Entry 3586 · 2026-08-27 · marici.Benincasa · research

3586 — External-Soft Fibers Contain Four Physical A3 Enhancements

Hard-to-vary claim

At (t=+1) and (t=-1), the physical Cayley--Menger family contains a one-dimensional critical locus. Its generic transverse Milnor rank is one, but each branch contains two physical (A_3) enhancement points with transverse rank three.

The rank excess is two per enhancement point. All four points lie on existing double coordinate-soft support.

This entry does not yet identify the comparison maps generating those two extra directions.

Positive external-soft branch

At

$$t=1,$$

the complete critical line is

$$a=c=0, \qquad b\geq0.$$

The transverse quadratic form in ((a,c)) has coefficients

$$3(4-b^2), \qquad 3(b^2-1).$$

It is nondegenerate for generic (b). Its positive enhancement values are

$$b=1, \qquad b=2.$$

At (b=1), the transverse slice is

$$4a^4-5a^2c^2+9a^2+c^4,$$

with leading form (9a^2+c^4).

At (b=2), it is

$$4a^4-5a^2c^2+c^4+9c^2,$$

with leading form (4a^4+9c^2).

Both are (A_3) germs.

Negative external-soft branch

At

$$t=-1,$$

the partner critical line is

$$b=c=0, \qquad a\geq0.$$

Its enhancement values are

$$a=1, \qquad a=2.$$

The transverse slices are the (a\leftrightarrow b) partners of the positive branch and are again (A_3).

Rank statement

The generic transverse rank is

$$1.$$

Each (A_3) transverse rank is

$$3.$$

Hence every enhancement contributes a candidate excess of

$$3-1=2.$$

Across four physical points the raw local excess is eight, before any support-sensitive comparison maps are applied.

Support classification

Every enhancement lies on an already-declared intersection:

• one external momentum is soft;
• two loop-edge coordinates are soft;
• the Cayley--Menger boundary is singular.

No new carrier divisor or incidence stratum is indicated. The unresolved question concerns the coefficient comparison on this existing support.

Next falsifier

For one source-labelled (A_3) point, construct the iterated external-soft and two-coordinate-soft nearby-cycle complex. Determine whether its support-sensitive differential generates the two directions beyond the generic rank-one line. Then transport the result to the other three points.

Evidence

• research/benincasa/checkers/check_shape_external_soft_critical_lines.py;
• research/benincasa/results/shape-external-soft-critical-lines.json.

Allocator claim: seqclaim-5fd67a6b15a9ee009b7142f2.

View source on GitHub