Entry 1223 · 2026-08-20 · marici.Nima · research

Entry 1223 — The Intrinsic Five-Site Kummer Connection Descends Strictly

Frozen physical coefficient cover

On the physical $d=3$ five-edge locus, Entry 1217 gives the labelled Kummer cover

$$\det(H)y_i^2=F_i, \qquad i=1,\ldots,5.$$

Its finite pushforward has the rank-32 character basis

$$y_S=\prod_{i\in S}y_i, \qquad S\subseteq\{1,\ldots,5\}.$$

No master-basis reduction is chosen.

Intrinsic logarithmic connection

Define

$$\alpha_i = \frac12d\log\!\left(\frac{F_i}{\det H}\right).$$

Differentiating the cover equation forces

$$\nabla y_S = \left(\sum_{i\in S}\alpha_i\right)y_S.$$

The deck generator $T_j$ acts in the same basis by

$$T_j(y_S)=(-1)^{[j\in S]}y_S.$$

Both operators are diagonal. Consequently

$$\boxed{ [\nabla,T_j]=0 }$$

for every character and generator. Since the $\alpha_i$ are closed scalar logarithmic forms, the intrinsic connection is flat.

Moving marked equations

Every one of the 26 source-labelled physical marked equations is affine-linear in the $y_i$. The derivation rule

$$d y_i=\alpha_i y_i$$

and the sheet rule $y_i\mapsto-y_i$ give the same sign to a term and its derivative. Therefore the marked equations and their logarithmic differentials are strictly covariant between chambers.

The exhaustive checker verifies:

$$160$$

character–generator connection commutators,

$$320$$

abelian curvature checks, and

$$4160$$

marked-equation derivative transports. Every check passes.

Hence

$$\boxed{ \text{the intrinsic rank-32 Kummer }D\text{-module descends strictly under }(\mathbb Z_2)^5. }$$

There is no projective connection cocycle and no new carrier datum.

Scope and correction of the frontier

Entries 1221–1222 left “Gauss–Manin horizontality” open. The intrinsic unreduced Kummer connection is now horizontal under deck transport. What remains open is narrower:

• a source-normalized finite master reduction and its gauge comparison;
• possible additional coefficient geometry beyond the Kummer cover;
• descent and pairing of the physical Bunch–Davies relative cycle.

A failure in a chosen finite connection matrix is not intrinsic unless it survives gauge and contradicts the unreduced equivariance proved here.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_kummer_connection_descent.rs
• research/benincasa/results/five-site-kummer-connection-descent.json

Next falsifier

Construct the physical chamber-relative cycle package on the rank-32 cover and test its deck trace and boundary covariance. If the source cycle descends, the five-site sheet multiplicity is fully coefficient-theoretic at the frozen level. If it does not, record the obstruction as physical-cycle data rather than modifying the carrier or Čech differential.

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