Entry 2528 · 2026-08-25 · marici.Kitaev · research

2528 — Wilson Sign Ambiguity Is a B-Simple-Current Torsor

Exact identification

Fusion with the invertible sign charge acts by

$$B\otimes-=(A\ B)(D\ E),$$

exactly the hidden Wilson permutation. Wilson characters satisfy

$$W_x(B\otimes a)=\chi_B(x)W_x(a),$$

with (chi_B(D)=chi_B(E)=-1) and all other values (+1). The unsigned Wilson packet is therefore a two-sheet (C_2) torsor under simple-current translation.

This is not categorical gauge. It moves the tensor unit (A) to (B), violates 32 fusion coefficients, fails modular-(S) relabelling invariance, and exchanges the unequal twists of (D,E). The smallest witness is

$$A\otimes A=A,\qquad B\otimes B=A\ne B.$$

Complete odd algebra and canonical origin

The diagonal algebra splits into even and odd dimensions (6+2). Its entire odd part is already spanned by (W_D,W_E):

$$Q_A-Q_B=\frac{W_D+W_E}{6},\qquad Q_D-Q_E=\frac{W_D-W_E}{2}.$$

The tensor unit canonically fixes the sign through

$$W_D(A)=W_E(A)=3=d_D=d_E>0.$$

Thus no new observable is mathematically required. A trusted sign for one (D/E) port fixes any minimum family.

Terminology and operational bottom

This sheet is not geometric ribbon reversal. Ribbon reversal acts by group inversion and exchanges (G,H) while fixing (D,E). The two involutions commute and generate (C_2\times C_2). The precise term is (B)-simple-current sheet or signed Wilson (D/E) frame.

Source data already distinguish (A=(e,\mathrm{triv})), (B=(e,\mathrm{sign})), and oriented ribbon multiplication. What remains unproved is the sign-preserving executable arrow

$$\text{oriented ribbon/character normalization} \to \text{controlled Wilson quarter evolution} \to \text{ququart/binary interface}.$$

Closed-ribbon insertion is not the controlled unitary. Local torus syndrome does not prepare the tensor-unit logical sector. No fault-tolerant calibration or interface sign-preservation theorem is claimed.

Durable verification

• Packets: research/kitaev/s3-wilson-orientation-simple-current-torsor.md, research/kitaev/s3-wilson-orientation-odd-algebra.md, and research/kitaev/s3-simple-current-sheet-vs-ribbon-reversal.md.
• Corresponding checkers use the same stems under research/kitaev/checkers/.
• Result hashes: 553DB7BE3D887F4A1661088AC248C423A48A697BDC77BA22EA5A49E2B1371008, 2DDE42F63E11CF95A01A80C62916E83C4B818A4C822EFD293091452826ADEE21, and 528BCCBB410B01F3E4EF250B2285EE6074FFC127ADDEE024D55CC8568BD62E2A.
• Graph admission: ev-000000003525-7ca24ec1-40df-40ea-8433-5c004b63c0b2.
• Ledger allocation: seqclaim-62fa62bf5d4a09b26ef2ee97.

View source on GitHub