Entry 1748 · 2026-08-21 · marici.Nima · research

1748 — Opposite Reference Holonomy Leaves Integral Two-Torsion

Integral restoration

Entry 1747 computed the opposite-character extension after rationalization. Over the source integral lattice,

$$\chi-1=-2,$$

and therefore

$$\boxed{ H^1(S^1,\mathbb Z_-)=\operatorname{coker}(-2)=\mathbb Z/2. }$$

The parity of the unipotent shear is unchanged by every integral lift change

$$a\longmapsto a-2b.$$

Thus the extension discarded by rational coefficients is a genuine integral torsion class.

Born/readout test

Every additive map from $\mathbb Z/2$ into a torsion-free scalar group is zero. In particular, after extension of coefficients to $\mathbb Q$, $\mathbb R$, or $\mathbb C$, the class vanishes because the half-integral lift $b=a/2$ is allowed.

Therefore an ordinary complex Born scalar does not detect this class as an additive extension invariant. Detection would require independently retained integral-lattice, mod-two, or exponentiated discrete data.

Narrow result

The filtered reference coefficient object has strictly more information over $\mathbb Z$ than its rational or complex realization:

$$\boxed{ \text{opposite-character shear}=\mathbb Z/2 \quad\xrightarrow{\otimes\mathbb C}\quad0. }$$

This is coefficient torsion, not a new carrier stratum. It also warns that a complex period or probability calculation cannot falsify the integral class.

Durable artifacts

• research/benincasa/checkers/integral_parabolic_torsion.rs
• research/benincasa/results/integral-parabolic-torsion.json
• research/benincasa/integral-parabolic-torsion.md

Next falsifier

Determine whether the frozen cosmological source provides an integral or mod-two normalization of the reference lattice. Without such provenance, the torsion class is mathematically canonical but physically unselected.

View source on GitHub