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Theorem · interface separation

FPRD-QA-MG-T02

Reachability and walk-count quotients differ

Exact statement

The simple orbit quotient preserves orbit reachability, while equitable quotient matrices preserve class-level walk counts; neither statement implies unique lifting.

StatusProved
External reviewNo documented external or specialist review of this Quotient Arsenal result is recorded.

Context

Orbit reachability, walk counts, unique lifting, and Hamiltonian preservation require different receipts. Stabilizer cosets control local choices, while voltage or monodromy controls closed lifts.

Proof or evidence

Path projection and equitable recurrence are proved independently.

Move-graph lifting and monodromy review →

Verification notes

The stabilizer-coset, equitable-partition, voltage, monodromy, and occupancy arguments were reconstructed. An independent implementation checked the 4-by-4, 5-by-5, and 6-by-6 half-turn quotients, walk counts through horizon four, the C4 parallel-edge control, and both 6-by-6 voltage outcomes.

Limitations

  • Hamiltonicity is not preserved by either fact alone.

Open work

Declare the intended observation before choosing a graph quotient.