Entry 1736 · 2026-08-21 · marici.Nima · research

1736 — Fubini–Study Normalization Removes Measurement-Jet Speed but Not Direction

Metric-normalization test

At the limiting measurement line $[e_0]$, write a transverse tangent as

$$v=ae_1+be_2.$$

The source-derived Fubini–Study metric gives

$$\|v\|_{\rm FS}^2=a^2+b^2$$

in this affine frame. For the state $|\psi\rangle=e_1$, Entry 1735's supported second coefficient becomes, after unit-speed normalization,

$$\boxed{ c_{\rm FS}([v]) =\frac{|\langle v|\psi\rangle|^2}{\langle v|v\rangle} =\frac{a^2}{a^2+b^2}. }$$

Invariances

For every nonzero scalar $q$,

$$c_{\rm FS}([qv])=c_{\rm FS}([v]).$$

Central phase also cancels between numerator and denominator. Thus the metric removes both parameter-speed rescaling and amplitude gauge.

The value still varies with $[a:b]$. Different projective tangent directions at the same normalized endpoint produce different supported Born coefficients.

Narrow result

Fubini–Study geometry canonically removes the radial measurement-jet ambiguity but cannot infer the projective tangent from endpoint data. A physical supported limit therefore requires:

1. the source measurement path, which supplies $[v]$;
2. the Fubini–Study metric, which fixes its speed normalization;
3. the normalized state, which evaluates the directional coefficient.

This is a typed coefficient/readout construction over the existing carrier; no new Cut stratum is needed.

Durable artifacts

• research/benincasa/checkers/fubini_study_tangent_normalization.rs
• research/benincasa/results/fubini-study-tangent-normalization.json
• research/benincasa/fubini-study-tangent-normalization.md

Next falsifier

Move around a loop in projective measurement directions and test whether the Fubini–Study-normalized supported coefficient has Berry holonomy or descends as a scalar function on projective tangent space.

View source on GitHub