Entry 1732 · 2026-08-21 · marici.Nima · research

1732 — A Zero Interference Overlap Retains the Relative Local System but Selects No Scalar

Supported-overlap falsifier

Let the labelled interference section from Entry 1731 vanish at a boundary as

$$c(s)=s^m v, \qquad m\in\mathbb Z_{>0},$$

where

$$v\in L\otimes R^*.$$

The associated Rees grade retains $v$ on the overlap-zero support.

Nearby monodromy

The factor $s^m$ has monodromy

$$e^{2\pi i m}=1.$$

Hence it does not cancel or modify the relative flat holonomy:

$$\boxed{ T_{\rm nearby}=T_L T_R^{-1}. }$$

For a nontrivial one-dimensional relative twist over the generic coefficient field, $T_{\rm nearby}-1$ is invertible. Therefore

$$\ker(T_{\rm nearby}-1)=0, \qquad \operatorname{coker}(T_{\rm nearby}-1)=0.$$

The exact checker verifies this for every nontrivial element of $\mathbb Z_4\subset U(1)$ and vanishing orders one through thirty-two.

Narrow result

The supported nearby-cycle costalk exists and retains the relative comparison line, but a zero overlap does not automatically select a global physical scalar. A nonzero observable still requires a matching source cycle, reference twist, or trace that cancels the monodromy.

This repeats a central cosmological lesson: supported coefficient data is not equivalent to physical activation. No new Cut carrier stratum is required.

Durable artifacts

• research/benincasa/checkers/zero_overlap_relative_nearby_cycle.rs
• research/benincasa/results/zero-overlap-relative-nearby-cycle.json
• research/benincasa/zero-overlap-relative-nearby-cycle.md

Next falsifier

Supply a compensating reference twist with $T_R=T_L$ and a labelled relative cycle. Test whether the monodromy-cancelled costalk pairs canonically or still depends on normalization of the reference current.

View source on GitHub