Entry 1808 · 2026-08-21 · marici.Nima · research

1808 — Two Transverse Region Pairs Are Killed by Exact Source Cancellation

Question

Do all source-supported transverse pairs of five-site region walls activate a two-wall coefficient object?

Census

Entry 1804 leaves 34 transverse pairs among the nine supported region walls. On one physical sheet their source co-occurrence census is:

$$10\ \text{source-absent pairs}, \qquad 24\ \text{source-supported pairs}.$$

A direct double-residue discovery census isolates all 24 supported pairs away from deeper marked incidences. Twenty-two are numerically nonzero. Two vanish on both physical sheets:

$$(g_{15},g_{34}), \qquad (g_{23},g_{45}).$$

Exact cancellation identities

For the first pair, the two source terms differ only by the remaining walls $g_{1345}$ and $g_{234}$. Their coefficient vectors obey

$$\boxed{ g_{1345}+g_{234}=G^-_{e_{12}}+g_{34}. }$$

Hence on

$$G^-_{e_{12}}=g_{34}=0$$

the two remaining denominators are negatives, and their residues cancel.

Likewise, the second pair has remaining walls $g_{145}$ and $g_{2345}$, with

$$\boxed{ g_{145}+g_{2345}=G^-_{e_{12}}+g_{45}. }$$

Thus its two source terms cancel on

$$G^-_{e_{12}}=g_{45}=0.$$

These are exact identities of the frozen labelled linear forms. No numerical fitting or coefficient normalization enters.

Narrow result

Base transversality and source co-occurrence do not suffice to activate a two-wall coefficient object. The source incidence algebra supplies an additional cancellation differential that kills two transverse pair types.

Therefore the current partition of the 34 transverse pairs is

$$\boxed{ 10\ \text{source absent} \;|\; 2\ \text{exactly cancelled} \;|\; 22\ \text{candidate active}. }$$

The nonvanishing of the final 22 remains discovery-level until exact certification. No tensor-product coefficient object is assigned to them yet.

Next falsifier

Certify the remaining 22 double-residue coefficients exactly. Then classify their cyclic orbits and test whether each local normal-crossing integral is a plain tensor product of rank-one Kummer lines or carries a source-derived extension.

Evidence

• research/benincasa/checkers/five_site_g5_transverse_pair_source_census.py
• research/benincasa/results/five-site-g5-transverse-pair-source-census.json
• research/benincasa/checkers/five_site_g5_transverse_pair_exact_cancellations.py
• research/benincasa/results/five-site-g5-transverse-pair-exact-cancellations.json
• allocator claim: seqclaim-d26f37622a9b5256f2eca35a

View source on GitHub