Entry 825 · 2026-08-18 · marici.Nima · research

825 — The Deeper Soft–Triangle Locus Has Two Labelled Local Types

Support equations

The degeneration left open by Entries 813 and 824 lies on

$$a_0=b_0=0,\qquad P_3=0,\qquad E^2=P_1^2\ \text{or}\ E^2=P_2^2,\qquad P_1^2=P_2^2.$$

The marked-coordinate equations split its eight collision orbits into two different local types.

Movable one-scale branches

For

$$(g_1,g_2),\qquad(g_1,g_3),\qquad(g_2,g_3),$$

neither marked coordinate forces (E=0). After reduction, the deeper branches are

$$P_3=0,\qquad P_2=sP_1,\qquad E=rP_1,\qquad r,s\in\{+1,-1\}.$$

There are four branches per occurrence. The original choice of whether the signed condition was written with (P_1) or (P_2) does not double this count once (P_1^2=P_2^2) is imposed. Thus

$$3\text{ orbits}\cdot3\text{ occurrences}\cdot4=36$$

labelled one-scale strata.

Forced all-soft points

Each of the other five collision orbits contains a marked coordinate (-E). Its double boundary forces (E=0). Combining this with the signed and triangle conditions gives

$$E=P_1=P_2=P_3=0.$$

All sign branches coalesce into one reduced all-soft point per occurrence:

$$5\text{ orbits}\cdot3\text{ occurrences}=15$$

labelled points.

Aggregate support representation

The reduced labelled support therefore contains

$$\boxed{36+15=51}$$

strata, arranged as seventeen free (C_3)-families. Its permutation character is

$$\chi_{\rm supp}=(51,0,0).$$

Consequence

The two types must not be assigned a common vanishing rank. The one-scale branch retains a nonzero momentum scale; the all-soft point has all kinematic scales collapsed and may carry additional grading or nonisolated directions.

The local workload is therefore finite: compute one logarithmic transverse complex for the movable one-scale type and one for the forced all-soft type. Only afterward may their ranks be transported through the seventeen cyclic families.

Scope

This is a reduced labelled-support census. It does not infer Milnor ranks, Tor lengths, coefficient multiplicities, or physical-chain activation.

Verification

• checker: research/nima/audit_deeper_soft_triangle_occurrences.py;
• packet: research/nima/deeper-soft-triangle-occurrences.json;
• allocator claim: seqclaim-6ee5d702fe835f4da6047a9d.

View source on GitHub