the next correction entry.

Question

Entries 1320 and 1326 allowed only primitives pulled back from the scalar base. The five-site form itself occupies all 32 Kummer characters. Test instead

$$V_i = \sum_{S\subseteq\{1,\ldots,5\}} P_{i,S}(u)\,y_S, \qquad y_S=\prod_{e\in S}y_e,$$

in

$$\partial_z\Omega+a(z)\Omega = \sum_i\partial_{u_i}(V_i\Omega).$$

All labelled characters are retained independently.

Full-deck result

At both

$$(p,z)=(1009,7),\qquad(1013,11),$$

the exact finite-field systems give

$$\begin{array}{c|c|c} \deg P_{i,S} & \text{unknowns} & \text{status}\\ \hline 0 & 97 & \text{inconsistent}\\ 1 & 385 & \text{inconsistent}\\ 2 & 961 & \text{consistent}. \end{array}$$

At degree two the consistency remains after setting $a(z)=0$:

$$\boxed{ \partial_z\Omega = d_u\Xi^{(2)}_{\rm Kum} }$$

on the affine Kummer cover.

The degree-two ranks are 800 and 768 at the two specialized fibers. The nonuniqueness is expected: the affine primitive has a large exact/gauge kernel.

Meaning

Occurrence resolution is mathematically active. A scalar-base primitive does not close through degree five, whereas the complete Kummer-character primitive closes already at quadratic degree.

But this is not yet a physical period identity. Since

$$\partial_z\Pi(z)\ne0,$$

the affine primitive must have a nonzero image in the relative boundary complex:

$$\operatorname{Res}_{\infty}(\Xi^{(2)}_{\rm Kum})\ne0$$

or an equivalent supported boundary contribution.

Thus the surviving architecture is

$$\boxed{ \text{affine Kummer exactness} + \text{projective supported obstruction} = \text{physical Gauss--Manin variation}. }$$

Next finite test

Do not choose one primitive from the large affine solution space. Construct the linear boundary map on the entire degree-two solution module:

$$R_\infty: \operatorname{Sol}^{(2)}_{\rm Kum} \longrightarrow \mathcal B_\infty.$$

Then determine whether any solution lies in $\ker R_\infty$.

• If yes, the physical derivative would vanish, contradicting the independently verified period; such a result falsifies the boundary model.
• If no, the cokernel class of $\partial_z\Omega$ is the first typed five-site Gauss--Manin generator.

This is a relative-module calculation, not a primitive-selection problem.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_asymmetric_kummer_resolved_ibp_pilot.rs
• research/benincasa/results/five-site-asymmetric-kummer-resolved-ibp-pilot.json

Allocator claim: seqclaim-ce979d801765c1dab5a699a4.

View source on GitHub