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Entry 1437 · 2026-08-21 · marici.Benincasa · research

1437 — Exceptional Deck Leading Units Have No Projective Cocycle

Status

Strict source-frame theorem with replicated exact modular leading-coefficient audits.

Source frame

Entry 1221 freezes the labelled chamber-relative deck package and proves that every elementary sheet flip preserves section labels, residue order, and ambient d3ℓd^3\ell orientation. Entry 1223 proves strict compatibility with the intrinsic Kummer connection.

Therefore the raw deck action has unit coefficient:

Ta(fS)=fSxora.T_a(f_S)=f_{S\mathbin{\rm xor}a}.

Entry 1435 adds only the exceptional power τΔa(S)\tau^{\Delta_a(S)} after valuation normalization.

Optional unit-leading normalization

Let cSc_S be the source-derived leading coefficient of sheet SS, and define

e^S=cS−1τ−o(S)fS\widehat e_S=c_S^{-1}\tau^{-o(S)}f_S

where cS≠0c_S\ne0. Then

Ta(e^S)=τΔa(S)ua(S)e^Sxora,T_a(\widehat e_S) = \tau^{\Delta_a(S)}u_a(S)\widehat e_{S\mathbin{\rm xor}a},

with

ua(S)=cSxoracS.u_a(S)=\frac{c_{S\mathbin{\rm xor}a}}{c_S}.

This unit system is an exact coboundary. In particular,

ua(S)ub(Sxora)=uaxorb(S),u_a(S)u_b(S\mathbin{\rm xor}a) = u_{a\mathbin{\rm xor}b}(S),

and every elementary square and commuting flip diagram closes identically.

Finite replication

All 3232 leading coefficients are nonzero at each of two independent exact finite-field radial points. The checker verifies per point

160 square identities160\text{ square identities}

and

800 ordered commutation identities.800\text{ ordered commutation identities}.

No projective defect appears.

Consequence

Combining Entries 1435–1437, the exceptional deck correspondence is completely typed at leading Rees order:

Ta:eS⟼τΔa(S)eS⊕a,\boxed{ T_a: e_S\longmapsto \tau^{\Delta_a(S)}e_{S\oplus a}, }

with strict valuation composition and no intrinsic unit or orientation cocycle.

The divisor cS=0c_S=0 appears only if one insists on the optional unit-leading normalization. It is not support of the raw source correspondence and must not be promoted to a carrier stratum.

Scope

The theorem concerns the leading exceptional lattice. It does not prove that the full subleading τ\tau-jets form finite Hecke modifications or that the physical positive current extends meromorphically across every correspondence.

Next finite falsifier

Compute the first subleading τ\tau-jet under one elementary flip between an order-nine and order-four sheet. Test whether the valuation-shifted transport remains regular to first jet or requires an additional extension supported on the exceptional divisor.

Durable verification

View source on GitHub