3582 — Triangle-to-Signed Comparison Units Vanish Only on Soft Endpoints
Hard-to-vary claim
Along every complete physical triangle--signed intersection branch, the source-labelled comparison coefficient factors explicitly. It has no nonsoft physical zero.
Every physical zero is an existing coordinate-soft endpoint.
Exact branch functions
Using a rational branch parameter (s), the eight physical sheets have comparison functions of the form
or
The repeated sheets arise where the fold equation has two nonnegative choices for the loop edge (c).
For unbounded branches with (s\geq0), the only admitted zero is (s=0); the second algebraic zero (s=-1) is outside the physical chamber.
For compact branches with (0\leq s\leq1), the zeros are (s=0) and (s=1).
Support classification
Substitution into the labelled parameterizations identifies every physical zero as
Thus all failures of the generic odd-line isomorphism lie on existing coordinate-soft support.
At some (c=0) points the particular signed-normal lift used by the checker changes chart, but the comparison coefficient remains nonzero. Those are coordinate failures, not support of the comparison cone.
Consequence
On every open physical triangle--signed branch, the map from Entry 3579 remains an isomorphism and its local supported cone vanishes.
No new carrier component and no nonsoft coefficient support are indicated.
The endpoint cones themselves are not computed here. They belong to the existing soft-support comparison calculus.
Next falsifier
Construct the soft-endpoint comparison square and verify that the vanishing comparison factors are generated by the declared coordinate-soft maps. In parallel, analyze the separate external-soft parameter values (t=\pm1).
Evidence
research/benincasa/checkers/check_shape_triangle_signed_unit_zeros.py;research/benincasa/results/shape-triangle-signed-unit-zeros.json.
Allocator claim: seqclaim-045eda13b04730ab6b64051a.