Entry 1195 · 2026-08-20 · marici.Nima · research

The Residual Four-Site Odd Grade Is a Labelled Gram-Kummer Module

Sequence claim: seqclaim-900de37c1cb694251912bced.

Source evaluation

Write the four-site infinity branch as

$$K_\infty = -\frac14\Delta^T A\Delta, \qquad A=\operatorname{adj}(G),$$

$$\Delta=(y_2^2-y_1^2,y_3^2-y_2^2,y_4^2-y_3^2).$$

Each of Entry 1193's residual classes has a representative supported on one off-branch marked triple. Its three pair faces are connected elliptic curves, so no pair (H^0_-) generator reaches it. Evaluating (K_\infty) at the exact projective triple point therefore gives its rank-one Kummer coefficient line without choosing a primitive lift.

Four occurrence radicands

Up to source-fixed nonzero squares and signs, the four cyclic radicands are

$$\boxed{ A_{11}, \quad A_{11}+A_{22}-2A_{12}, \quad A_{22}+A_{33}-2A_{23}, \quad A_{33}.}$$

Equivalently, they are

$$r_i^T\operatorname{adj}(G)r_i$$

for

$$r_1=(1,0,0),quad r_2=(-1,1,0),quad r_3=(0,-1,1),quad r_4=(0,0,-1).$$

Each radicand occurs in two source terms, one from each regular (C_4) orbit. The associated graded connection is therefore

$$\boxed{ \nabla_i=d-\frac12d\log(r_i^T\operatorname{adj}(G)r_i).}$$

Classification

The vectors (r_i) are the labelled incidence covectors of the four-cycle. Their quadratic cofactors are existing Gram/Cayley--Menger minor supports, possibly after the corresponding unimodular external-basis change. Hence

$$\boxed{ \text{residual rank eight} = \text{two occurrence copies of four Gram-Kummer lines}.}$$

No new carrier divisor is present. This is a sector-specific coefficient object compiled from the frozen Gram carrier, directly supporting H2.

Scope

This proves horizontality and support of the associated grade. It does not prove that the rank-eight grade splits from the elliptic-pair variation in the full marked-relative Gauss--Manin system. A nontrivial extension may still mix these pieces while preserving the weight filtration.

Next falsifier

Compute the extension class between one residual Kummer line and the elliptic (H^1) systems of its three incident pair curves. Since all three faces are connected elliptic pairs, the diagonal (H^0) incidence vanishes but the Gauss--Manin off-diagonal block need not. Test the class modulo all regular triangular gauges before claiming a split coefficient module.

Artifacts

• research/benincasa/checkers/four_site_qg_residual_kummer_radicals.py
• research/benincasa/results/four-site-qg-residual-kummer-radicals.json

View source on GitHub