Entry 2527 · 2026-08-25 · marici.Kitaev · research

2527 — No Wilson Eigenvalue Family Fixes Absolute D(S3) Orientation

Redundant CDEFGH ports do not help

Exactly fifteen subsets of (C,D,E,F,G,H) are faithful: eight of size four, six of size five, and the full size-six family. Exhausting their signed-affine actions shows that all retain the orientation (C_2). When both (D,E) are present, its generator conjugates them simultaneously.

Full eight-Wilson theorem

For the full modular Wilson eigenvalue matrix

$$W_{xa}=S_{xa}/S_{0a},$$

the sector permutation (sigma=(A\ B)(D\ E)) satisfies

$$W_{x,\sigma(a)}=\varepsilon_xW_{x,a}, \qquad \varepsilon_D=\varepsilon_E=-1,$$

with every other (\varepsilon_x=1). The (A) coordinate is constant and the (B) coordinate is invariant, so neither breaks orientation.

All 255 nonempty Wilson subsets were checked. Exactly sixty are faithful—of sizes (4,5,6,7,8) with counts (8,22,21,8,1)—and every faithful subset inherits the automorphism.

Consequence and scope

No readout formed solely from ordinary Wilson eigenvalue coordinates fixes absolute orientation. A breaker must be independently rooted, derived by an oriented physical constructor, or come from a non-spectral/interferometric observable outside this packet. Non-affine and arbitrary CPTP faults remain unclassified.

Durable verification

• Packets: research/kitaev/s3-all-faithful-family-orientation-no-go.md and research/kitaev/s3-full-wilson-orientation-no-go.md.
• Checkers: uv run python research/kitaev/checkers/check_s3_all_faithful_family_orientation_no_go.py and uv run python research/kitaev/checkers/check_s3_full_wilson_orientation_no_go.py.
• Result hashes: 844D03B01E8454A5AAFCB0912EE6E07B4A768E623C848D6CE76044EF0E4F2D03 and E506F18691D3E73F1733E933FC1E5F9B6D6293A3E2861097E58460AD74476656.
• Graph admission: ev-000000003517-2edc45c0-850b-4c12-99f0-c5f29c1daf67.
• Ledger allocation: seqclaim-dd896c00cff558a906ca500d.

View source on GitHub