# LIB2-084 — The mirror ladder reproduces an NS/R parity pattern, not a spin structure

- class: CLAIM · review_state: PUBLISHED · house_tier: not yet assigned
- labels as printed: math_status NOT_PRINTED · attachment NOT_PRINTED · scorecard tier word none (PDF p.269)
- basis: {"pdf_sha256_prefix": "b64ddcd9e16a6dcd", "release": "Rev33.1_S369"}

## Statement

The signed trace over the banked mirror/Lucas ladder reproduces the antiperiodic NS and periodic R parity pattern of the three-torus sector, with the Freudenthal bit playing the holonomy-sign role.

## Caution

The parity pattern does not construct a spin structure, a principal Spin bundle or a lift of a frame bundle.

## Primary source

Appendix P, §2, Theorem 2.3: signed-trace pattern, PDF p.284: "The signed trace over the banked mirror (Lucas ±) ladder reproduces the antiperiodic (NS) / periodic (R) parity pattern of the three-torus sector TS3 , with the Freudenthal bit playing the role of the holonomy sign."

## Machine

- url: /library-v2/LIB2-084/
- content_sha256: 1b546a35f70d5947c5925560f5873506e8c8621457238f6623ddd3d9e5c63f75
- candidate_ids: C2-027

## Freshness

- BUILD_STAMP S371a · 2026-09-26T22:53Z · master 01d1a5873ede42f5 · cut 9dcc6ce9a8ee3e02
- Declared volatile fields (R36): BUILD_STAMP