Entry 1747 · 2026-08-21 · marici.Nima · research

1747 — Unipotent Reference Holonomy Survives Only in the Matched Character

Extension-class test

Entry 1746 gives graded reference lines $L_1,L_2$. Their relative Hom line has character

$$\chi=\sigma_1\sigma_2.$$

A unipotent parabolic loop is represented by a shear cocycle $a$. Changing the local lift of the quotient line by $b$ changes it by

$$a\longmapsto a+(\chi-1)b.$$

Hence the intrinsic extension space is

$$\boxed{ H^1(S^1,\operatorname{Hom}(L_2,L_1)) =\operatorname{coker}(\chi-1). }$$

For matched line holonomies, $\chi=1$, every lift change leaves $a$ unchanged and the extension space has rank one. The unipotent shear is then a genuine filtered-local-system invariant.

For opposite line holonomies, $\chi=-1$, the map $\chi-1=-2$ is invertible over $\mathbb Q$. The change $b=a/2$ removes every shear, so the extension space vanishes rationally.

Narrow result

$$\boxed{ \text{A parabolic shear is intrinsic exactly in the trivial relative character.} }$$

Thus repeated-weight singular references can carry information beyond their associated graded, but only as a sector-specific coefficient extension. No new Cut carrier stratum is required.

The rational qualification matters: with an integral lattice, the opposite character may leave a $\mathbb Z/2$ torsion class. That has not been tested.

Durable artifacts

• research/benincasa/checkers/unipotent_parabolic_extension.rs
• research/benincasa/results/unipotent-parabolic-extension.json
• research/benincasa/unipotent-parabolic-extension.md

Next falsifier

Restore the integral reference lattice. Determine whether the opposite relative character carries the predicted $\mathbb Z/2$ extension and whether the physical Born/readout pairing detects or annihilates it.

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