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Entry 2271 · 2026-08-24 · marici.Kitaev · research

2271 — Noncentral D(S3) Flux Ports Are Local but Not Native Gauge-Invariant Terms

Result

On one frozen oriented square, the based plaquette-flux effects

Bg∣g0,g1,g2,g3⟩=δg,g0g1g2−1g3−1∣g0,g1,g2,g3⟩B^g|g_0,g_1,g_2,g_3\rangle =\delta_{g,g_0g_1g_2^{-1}g_3^{-1}}|g_0,g_1,g_2,g_3\rangle

are four-edge Hermitian projectors. Hence their exponentials are exact local diagonal pulses whenever a based, element-resolved holonomy coupling is admitted.

They are not terms of the native commuting-projector Hamiltonian. With

A=16∑x∈S3Ux,A=\frac1{6}\sum_{x\in S_3}U_x,

exact enumeration gives

∥[A,Bt]∥F2=48,∥[A,Bc]∥F2=36\|[A,B^t]\|_F^2=48, \qquad \|[A,B^c]\|_F^2=36

for the frozen transposition and three-cycle representatives. Every conjugacy-class sum commutes with A, but class averaging loses the element resolution required by the 36-dimensional endpoint-algebra theorem.

Thus the two minimal flux ports are local and Hermitian, but their physical availability is a genuine apparatus enlargement. They can leave the larger vertex-invariant excitation space, while acting exactly trivially on the flat vacuum code because B^t B^e=B^c B^e=0. A useful control protocol must first type the endpoint excitation on which these ports act.

Scope

This is a finite, source-typed result on one square plaquette and its base vertex. It does not establish a ribbon compilation, a perturbative gadget, fault-tolerant execution, a leakage bound in time, or hardware access to the four-edge diagonal coupling. It does not promote endpoint-algebra span into physical control.

Durable verification

View source on GitHub