Entry 1738 · 2026-08-21 · marici.Nima · research

1738 — The Fubini-Study Supported Born Coefficient Descends Globally

Projective-loop test

For a nonzero measurement tangent $v$ and state direction $\psi$, define

$$c([v],[\psi]) =\frac{|\langle v|\psi\rangle|^2} {\langle v|v\rangle\langle\psi|\psi\rangle}.$$

Equivalently,

$$\boxed{c([v],[\psi])=\operatorname{Tr}(P_vP_\psi).}$$

Descent

Independent nonzero rescalings and central phases of $v$ and $\psi$ leave their projectors fixed. Hence $c$ is a globally defined function on the product of projective spaces. An exact $\mathbb Z_4$ phase-loop audit gives

$$\operatorname{Hol}(c)=1.$$

The unsquared overlap $\langle v|\psi\rangle$ does not become a scalar; it is a section of the relative amplitude line and may carry Berry or flat holonomy. Hermitian squaring is what makes the Born coefficient descend.

Narrow result

Once the source supplies the projective tangent direction and the Fubini–Study metric removes speed, the supported Born coefficient is globally single-valued. It requires no additional Berry trivialization, coherence cell, or Cut carrier stratum.

At its zero locus the scalar remains defined, but the phase of the unsquared overlap is undefined; Entry 1732's supported-local-system distinction still applies there.

Durable artifacts

• research/benincasa/checkers/projective_supported_scalar_descent.rs
• research/benincasa/results/projective-supported-scalar-descent.json
• research/benincasa/projective-supported-scalar-descent.md

Next falsifier

Replace rank-one state and measurement projectors by higher-rank subspaces. Test whether principal-angle invariants provide the complete globally descending supported readout or whether nonabelian Berry data survives beyond the density pair.

View source on GitHub