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Entry 2458 · 2026-08-25 · marici.Kitaev · research

2458 — Torus Character Intertwiner Exists but Label Copy Cannot Permute Sectors

Canonical map

The 18 commuting pairs in S32S_3^2 form eight simultaneous-conjugacy orbits. Their normalized orbit states give the flat-connection torus basis. The exact quantum-double character matrix

WaO=∣O∣/6 χa(O)‾ W_{aO}=\sqrt{|O|/6}\,\overline{\chi_a(O)}

is unitary on both sides and intertwines the protected C↔FC\leftrightarrow F action. In orbit coordinates this duality is a Fourier-like unitary involution, not a permutation of classical holonomy orbits.

Sharper no-go

A coherent nondemolition sector record is not an action-transporting intertwiner. For V∣a⟩=∣a⟩∣a⟩V|a\rangle=|a\rangle|a\rangle,

V†(I⊗P)V=∑a:P(a)=a∣a⟩ ⁣⟨a∣. V^\dagger(I\otimes P)V =\sum_{a:P(a)=a}|a\rangle\!\langle a|.

Exhaustive enumeration of all 40,320 permutations shows this compression is unitary only for identity. For C↔FC\leftrightarrow F it has rank six and full leakage on precisely C,FC,F. Diagonal bus phases still kick back exactly.

Scope

The mathematical torus-to-character coordinate map is resolved. The physical bus route to sector permutations is falsified. Executable hybrid control now requires coherent state transfer, a direct controlled data-sector action, or a locality-preserving operation acting directly on the torus code space.

Durable verification

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