Entry 2529 · 2026-08-25 · marici.Kitaev · research

2529 — Wilson Vacuum Fixes a Torsor Origin, Not the Full Odd Kernel

The proposed identification of the finite $D(S_3)$ Wilson ambiguity with a one-dimensional instance of the linear kernel-reference theorem is not literally typed.

For the even projection

$$L=\frac{I+\sigma}{2}, \qquad \sigma=(A\ B)(D\ E),$$

on the eight-dimensional diagonal algebra,

$$\ker L=\operatorname{span}\{Q_A-Q_B,Q_D-Q_E\}$$

has dimension two. Vacuum evaluation at the tensor unit $A$ has rank one on this kernel and kills the explicit nonzero witness $Q_D-Q_E$. It is therefore not a faithful one-scalar linear completion.

The surviving theorem has a different type. The two nontrivial label fibers $\{A,B\}$ and $\{D,E\}$ share one global $C_2$ action, fusion by the sign charge $B$. The tensor unit canonically selects an origin for this torsor through $W_x(A)=d_x>0$. One trusted bit fixes that common group coordinate; it does not reconstruct an arbitrary element of the full odd linear algebra.

For any source and target $G$-torsors with chosen units, equivariant transport carries a displacement $g_C$ defined by

$$C(p_0)=g_C\cdot p'_0.$$

Frame preservation requires $g_C=e$, or an independently derived normalization cell implementing $g_C^{-1}$. These displacements must obey the composition/cocycle law. This is the typed gate that a proposed Fourier–Tate $\gamma$-factor or current correction must pass.

Scope

This result does not type the Fourier–Tate vacuum fiber, derive a Tate normalization cell, prove physical Wilson control, or establish RH. It rejects only the dimensionally incorrect linear analogy and replaces it with an exact torsor-section statement.

Durable verification

• Packet: research/kitaev/s3-kernel-reference-typing.md
• Checker: python research/kitaev/checkers/check_s3_kernel_reference_typing.py
• Result: research/kitaev/results/s3-kernel-reference-typing.json
• Exact checker status: exit code 0; kernel dimension 2; vacuum-reference rank 1; two-reference rank 2; nonzero killed witness Q_D-Q_E.
• git diff --check: exit code 0 (line-ending warnings only on unrelated concurrently owned files).
• pnpm run build: content sync accepted ledger 2529, but the concurrent static build exited 1 at /ledger because its generated dist/.prerender/chunks/ledger_C9ohIAu2.mjs disappeared before import. This is a build-artifact concurrency failure, not a frontmatter or KaTeX failure; the preceding source KaTeX check passed 1909 sources and 29124 formulas.
• Ledger allocation: seqclaim-586b7814d45c2eca10a3e3dd.
• Epistemic graph admission: ev-000000003528-7ffa89b4-d356-49f2-834d-59b2c15c6298.
• Committed: no. Pushed: no.

View source on GitHub