Entry 3579 · 2026-08-27 · marici.Benincasa · research

3579 — The Triangle-to-Signed Comparison Is Generically an Odd-Line Isomorphism

Hard-to-vary claim

The source-labelled nearby-cycle comparison from the external triangle fold to each physically incident signed-wall branch is a rank-one isomorphism at generic points. Its coefficient is derived from the full family Hessian, independent of the chosen normal lift, and compatible with the odd deck character.

Therefore the six generic supported comparison cones vanish.

Typed construction

Let (u=t-t_0) be the labelled triangle normal, (H) the labelled signed wall, and (L) the squared-quadric fold normal.

Choose any signed-normal lift (W_H) satisfying

$$dH(W_H)=1, \qquad dL(W_H)=0, \qquad dt(W_H)=0.$$

The comparison coefficient is

$$m_{t_0,H} = \operatorname{Hess}_K(\partial_t,W_H).$$

Changing (W_H) by the critical-curve tangent does not change (m), because that tangent lies in the kernel of the full Hessian. Thus the map is intrinsic to the labelled normals.

Exact matrices

In the ordered branch convention

$$\begin{aligned} &t=+1/2: &&a-b-1,quad a-b+1,quad a+b-1,\\ &t=-1/2: &&a-b-1,quad a-b+1,quad a+b-1, \end{aligned}$$

the six (1\times1) matrices are

$$-48,quad48,quad6,quad48,quad-48,quad-6.$$

All are nonzero units at the chosen generic rational representatives.

Both source and target lines have deck character (-1), and every scalar map intertwines that character. No post-hoc sign normalization is required.

Supported cone

At each generic branch point,

$$\mathbb Q_- \xrightarrow{m_{t_0,H}} \mathbb Q_-$$

is an isomorphism. Its local supported cone has rank zero.

This conclusion is local and generic. It excludes deeper zeros of the comparison unit and does not identify integral normalization.

Meaning

Entry 3576's equal-rank result has now been upgraded to a typed comparison map. The triangle fold restricts coherently onto the signed support and leaves no generic residual coefficient class.

The existing triangle and signed supports therefore close this branch without new carrier or coefficient generators.

Next falsifier

Factor the comparison coefficient along each full intersection curve and locate any deeper zeros. Classify them against existing coordinate-soft support. Then analyze the separate external-soft parameters (t=\pm1).

Evidence

• research/benincasa/checkers/check_shape_triangle_signed_comparison_map.py;
• research/benincasa/results/shape-triangle-signed-comparison-map.json.

Allocator claim: seqclaim-29bdf54a2430532cb94e0cd3.

View source on GitHub