3576 — Triangle–Signed Intersections Have No Transverse Rank Excess
Hard-to-vary claim
At every physically incident intersection of the two external triangle branches with the signed loop-wall arrangement, the complete Cayley--Menger family is Morse--Bott with a one-dimensional critical curve and a nondegenerate three-dimensional normal Hessian.
The transverse vanishing rank remains one. No additional transverse coefficient direction appears.
This is a rank and local-type theorem. The supported comparison map itself has not yet been constructed.
Labelled physical branches
At each of
three signed walls meet the positive loop-edge chamber:
The fourth factor
has no positive-loop-edge incidence.
One exact positive rational point was tested on each of the six branches.
Morse--Bott calculation
For the complete function
the (4\times4) Hessian at every tested branch point has rank three. Its one-dimensional kernel annihilates:
- the triangle parameter normal;
- the signed-wall normal;
- the squared-quadric fold normal.
Hence the kernel is tangent to the one-dimensional support intersection, and the normal Hessian is nondegenerate.
The generic triangle fold has transverse rank one before restriction. The signed restriction also has transverse rank one. Therefore the transverse rank excess is zero.
Epistemic boundary
Equal ranks do not establish an isomorphism of supported objects. In particular, this computation does not yet prove that the triangle-to-signed restriction map has zero cone. It proves only that no additional transverse Milnor direction is available.
Any residual comparison class would have to arise from orientation, inertia, integral lattice, or extension data—not from an uncounted vanishing cycle.
Meaning
The first hostile test favors a clean restriction of the existing triangle fold along existing signed support. It supplies no evidence for a new carrier stratum or a higher-rank coefficient object.
Next falsifier
Construct one source-labelled triangle-to-signed nearby-cycle restriction map, including orientation and deck character. Transport it across all six branches. Only then compute its supported cone.
Afterward test the separate external-soft parameters (t=\pm1).
Evidence
research/benincasa/checkers/check_shape_triangle_signed_morse_bott.py;research/benincasa/results/shape-triangle-signed-morse-bott.json.
Allocator claim: seqclaim-ecd097c50328b919af0195d9.