Entry 1225 · 2026-08-20 · marici.Nima · research

Entry 1225 — The Five-Site Coefficient–Betti Pairing Is Deck Invariant

Typed factors

Entries 1223–1224 produced two rank-32 deck objects:

1. the finite pushforward of the five-site Kummer coefficient system;
2. the regular orbit of the locally selected physical cycle $\Gamma_+$.

Neither the positive coefficient branch nor $\Gamma_+$ is separately invariant. The physical question is whether their pairing is invariant under simultaneous continuation.

Sheet-basis pairing

Let $e_g$ denote the coefficient sheet basis and $\Gamma_h$ the Betti orbit basis for $g,h\in G=(\mathbb Z_2)^5$. Freeze

$$\langle e_g,\Gamma_h\rangle=\delta_{g,h}.$$

Deck transport is regular translation on both factors. Hence

$$\langle T_ke_g,T_k\Gamma_h\rangle = \delta_{g\oplus k,h\oplus k} = \delta_{g,h}.$$

Thus the pairing is strictly deck invariant.

Character-basis form

Fourier transform by the real character table

$$H_{S,g}=(-1)^{|S\cap g|}$$

gives

$$HH^T=32I.$$

Both coefficient and Betti factors carry the same order-two character. Their diagonal character pairing is therefore invariant because every character square is one.

The checker verifies:

$$1024$$

Hadamard orthogonality identities,

$$5120+5120$$

sheet- and character-basis covariance identities, and

$$1024$$

joint chamber-evaluation identities. Every check passes.

Therefore

$$\boxed{ \langle T c,T\Gamma\rangle = \langle c,\Gamma\rangle. }$$

A scalar evaluated in the physical chamber is continuation-covariant when its coefficient branch and relative cycle are transported together.

Classification

The five-site square-root multiplicity is now accounted for by

$$\boxed{ \text{Kummer coefficient local system} + \text{regular Betti orbit} + \text{invariant pairing} }$$

over the unchanged physical energy/Cut carrier. No carrier enlargement is indicated.

Scope

This proves representation-theoretic deck descent of the pairing. It does not determine period normalization, endpoint regularization, or monodromy around discriminants not generated by the five Kummer sheet changes.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_kummer_betti_pairing.rs
• research/benincasa/results/five-site-kummer-betti-pairing.json

Next falsifier

Compute the remaining discriminant arrangement of the physical five-site marked cover after removing the known Kummer branch divisors $F_i=0$ and $\det H=0$. Test whether continuation around any residual divisor preserves the coefficient–Betti pairing or forces an independently derived coefficient extension. Only an incidence divisor absent from the frozen carrier would challenge H2 at carrier level.

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