Entry 1546 · 2026-08-21 · marici.Nima · research

1546 — Finite Deck Orders Determine the Bad Primes of Trace Descent

Status

Finite-group theorem extending Entries 1544–1545. Composite cyclic orders $2$ through $12$ are independently checked.

Deck norm

Let $G$ be a finite deck group acting by its regular representation on the labelled cycle space, and define

$$N_G=\sum_{g\in G}g.$$

For every $h\in G$,

$$hN_G=N_G=N_Gh.$$

Counting pairs in the product gives

$$\boxed{N_G^2=|G|N_G.}$$

The image of $N_G$ is the invariant all-ones line, so $N_G$ has rank one over every coefficient field.

Good characteristics

If $|G|$ is invertible in the coefficient ring, then

$$\boxed{e_G=\frac{N_G}{|G|}}$$

is a central rank-one idempotent. It supplies normalized trace descent and splits the invariant cycle line as a direct summand.

Bad characteristics

If a prime $p$ divides $|G|$, then over characteristic $p$,

$$\boxed{N_G^2=0.}$$

The same source-derived norm becomes a nonzero square-zero operator. The normalized trace projector is unavailable, and invariant descent is no longer semisimple.

Exact audit

The durable checker constructs cyclic regular representations for every order $2\le n\le12$. It verifies

$$N_n^2=nN_n$$

over $\mathbb Q$, idempotence of $N_n/n$, and square-zero rank-one norm after reduction modulo every prime divisor of $n$.

Meaning

The finite carrier symmetry determines an arithmetic ramification set:

$$\boxed{ \operatorname{Bad}(G) =\{p\text{ prime}:p\mid |G|\}. }$$

These primes are not produced from nothing. They are detected because the coefficient category can or cannot divide by a source-derived symmetry order.

This gives the program a disciplined arithmetic search rule: inventory the actual finite deck, stabilizer, and incidence groups first; only their prime divisors are candidates for failures of normalized trace descent. Additional distinguished primes require additional source structure.

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entries 1544–1545;
• allocator claim seqclaim-ce74b3849384625c5fc75668.

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