Entry 1809 · 2026-08-21 · marici.Nima · research

1809 — Exactly Twenty-Two Transverse Region Pairs Are Source-Active

Claim

Among Entry 1808's 34 transverse pairs of supported five-site region walls, exactly 22 have nonzero source double-residue coefficients at the frozen physical threshold.

Frozen calculation

For each source-supported pair, keep

$$G^-_{e_{12}}=g_5=g_A=g_B=0$$

and preserve total energy while solving the two labelled wall equations with three source-energy coordinates. The adjustment triple is chosen only to isolate the intersection from all remaining marked walls; it does not alter the source sum.

For each of the two physical loop sheets, exact rational interval arithmetic then evaluates the complete double-residue coefficient after removing the four frozen denominators. Every remaining denominator is certified away from zero.

Result

After excluding the ten source-absent pairs and Entry 1808's two exact source cancellations, all remaining coefficients obey

$$0\notin \operatorname{Res}_{A,B}$$

on both physical sheets. The certificate count is

$$22\times2=44.$$

Thus the exact partition is

$$\boxed{ 34 =10\ \text{source absent} +2\ \text{source cancelled} +22\ \text{source active}. }$$

This certifies activation only. It does not yet identify the integrated local coefficient object with a tensor product of the two one-wall Kummer systems.

Architectural consequence

The carrier incidence, source term incidence, and source residue differential are three distinct gates:

$$\text{transverse carrier pair} \not\Rightarrow \text{source co-occurrence} \not\Rightarrow \text{nonzero source coefficient}.$$

All three gates are now explicitly resolved for this finite pair census.

Next falsifier

Compute the labelled $C_5$-orbit decomposition of the 22 active pair types. For one representative of each orbit, derive the two-normal local integral and test whether its coefficient system splits as a Kummer tensor product or contains a nontrivial extension.

Evidence

• research/benincasa/checkers/five_site_g5_transverse_pair_exact_coefficients.py
• research/benincasa/results/five-site-g5-transverse-pair-exact-coefficients.json
• allocator claim: seqclaim-74050128f126d049f53c1a30

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