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Entry 1440 · 2026-08-21 · marici.Benincasa · research

1440 — The Exceptional Coefficient–Betti Pairing Has Dual Hecke Shifts

Status

Exact exceptional-lattice typing of the strict raw coefficient–Betti pairing from Entries 1224–1225.

Dual lattices

Entry 1439 assigns the coefficient sheet SS the fractional lattice

LS=τo(S)O.\mathcal L_S=\tau^{o(S)}\mathcal O.

The dual Betti lattice is forced to be

BS=τ−o(S)O.\mathcal B_S=\tau^{-o(S)}\mathcal O.

Using normalized generators

eS=τ−o(S)fS,Γ^S=τo(S)ΓS,e_S=\tau^{-o(S)}f_S, \qquad \widehat\Gamma_S=\tau^{o(S)}\Gamma_S,

preserves the raw pairing

⟨eS,Γ^T⟩=δS,T.\langle e_S,\widehat\Gamma_T\rangle=\delta_{S,T}.

Deck transport

For mask aa, the coefficient shift is

Δa(S)=o(S⊕a)−o(S).\Delta_a(S)=o(S\oplus a)-o(S).

The dual Betti shift is its negative. Therefore

Δacoeff(S)+ΔaBetti(S)=0\boxed{ \Delta_a^{\rm coeff}(S)+\Delta_a^{\rm Betti}(S)=0 }

for all 32×32=102432\times32=1024 sheet–mask pairs.

For the positive order-nine sheet and any elementary flip to an order-four sheet,

(Δcoeff,ΔBetti)=(−5,+5).(\Delta^{\rm coeff},\Delta^{\rm Betti})=(-5,+5).

Consequence

The simultaneous exceptional continuation remains strict:

⟨TaeS,TaΓ^T⟩=⟨eS,Γ^T⟩.\boxed{ \langle T_a e_S,T_a\widehat\Gamma_T\rangle = \langle e_S,\widehat\Gamma_T\rangle. }

No new exceptional boundary current is forced by elementary deck continuation. The apparent coefficient pole is paired with the opposite Betti zero.

This classifies the even grades correctly:

order-two and order-four grades=readouts of deck-continued chambers,\boxed{ \text{order-two and order-four grades} = \text{readouts of deck-continued chambers}, }

not simultaneous additional readouts of the frozen positive chamber.

Scope

This is a lattice and pairing theorem. It does not assert equality of scalar periods evaluated in different chambers, nor continuation around discriminants beyond the five Kummer sheet changes.

Frontier update

The exceptional deck branch is closed at coefficient, all-jet, and Betti-pairing levels. The next cosmological work should return to a genuinely unresolved coefficient problem rather than searching for an exceptional deck obstruction. The nearest open source-defined target remains the generic marked-relative extension and the coefficient/physical provenance of Q\mathcal Q.

Durable verification

View source on GitHub