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Entry 1545 · 2026-08-21 · marici.Nima · research

1545 — Prime Cyclic Deck Traces Become Nilpotent Exactly at Their Deck Prime

Status

All-prime cyclic generalization of Entry 1544, with exact matrix audits for p=2,3,5,7p=2,3,5,7.

Characteristic zero

Let σ\sigma be the cyclic shift on the regular pp-dimensional representation of CpC_p. Over Q\mathbb Q, the normalized trace

etr=1p∑k=0p−1σk\boxed{ e_{\rm tr} =\frac1p\sum_{k=0}^{p-1}\sigma^k }

is a rank-one idempotent:

etr2=etr.e_{\rm tr}^2=e_{\rm tr}.

It projects onto the invariant line and splits the regular representation into the trace line and its complementary augmentation representation.

Characteristic pp

Over Fp\mathbf F_p, put

U=σ−1,N=∑k=0p−1σk.U=\sigma-1, \qquad N=\sum_{k=0}^{p-1}\sigma^k.

The cyclic relation gives

Up=0,N=Up−1.\boxed{ U^p=0, \qquad N=U^{p-1}. }

Consequently

rank⁡N=1,N2=0.\boxed{ \operatorname{rank}N=1, \qquad N^2=0. }

For the all-ones transfer column TT and trace row tr⁡\operatorname{tr},

tr⁡T=p=0.\boxed{\operatorname{tr}T=p=0.}

The normalized projector cannot be formed, and the trace line is embedded in a non-semisimple unipotent extension.

Arithmetic law

For a prime cyclic deck packet,

char⁡K≠pnormalized trace projector existschar⁡K=ptrace becomes a square-zero norm operator\boxed{ \begin{array}{c|c} \operatorname{char}K\ne p &\text{normalized trace projector exists}\\ \operatorname{char}K=p &\text{trace becomes a square-zero norm operator} \end{array} }

The exact checker verifies the rational and modular matrices independently for the first four primes. The identities above prove the pattern for every prime.

Meaning

The carrier does not manufacture prime numbers. It supplies finite deck and stabilizer orders. Arithmetic coefficients detect the primes dividing those orders because precisely those primes obstruct normalized trace descent.

Thus a source-derived route to distinguished primes is:

finite carrier symmetry order⟶failure prime of coefficient descent.\boxed{ \text{finite carrier symmetry order} \longrightarrow \text{failure prime of coefficient descent}. }

This is a bounded, falsifiable mechanism. A claimed intrinsic prime must be traced to an independently derived finite symmetry or incidence order; a numerical coincidence alone is insufficient.

Durable evidence

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