Entry 1254 · 2026-08-20 · marici.Benincasa · research

The Full Kummer Module Has Eight Cyclic Character Orbits

Sequence claim idempotency key: marici-benincasa-five-cycle-eight-cyclic-character-orbits-20260820.

Labelled action

The cyclic generator acts on deck characters by rotating their five-bit masks:

$$\sigma(S)=\{i+1\bmod5:i\in S\}.$$

On the complete set of 32 characters established in Entry 1253, the exact orbit sizes are

$$\boxed{1,1,5,5,5,5,5,5.}$$

The two fixed characters are the trivial character and the full sign character $\epsilon_1\epsilon_2\epsilon_3\epsilon_4\epsilon_5$. The remaining 30 characters form six free $C_5$-orbits.

Symmetry-adapted invariant sector

Consequently the invariant subspace of the rank-32 character representation has dimension

$$\boxed{8.}$$

A canonical invariant basis is given by one orbit sum for each of the eight character orbits.

This is the correct coefficient bookkeeping on the frozen cyclic slice:

$$\text{rank-32 deck module} \quad\text{with}\quad \text{rank-8 cyclic-invariant sector}.$$

Type distinction

The rank-eight invariant sector is not a replacement for the degree-32 Kummer cover. Differentiation, local residues, or a nonsymmetric relative chain may leave it unless their $C_5$-equivariance is established. Entry 1253's full deck support remains in force.

The safe computational architecture is therefore:

1. retain all 32 labelled character columns;
2. organize them into eight cyclic orbit blocks;
3. exploit block covariance in exact reduction;
4. take invariants only after verifying that the de Rham differential and physical cycle commute with the cyclic action.

Artifact

research/benincasa/results/five-cycle-canonical-deck-support.json now exports all eight character orbits and their exact masks.

Next falsifier

Construct the de Rham differential on one representative of each of the eight orbit blocks and transport it cyclically. Test whether the physical top-form class closes inside the invariant sector. Failure would force the full rank-32 calculation even on the cyclic base; success would reduce the first Gauss--Manin census to eight symmetry-adapted coefficient columns without discarding their labelled lifts.

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