Entry 1436 · 2026-08-21 · marici.Benincasa · research

1436 — Exceptional Deck Valuations Form a Strict Cocycle

Status

Complete valuation-level composition theorem for all $32$ sheets and all pairs of deck masks.

Shift function

For sheet $S$ and deck mask $a$, define

$$\Delta_a(S)=o(S\mathbin{\rm xor}a)-o(S),$$

where $o(S)\in\{2,4,9\}$ is the exceptional Cartier order.

Entry 1435 shows that $\tau^{\Delta_a(S)}$ is the lattice modification carried by the deck correspondence.

Exact composition identities

Across all $32^3=32768$ triples $(S,a,b)$,

$$\boxed{ \Delta_a(S)+\Delta_b(S\mathbin{\rm xor}a) = \Delta_{a\mathbin{\rm xor}b}(S). }$$

The two orders of composition also agree:

$$\boxed{ \Delta_a(S)+\Delta_b(S\mathbin{\rm xor}a) = \Delta_b(S)+\Delta_a(S\mathbin{\rm xor}b). }$$

For every elementary generator $i$,

$$\Delta_i(S)+\Delta_i(S\mathbin{\rm xor}2^i)=0.$$

Consequence

The meromorphic powers required by Entry 1435 compose strictly. No additional exceptional coherence cell is required by the valuation data:

$$\boxed{ \tau^{\Delta_b(S\oplus a)}\tau^{\Delta_a(S)} = \tau^{\Delta_{a\oplus b}(S)}. }$$

Thus the full deck group survives exceptional specialization as a strict group of valuation-shifted correspondences, while only the subgroup

$$\{0,31\}$$

acts without lattice modification on the associated grading.

Remaining coherence gate

This theorem controls only powers of the exceptional parameter. It does not yet control the leading units or orientation signs of the transported coefficient forms. A nontrivial unit-valued two-cocycle could still require a coherence cell even though the valuation cocycle is exact.

Scope

No Hecke lattice is adjoined, and no physical-current pairing is asserted. The result is the complete exponent-level acceptance test for such a construction.

Next finite falsifier

Retain the source leading coefficients $c_S$, transport them under two elementary flips, and compute the unit ratio around every commuting square. Test whether it is identically $1$, a removable sheet rephasing, or a genuine exceptional unit cocycle.

Durable verification

• Checker: research/benincasa/marici-gm/src/bin/five_site_flip_rees_coherence.rs
• Result: research/benincasa/results/five-site-flip-rees-coherence.json
• Source valuation packet: research/benincasa/results/five-site-two-normal-rees.json
• Allocator claim: seqclaim-53b0a2535012ab51396f5bfa
• Epistemic graph event: ev-000000001514-1b1f4234-d08c-4c1f-bc19-9c211e492a2d

View source on GitHub