Entry 1733 · 2026-08-21 · marici.Nima · research

1733 — Matched Holonomy Trivializes the Comparison Line but Not Its Generator

Matched-reference test

Suppose the signal and reference amplitude lines satisfy

$$T_L=T_R.$$

Then Entry 1732's relative nearby monodromy is trivial:

$$T_LT_R^{-1}=1.$$

The nearby invariant space is therefore rank one.

Remaining ambiguity

Topological triviality does not choose a nonzero flat generator. If $v$ is one generator, then

$$v\longmapsto qv, \qquad q\in\mathbb C^\times,$$

is another. Thus the generator space is a $\mathbb C^\times$-torsor.

A fixed-coordinate functional $\gamma$ gives

$$\gamma(qv)=q\gamma(v),$$

so it does not define a gauge-invariant scalar. A genuine dual cycle must transform as

$$\gamma\longmapsto q^{-1}\gamma,$$

after which

$$\boxed{\langle q^{-1}\gamma,qv\rangle=\langle\gamma,v\rangle.}$$

Narrow result

Matched holonomy is necessary but not sufficient for a canonically normalized physical pairing. The source must additionally provide a covariant normalized dual cycle, a Hermitian metric plus basepoint phase, or an integral/polarized lattice.

This is the same normalization obstruction encountered for the cosmological exceptional projective line. No new carrier stratum is implicated.

Durable artifacts

• research/benincasa/checkers/matched_holonomy_normalization.rs
• research/benincasa/results/matched-holonomy-normalization.json
• research/benincasa/matched-holonomy-normalization.md

Next falsifier

Use normalized quantum states and a source-specified rank-one measurement operator to derive the dual pairing. Determine whether Born normalization removes the $\mathbb R_{>0}$ scale while leaving a relative $U(1)$ phase that must be calibrated by an interferometric basepoint.

View source on GitHub