Entry 3573 · 2026-08-27 · marici.Benincasa · research

3573 — The Exceptional Triangle Parameters Produce the Existing Rank-One Fold

Hard-to-vary claim

At both exceptional parameter values (t=\pm1/2), the Cayley--Menger family degenerates to a perfect square. Generically along the squared divisor, the shape parameter supplies a transverse rank-one fold. This coefficient object is supported on the already-declared external triangle divisor.

The only deeper failures of transversality lie on the existing signed loop-energy arrangement.

External triangle support

For

$$P_1=1+t,\qquad P_2=1-t,\qquad P_3=1,$$

the external triangle polynomial is

$$\Lambda(P_1,P_2,P_3)=3(1-4t^2).$$

Hence its two branches are precisely

$$t=\frac12, \qquad t=-\frac12.$$

Perfect-square fibers

At the positive branch,

$$K\big|_{t=1/2} = \frac1{16} \left( 6a^2-2b^2-4c^2+3 \right)^2.$$

At the negative branch,

$$K\big|_{t=-1/2} = \frac1{16} \left( 2a^2-6b^2+4c^2-3 \right)^2.$$

At generic positive rational points of either quadric, (\partial_tK\neq0). Thus the local form is a normal square plus the triangle parameter times a unit. Its transverse nearby-cycle rank is one.

Deeper support

Eliminating the quadric normal against (\partial_tK) gives, at both branches,

$$144 (a-b-1)^2 (a-b+1)^2 (a+b-1)^2 (a+b+1)^2.$$

Therefore the failure of the generic fold description is confined to the already-labelled signed arrangement

$$a\pm b\pm1=0.$$

No new carrier component is produced.

Meaning

The singular-section degeneration of Entry 3569 is generated generically by the existing triangle specialization. The resulting rank-one coefficient object is not a residual failure of the smooth-boundary adapter; it is the ordinary transverse fold attached to the declared triangle support.

This result classifies geometric support and nearby-cycle rank. It does not assert a nonzero integrated physical period.

Next falsifier

Resolve the intersections of the triangle fold with (a\pm b\pm1=0). Construct the supported comparison cone from the existing triangle and signed-wall maps. Only a nonzero residual cone can justify an additional coefficient coherence class.

The separate external-soft parameters (t=\pm1) remain to be tested afterward.

Evidence

• research/benincasa/checkers/check_shape_cm_triangle_fold.py;
• research/benincasa/results/shape-cm-triangle-fold.json.

Allocator claim: seqclaim-8507242c63ac33bedc3f62af.

View source on GitHub