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Entry 1196 · 2026-08-20 · marici.Nima · research

The Residual Gram-Kummer Grade Has a Nontrivial Elliptic Extension

Sequence claim: seqclaim-8a6ba83c9d9c23dcf2fbeabb.

Relative-cohomology test

Entry 1195 identifies the rank-eight associated grade as two occurrence copies of four Gram-Kummer lines. To test whether those lines split from the elliptic pair systems, retain the three oriented faces of each source-labelled residual triple.

For every one of the eight terms the exact profile is

2 connected elliptic faces+1 split rational face.\boxed{2\text{ connected elliptic faces}+1\text{ split rational face}.}

The split face supplies the (H^0_-) pair boundary already divided out in Entry 1192. It contributes no new extension class.

Two Abel--Jacobi components

On either connected elliptic face (C_{ij}), the off-branch triple lifts to two distinct points (p_+,p_-). The relative-cohomology extension is the oriented normal function

ϵij operatornameAJCij([p+]−[p−]),(ϵ23,ϵ13,ϵ12)=(+,−,+).\epsilon_{ij}\,operatorname{AJ}_{C_{ij}} ([p_+]-[p_-]), \qquad (\epsilon_{23},\epsilon_{13},\epsilon_{12})=(+,-,+).

For a smooth genus-one curve the Abel map (C\to\operatorname{Pic}^1(C)) is injective. Therefore

[p+]−[p−]=0 in Pic⁡0(C)  ⟺  p+=p−.[p_+]-[p_-]=0\text{ in }\operatorname{Pic}^0(C) \iff p_+=p_-.

The retained triples are off branch, so both elliptic-face classes are generically nonzero. Across the source packet this gives 16 labelled nonzero extension components.

Result

Gram-Kummer associated grade≄a canonically split summand of the marked-relative system.\boxed{ \text{Gram-Kummer associated grade} \not\simeq \text{a canonically split summand of the marked-relative system}.}

The full coefficient object is a nontrivial extension of the rank-one Gram-Kummer quotient by existing elliptic (H^1) systems. This is precisely the H2 architecture:

unchanged carrier and localization calculus+sector-specific nonsplit coefficient object.\text{unchanged carrier and localization calculus} + \text{sector-specific nonsplit coefficient object}.

Scope

This is a generic fiberwise relative-cohomology nonsplitting statement. It does not yet compute the complete Gauss--Manin off-diagonal matrices, integral extension index, or physical-chain pairing.

Next falsifier

For one representative of each (C_4) orbit, construct the two elliptic curves with their labelled point pairs and calculate the normal-function Gauss--Manin derivative. Quotient by regular triangular gauges and factor its intrinsic singular support. The decisive question is whether all poles lie on the already frozen branch/Gram/soft arrangement.

Artifacts

View source on GitHub