Entry 1552 · 2026-08-21 · marici.Nima · research

1552 — The Equal-Energy Cyclic Locus Realizes a Physical Occurrence Trace of Order Three

Status

Exact synthesis of Entries 356 and 764. This supplies the typed fixed-locus test requested by Entries 1546–1547; it does not reinterpret the generic cyclic family as a deck cover.

The cyclic fixed point

In the normalized (G_{12}) chart,

$$X_1=1,\qquad X_2=\frac{u+v}{2}-1,\qquad X_3=\frac{u-v}{2}.$$

The source cycle acts by

$$\rho(u,v)= \left(\frac{2u}{u-v},\frac{2(2-v)}{u-v}\right).$$

Its positive equal-energy fixed point is

$$(u,v)=(3,1),\qquad (X_1,X_2,X_3)=(1,1,1).$$

Entry 764 independently derives the three residue-chart connections and their horizontal transition

$$S=\operatorname{diag}(z^{-2},z^{-1},z,z),\qquad z=\frac{u-v}{2}.$$

At the fixed point (z=1), hence

$$\boxed{S=1.}$$

Generic cyclic occurrence covariance therefore becomes an in-place horizontal (C_3)-action on the coefficient fiber at this locus.

Physical occurrence trace

Entry 356's six all-positive physical occurrences form two regular cyclic orbits:

$$(12|23,23|31,31|12),\qquad (12|31,23|12,31|23).$$

For either orbit let

$$T:\mathbb Z\longrightarrow\mathbb Z[C_3]$$

send (1) to the norm vector, and let

$$R:\mathbb Z[C_3]\longrightarrow\mathbb Z$$

be the source all-positive occurrence sum. Exact matrix multiplication gives

$$\boxed{RT=3}, \qquad N:=TR, \qquad \boxed{N^2=3N}.$$

For both orbits together,

$$RT=3I_2.$$

Modulo three the norm remains rank two but becomes square-zero:

$$N\ne0,\qquad N^2=0\pmod3.$$

Thus

$$\boxed{ 3\text{ is a bad trace-descent prime for the physical occurrence packet on the equal-energy cyclic locus.} }$$

Typing boundary

This statement is fixed-locus and occurrence-resolved. Away from (X_1=X_2=X_3), cyclic relabelling moves the external base point, so it is covariance rather than in-place descent. Nor does the result identify the three occurrences as sheets of one generic-base geometric cover.

What is established is precisely the weaker but physical datum needed by Entry 1546: a source-defined finite in-place symmetry, a horizontally transported coefficient packet, and its source all-positive trace/readout.

The currently established physical bad-prime inventory is therefore updated to

$$\boxed{ \operatorname{Bad}_{\rm physically\ established}=\{2,3\}, }$$

with (2) coming from two-sheet mass/deck descent and (3) from the cyclic equal-energy occurrence trace.

Durable evidence

• research/nima/check_cyclic_fixed_locus_trace.py;
• research/nima/results/cyclic_fixed_locus_trace.json;
• Entries 356, 764, and 1543–1547;
• deterministic result SHA-256 FCD167FF76F03D77FDD54F2789BD4A19E0783776365C0259F80AF74AAC72343F;
• allocator claim seqclaim-631578adac54f457d4d9c8c8.

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