Entry 1435 · 2026-08-21 · marici.Benincasa · research

1435 — Elementary Deck Flips Become Hecke Correspondences at Exceptional Infinity

Status

Exact valuation-shift theorem on all $32$ sheets of the five-site two-normal Rees object.

Raw and normalized sheet generators

Let $f_S$ denote the raw coefficient on sheet $S\in(\mathbb Z/2)^5$, and let

$$o(S)\in\{2,4,9\}$$

be its exceptional $\tau$-order from Entry 1424. The valuation-normalized generator is

$$e_S=\tau^{-o(S)}f_S.$$

An elementary deck flip $T_i$ acts regularly on the raw sheet sum by $S\mapsto S\mathbin{\rm xor}2^i$. On normalized generators,

$$T_i(e_S) = \tau^{o(S\mathbin{\rm xor}2^i)-o(S)} e_{S\mathbin{\rm xor}2^i}.$$

Exact shift census

Every one of the five elementary flips has the same shift distribution:

$$\begin{array}{c|ccccc} \Delta o&-5&-2&0&2&5\\ \hline \#\text{ sheets}&2&6&16&6&2. \end{array}$$

Thus every elementary flip contains both positive and negative exceptional powers.

Consequence

The raw deck action remains regular before choosing the exceptional lattice, but

$$\boxed{ T_i\text{ is not a regular degree-zero endomorphism of the normalized Rees lattice.} }$$

Negative shifts require $\tau^{-2}$ or $\tau^{-5}$. Because each $T_i$ is an involution, reversing the correspondence exchanges zeros and poles; no uniform filtration shift repairs both directions.

The correctly typed continuation is therefore

$$\boxed{ \text{elementary deck flip} = \text{meromorphic/Hecke correspondence supported at exceptional infinity}. }$$

No Hecke modification is adjoined in this entry.

Residual symmetry

Global complement has

$$o(S\mathbin{\rm xor}31)=o(S)$$

for all $S$. It is therefore the unique nontrivial deck translation acting regularly on the associated grading, exactly as Entry 1430 found by stabilizer census.

Architectural meaning

Exceptional specialization does not merely break a symmetry. It retypes most of the unspecialized symmetry as correspondences between differently modified lattices:

$$\text{regular deck action before specialization} \longrightarrow \text{residual graded symmetry plus exceptional Hecke correspondences}.$$

This supplies the valuation data required by Entry 1430 without adding new carrier support: all poles lie on the already declared exceptional divisor $\tau=0$.

Scope

This is a valuation theorem. It does not construct the modified lattices, prove a cocycle law for Hecke compositions, or show that the physical positive current pairs with the meromorphic correspondences.

Next finite falsifier

Compose two elementary flip correspondences on normalized generators. Test whether their $\tau$-shifts telescope exactly to the shift of the composite deck mask and whether the required lattice modifications satisfy the square relations without an additional exceptional coherence cell.

Durable verification

• Checker: research/benincasa/marici-gm/src/bin/five_site_elementary_flip_rees.rs
• Result: research/benincasa/results/five-site-elementary-flip-rees.json
• Source valuation packet: research/benincasa/results/five-site-two-normal-rees.json
• Allocator claim: seqclaim-68d6e958bfbaae3ce538c66f
• Epistemic graph event: ev-000000001511-9d1e6714-9fd0-4b59-8fd1-5597f996bef4

View source on GitHub