Entry 1197 · 2026-08-20 · marici.Nima · research

The Residual Elliptic Extensions Have Only Frozen Relative Discriminant Support

Sequence claim: seqclaim-12e57ebc9ec66714d004d956.

Two representative pair families

Entry 1196's sixteen Abel--Jacobi components form two labelled (C_4) orbits. It is therefore enough to restrict

$$K_\infty=\Delta^T\operatorname{adj}(G)\Delta$$

to the two representative pair lines

$$(G\setminus e_{12},g_3), \qquad (G\setminus e_{12},g_{34}).$$

On either line, write

$$\Delta=uS+vT, \qquad S=s^2,\quad T=t^2.$$

The restricted branch quartic is

$$\alpha s^4+2\beta s^2t^2+\gamma t^4,$$

where

$$\alpha=u^TAu,\qquad \beta=u^TAv,\qquad \gamma=v^TAv,\qquad A=\operatorname{adj}(G).$$

Exact factorization

For both source representatives, exact characteristic-zero Symbolica reduction gives

$$\boxed{ \alpha\gamma-\beta^2 = \det(G)\,n^TGn, }$$

with

$$n=(0,-1,0)$$

for ((G\setminus e_{12},g_3)), and

$$n=(0,1,1)$$

for ((G\setminus e_{12},g_{34})).

Since

$$\operatorname{Disc} (\alpha s^4+2\beta s^2t^2+\gamma t^4) \sim \alpha\gamma(\alpha\gamma-\beta^2)^2,$$

the complete support is

$$\boxed{ \alpha\gamma\,\det(G)^2\,(n^TGn)^2=0. }$$

The marked residual section evaluates in both representative families to

$$G_{22}G_{33}-G_{23}^2,$$

the expected principal Gram cofactor.

Classification

Every factor has source provenance:

• $\alpha=0$ and $\gamma=0$ are endpoint collisions with the frozen branch divisor;
• $\det G=0$ is the existing total Gram divisor;
• $n^TGn=0$ is the labelled pair-normal Gram degeneration;
• the marked-section collision is the existing cofactor divisor.

Thus no new marked incidence or cosmology-specific carrier stratum is required. Any singularity of the Abel--Jacobi extension is coefficient support produced by the frozen relative geometry.

Consequence

Entries 1192--1197 now isolate a coherent finite object:

$$\boxed{ \text{two regular }C_4\text{ orbits of Gram-Kummer grades} \text{ nonsplit over elliptic }H^1, }$$

with all generic discriminant support generated by the unchanged Cayley--Menger/marked carrier. This is a direct nontrivial success of H2.

Remaining qualification

The exact normal-function connection matrices and physical relative-chain pairing remain unconstructed. The present result classifies the algebraic support of their source-defined relative geometry; it does not establish physical activation.

Artifacts

• research/benincasa/marici-gm/src/bin/four_site_residual_pair_discriminants.rs
• research/benincasa/results/four-site-qg-residual-pair-discriminants.json

View source on GitHub