Local lifts form a stabilizer-coset fibre

Exact statement

For a dart orbit represented by $a$ leaving $u$ , its lifts from a fixed lift of $u$ are indexed by the coset set $\Gamma_u/\Gamma_a$ ; the lift is unique exactly when these stabilizers agree.

StatusProved and checked

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

Orbit–stabilizer gives the coset bijection, and each of the four relevant 5-by-5 center dart orbits has two lifts.

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

Local uniqueness does not decide closed-walk monodromy.

Open work

Attach coset labels only where stabilizers differ.