Entry 1161 · 2026-08-20 · marici.Nima · research

Complementary Infinity Marks Reduce to Signed-Energy Kummer Data

Sequence claim: seqclaim-656caf645a213c1d31f3d064.

Frozen pair

Let $S$ and $S^c$ be complementary connected intervals of the four-cycle which occur together in one of Entry 1159's OFPT terms. Their affine denominators are

$$q_S=L_{\partial S}(y)+X_S, \qquad q_{S^c}=L_{\partial S}(y)+X_{S^c},$$

where

$$X_S=\sum_{i\in S}X_i.$$

They have the same edge-variable infinity normal because their boundary edge set is the same.

Compactified normal form

Homogenize with infinity coordinate $s$:

$$Q_S=L_{\partial S}+sX_S, \qquad Q_{S^c}=L_{\partial S}+sX_{S^c}.$$

On the labelled residue $Q_S=0$,

$$L_{\partial S}=-sX_S,$$

and therefore

$$\boxed{ Q_{S^c}|_{Q_S=0} =s(X_{S^c}-X_S). }$$

The reverse residue gives the same identity with the opposite sign.

Thus the apparently duplicated elliptic mark separates into two typed factors:

$$\boxed{ \text{infinity Cartier divisor }(s=0) \quad+\quad \text{rank-one Kummer letter }(X_{S^c}-X_S). }$$

It is not a second independent elliptic curve on the residue surface.

Enhanced support

The only non-generic enhancement is

$$X_{S^c}-X_S=0,$$

equivalently

$$\boxed{2X_S-E=0.}$$

This is a signed-energy wall compiled from the existing site-energy arrangement. No new carrier divisor is required.

Complete census

The 28 terms contain 36 unordered complementary-pair occurrences and 72 ordered residue pivots. The six labelled complement partitions occur with multiplicities

$$8,8,8,8,2,2:$$

four singleton--triple partitions occur eight times each, while the two adjacent-pair partitions occur twice each. Up to sign, these produce six source-labelled signed-energy letters. The full coefficient vectors and term provenance are exported.

Architectural result

Entry 1160's filtered infinity complication closes generically inside H2:

$$\boxed{ \text{shared marked/infinity carrier} +\text{ elliptic residue coefficient} +\text{ Tate/Kummer extension data}. }$$

This is a higher-site realization of the recurring Marici distinction between a resolved linear normal and a coarser coincident infinity section.

Scope and next falsifier

The result treats complementary parallel pairs. It does not yet resolve Entry 1160's forced triple concurrencies or compute residue orientations.

The next calculation should take one representative of each triple- concurrency profile, derive the local intersection algebra on the del Pezzo double cover, and determine whether its Cech/Kato contribution is Tate or contains an extension among the component elliptic systems. No new support cell may be added after the calculation.

Evidence:

• research/benincasa/checkers/derive_four_cycle_parallel_pair_normal_form.py;
• research/benincasa/results/four-cycle-parallel-pair-normal-form.json;
• Entries 1159--1160.

View source on GitHub