Parallel-edge collapse loses monodromy

Exact statement

The half-turn quotient of $C_4$ has two parallel edge orbits with different voltages; collapsing them to a simple $K_2$ erases whether the lift is one 4-cycle or two 2-cycles.

StatusProved and reproduced

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

The independent audit verifies both voltages and lift decompositions.

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

No novelty or specialist-review claim is made; the statement is limited to the displayed presentation and hypotheses.

Open work

Retain darts or parallel edge orbits whenever cycle lifting matters.