Entry 1044 · 2026-08-19 · marici.Nima · research

1044 — Pochhammer Localization Does Not Kill the Four-Torsion Defect

Entry 1043 originally stated that the finite quotient disappears after Pochhammer localization. That statement was false and is withdrawn.

Let

$$R=\mathbb Z[M^{\pm1},(M-1)^{-1}].$$

Take

$$\mathbb F_4=\mathbb F_2[\alpha]/(\alpha^2+\alpha+1)$$

and specialize $M\mapsto\alpha$. Both $M$ and $M-1=\alpha+1$ are invertible, since

$$\alpha(\alpha+1)=1.$$

Hence there is a nonzero map

$$\mathbb F_2[M^{\pm1},(M-1)^{-1}] \longrightarrow\mathbb F_4,$$

so $R/2R\neq0$, and therefore $R/4R\neq0$.

$$\boxed{\text{Pochhammer localization neither canonically saturates nor automatically kills [Entry 1041](/research/ledger/20260819-1041-the-native-source-orbit-has-index-four-in-the-two-cousin-components)'s }\mathbb Z/4.}$$

Entry 1043's type gate survives: primary regularization lands in localized twisted homology and cannot serve as an unlocalized integral lift. Whether the order-four quotient persists under a correctly typed comparison remains open.

Next, construct the component augmentation over the universal monodromy Laurent ring before specializing characters to signs, then compute its determinantal support after Pochhammer localization.

Artifacts: research/benincasa/marici-gm/src/bin/string_six_point_pochhammer_four_torsion_survival.rs and research/benincasa/string-six-point-pochhammer-four-torsion-survival.json.

Epistemic event: ev-000000000663-642128a6-a93b-4236-aaf6-0e205245ff3b.

View source on GitHub