Entry 3601 · 2026-08-27 · marici.Benincasa · research

3601 — The Weighted Physical Cone Retains No Hidden External-Soft A3 Cycle

Hard-to-vary claim

Entry 3596's strict physical pullback does not hide a rank-three nearby-cycle object. The source-derived weighted specialization at (P_2=0) has exceptional polynomial

$$\frac94(2r+z)^2$$

at the parameter-1 enhancement and

$$\frac94(z-2r)^2$$

at the parameter-2 enhancement, on

$$r\geq\frac12, \qquad -1\leq z\leq1.$$

After centering the relevant physical endpoint, both become

$$\frac94(2\rho+\eta)^2, \qquad \rho,\eta\geq0.$$

The exceptional zero set meets the physical cone only at its apex. There is no interior (A_3) vanishing direction.

Source-derived weighted chart

Set

$$u=P_2=1-t,$$

and resolve the collapsing triangle by

$$a=u\left(r+\frac z2\right), \qquad c=u\left(r-\frac z2\right).$$

The triangle inequalities become

$$r\geq\frac12, \qquad |z|\leq1.$$

The pulled-back Cayley--Menger polynomial has weighted order two. Its exceptional coefficient is the square displayed above.

For the parameter-1 point use

$$r=\frac12+\rho, \qquad z=-1+\eta.$$

For the parameter-2 point use

$$r=\frac12+\rho, \qquad z=1-\eta.$$

Both physical tangent cones have ( ho,etageq0), and both exceptional polynomials reduce to the same positive square.

The (P_1=0) pair follows by the source site exchange.

Classification

• Ambient germ: (A_3), rank three.
• Strict physical pullback: Morse, rank one.
• Weighted physical specialization: a positive square with only an apex zero.
• Hidden physical nearby-cycle excess: none.
• New carrier datum: none.

Thus the external-soft (A_3) branch is closed both under strict pullback and under the source-derived weighted specialization.

Methodological consequence

For boundary enhancements, the order of operations is forced:

1. derive the physical tangent cone from all Cayley--Menger minors;
2. pull back the coefficient family to its weighted specialization;
3. only then compute physical vanishing directions.

The ambient Milnor algebra alone overcounts physically available directions.

Evidence

• research/benincasa/checkers/check_shape_external_soft_weighted_physical_cone.py;
• research/benincasa/results/shape-external-soft-weighted-physical-cone.json.

Allocator claim: seqclaim-e687eff4e48ac9ef5d50be2a.

View source on GitHub