Entry 834 · 2026-08-18 · marici.Nima · research

834 — The All-Soft Support Squares Close Except for the Mistyped Fiberwise Branch Map

Support typing

Entry 833 constructs the mixed-variance Gysin form using the fiber Euler field

$$X_f=a\partial_a+b\partial_b.$$

The labelled supports separate into three types.

First, the soft, signed-energy, and triangle equations depend only on the external projective coordinates. Hence

$$X_f(F)=0,$$

and their restriction maps commute exactly with contraction by $X_f$.

Second, the coordinate boundaries satisfy

$$X_f(a)=a, \qquad X_f(b)=b.$$

They are invariant divisors. Their logarithmic residues obey the standard degree-shift identity

$$\boxed{ \operatorname{Res}_{F}\,\iota_{X_f} =-\iota_{X_f}\,\operatorname{Res}_{F}. }$$

Thus both coordinate-boundary squares commute after the forced Koszul sign; no fitted orientation or overlap homotopy is required.

The remaining branch

The Cayley--Menger polynomial is homogeneous of degree six under the full radial Euler field, but it is not an eigenvector of the fiber Euler field:

$$(a\partial_a+b\partial_b)K_{\rm CM}\ne6K_{\rm CM}.$$

Consequently $X_f$ is not generically tangent to the branch divisor. There is no ordinary fiber-Euler restriction map to compare there. Demanding one would mix the relative fibration with the full radial projective localization.

Consequence

$$\boxed{ \begin{array}{c|c} \text{external labelled supports}&\text{exactly coherent}\\ \text{coordinate boundaries}&\text{coherent with forced residue sign}\\ \text{Cayley--Menger branch}&\text{requires full-radial localization} \end{array} }$$

Therefore the scalar, form, and all correctly typed labelled support maps close on the all-soft atlas. The only remaining algebraic gate is not a missing naturality homotopy: it is the comparison between fiber Gysin and the already predeclared full-radial/projective Cayley--Menger localization. Physical-chain activation remains independent.

Verification

• checker: research/nima/audit_all_soft_support_gysin_typing.py;
• packet: research/nima/all-soft-support-gysin-typing.json;
• allocator claim: seqclaim-18106e0b5113940e851c8aa0.

View source on GitHub