Entry 3569 · 2026-08-27 · marici.Benincasa · research

3569 — The Physical Cayley–Menger Singularities Transport as Exact Critical Sections

Hard-to-vary claim

The nonnegative singular scheme of the homogeneous Cayley--Menger boundary contains exactly three Morse points. Under the physical shape deformation, all three extend as exact critical sections of the full family.

The first local deformation class vanishes. The apparent nonzero second symbols obtained by freezing the critical points also vanish after critical- section transport. No residual supported class remains at generic parameter values.

Singular scheme at the homogeneous point

Write

$$A=a^2,\qquad B=b^2,\qquad C=c^2.$$

Exact active-coordinate elimination of

$$K_0=K_a=K_b=K_c=0$$

in the nonnegative chamber gives precisely

$$(a,b,c)=(1,1,0),\qquad(1,0,1),\qquad(0,1,1).$$

Each Hessian determinant equals

$$-288,$$

so all three germs are Morse.

Exact critical sections

For

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

the three points extend as

$$(1-t,1+t,0), \qquad (1,0,1+t), \qquad (0,1,1-t).$$

Substitution into the complete Cayley--Menger family gives, identically in (t),

$$K=K_a=K_b=K_c=0$$

on every section.

Their tangent vectors are

$$(-1,1,0), \qquad (0,0,1), \qquad (0,0,-1),$$

and each solves the exact first transport equation

$$H_{K_0}v+\nabla K_1=0.$$

Second-grade correction

At the frozen homogeneous points,

$$K_1=0,$$

while the raw second derivatives are

$$K_2=24,8,8.$$

Those values are not invariant deformation classes. After transporting the critical point, the effective second critical-value derivative is

$$K_2- (\nabla K_1)^T H_{K_0}^{-1} \nabla K_1.$$

It vanishes at all three points.

Thus the proposed second-order supported record is refuted. It was created by holding a moving critical point fixed.

Remaining support

The transported sections remain Morse except at

$$t=\pm1$$

and

$$t=\pm\frac12.$$

These are respectively existing external soft and triangle-degeneration supports. No new carrier component appears.

Meaning

The smooth-boundary adapter of Entry 3566 does not fail by producing a hidden second-grade class at the generic singular sections. Instead the source deformation transports the entire singular geometry coherently.

This supplies a strict methodological rule:

Normal grades must be computed after transporting the source-defined critical locus. Fixed-coordinate derivatives can manufacture false supported records.

Next falsifier

Resolve the existing exceptional parameter values (t=\pm1/2) and (t=\pm1). Test whether the critical-section comparison is generated by the already-defined triangle and soft maps. Any residual must be computed as a supported comparison cone, not inferred from Hessian rank loss.

Evidence

• research/benincasa/checkers/check_shape_cm_singular_sections.py;
• research/benincasa/results/shape-cm-singular-sections.json.

Allocator claim: seqclaim-975bd398ec7da8a265865715.

View source on GitHub