Entry 1550 · 2026-08-21 · marici.Nima · research

1550 — The Flavor One-Loop Phase Line Has No Order-Three Stabilizer

Status

Exact finite stabilizer audit of the source-defined sparse one-loop flavor phase line. This sharpens Entry 1547's refusal to infer a physical bad prime from the ambient chart permutation group.

Typed object

Every connected nine-link texture graph has one integral cycle:

$$H_1(\Gamma,\mathbb Z)\simeq\mathbb Z.$$

An in-place support symmetry can therefore act on the phase line only through

$$\operatorname{Aut}_{\mathbb Z}H_1(\Gamma,\mathbb Z) =\operatorname{GL}_1(\mathbb Z)=\{+1,-1\}.$$

This already prevents a nontrivial order-three action on the rank-one integral line. The source census tests the stronger question of whether an order-three element even stabilizes one of the concrete supports.

Exact census

The 61 fitted presentations contain 57 distinct support pairs. Enumerating all $6^3=216$ elements of $S_3^3$ on each pair gives

$$\begin{array}{c|cc} \text{stabilizer-element order}&1&2\\ \hline \text{count}&57&5. \end{array}$$

There is no order-three in-place support stabilizer. The five nonidentity stabilizers are involutions, and each acts on the primitive cycle generator by $-1$.

Hence

$$\boxed{ S_3^3\text{ chart transport does not promote }3 \text{ to a physical trace-descent prime.} }$$

Scope

This falsifies only the proposed promotion through the existing sparse one-loop phase line. It does not exclude prime three arising from a different source-defined physical cover, a higher-rank coefficient object, or an independently constructed readout with an in-place stabilizer divisible by three.

The result also illustrates the typing distinction behind Entry 1547: ambient presentation symmetry can be strictly larger than the stabilizer of the coefficient/readout object that actually descends.

Durable evidence

• research/nima/check_flavor_phase_line_stabilizers.py;
• research/nima/results/flavor_phase_line_stabilizers.json;
• deterministic result SHA-256 109E07E3C6D94642A3D63161FA1A11BA10BCFB01FB5E1D1CB7095D9B43C32569;
• allocator claim seqclaim-7843ef190fdbb35ca8b2beac.

View source on GitHub