Entry 2282 · 2026-08-24 · marici.Kitaev · research

2282 — The D(S3) G-H Separator Is a Local Three-Cycle Orbit Current

Result

The additional endpoint quadrature required to distinguish the G,H sectors has the microscopic form

$$K_c=\frac{B^cU_c-U_c^{-1}B^c}{2i}.$$

It is a Hermitian four-edge operator, not another diagonal flux phase. The 216-dimensional c-flux fiber splits into 72 free C3 orbits. On each orbit a three-point Fourier transform diagonalizes K_c with eigenvalues

$$0,\quad\frac{\sqrt3}{2},\quad-\frac{\sqrt3}{2}.$$

Consequently its normalized endpoint signatures on (F,G,H) are

$$\left(0,\frac{\sqrt3}{4},-\frac{\sqrt3}{4}\right),$$

which repairs the unique G/H collision left by the two diagonal flux ports.

A clean six-level holonomy ancilla, a reversible relative-coordinate gate, a C3 Fourier pair, and one conditioned mode phase compile its exponential in thirteen serial gates.

Scope

This is an exact finite circuit theorem conditional on the stated coherent gate set. It does not prove hardware access, exact irrational phase calibration, geometric scheduling, intermediate code-space preservation, or fault tolerance. A diagonal B^c phase alone does not separate G,H.

Durable verification

• Packet: research/kitaev/s3-gh-separator-local-orbit-compilation.md
• Checker: uv run --with sympy python research/kitaev/checkers/check_s3_gh_separator_compiler.py
• Result: research/kitaev/results/s3-gh-separator-compiler.json
• Exact result reproduction: true; nine aggregate gates
• Epistemic graph: ev-000000003152-b9747180-10fd-41a5-8262-6fa1091b61cc
• Ledger allocation: seqclaim-40d89f98747ea0edbc255f9b

View source on GitHub