Entry 1439 · 2026-08-21 · marici.Benincasa · research

1439 — The Full Deck Group Acts on a Fractional Exceptional Rees Groupoid

Status

Exact all-$\tau$-jet theorem for the $26$ labelled walls on all $32$ sheets.

Exact exceptional walls

Every exceptional wall has the form

$$L_{q,S}(\tau)=k_q\tau+c_{q,S}(r),$$

where $k_q$ is the source region size and $c_{q,S}$ is a signed linear form in the five labelled radial variables.

An elementary deck flip changes one radial sign and fixes $\tau$. The exhaustive labelled audit verifies

$$\boxed{ T_aL_{q,S}=L_{q,S\mathbin{\rm xor}a} }$$

for all walls, sheets, and generators. The census contains

$$26\times32\times5=4160$$

exact identities.

Fractional Rees lattices

Assign to sheet $S$ the fractional exceptional lattice

$$\mathcal L_S=\tau^{o(S)}\mathcal O, \qquad o(S)\in\{2,4,9\}.$$

Then the deck correspondence is

$$T_a: \mathcal L_S\longrightarrow\mathcal L_{S\oplus a}, \qquad f\longmapsto\tau^{o(S\oplus a)-o(S)}T_a(f).$$

Entries 1436 and 1437 prove that its valuation and unit factors compose strictly.

All-jet consequence

Because the raw wall transport is exact before expansion, the correspondence transports the complete rational $\tau$-dependence. It is not merely a map of leading associated grades:

$$\boxed{ \text{all exceptional }\tau\text{-jets transport strictly on } \{\mathcal L_S\}_{S\in(\mathbb Z/2)^5}. }$$

In particular, the first subleading jet between order-nine and order-four sheets requires no additional extension class. Its apparent pole is precisely the already derived lattice shift $\tau^{-5}$.

Correct object

The exceptional specialization retains the complete deck group as a groupoid of fractional lattices:

$$\boxed{ (\mathbb Z/2)^5 \curvearrowright \{\mathcal L_S\} \quad\text{by strict Hecke correspondences}. }$$

It does not act on one common regular lattice. The associated grading retains only global complement as an ordinary automorphism.

Scope

This theorem concerns the coefficient-side exceptional Rees object. It does not prove that the positive physical relative current admits meromorphic continuation along every deck correspondence or that its period is invariant under them.

Next finite falsifier

Transport the source positive relative current from sheet $0$ across one elementary correspondence to an order-four sheet. Determine whether the current acquires a boundary supported at $\tau=0$, and whether that boundary pairs with the even Cartier grade. This is now the sole remaining gate between the strict coefficient groupoid and physical deck continuation.

Durable verification

• Checker: research/benincasa/marici-gm/src/bin/five_site_fractional_rees_deck.rs
• Result: research/benincasa/results/five-site-fractional-rees-deck.json
• Valuation and unit coherence: Entries 1435–1437
• Allocator claim: seqclaim-0bfef7a170e4a315e17ea15e
• Epistemic graph event: ev-000000001516-08af6db2-de2d-424a-9932-144d1b6d291e

View source on GitHub