Origami expanders

The paper proves Theorem A—on every origami surface Σ there is a Lipschitz, measure‑preserving F2-action with spectral gap—by (i) establishing spectral gap for the torus action generated by ak,bk (Theorem 5.3), (ii) constructing an explicit equivariant action on Σ, and (iii) lifting measure expansion through the finite branched covering via Theorem 4.1 after verifying ergodicity (Theorem 5.7) . The candidate solution hinges on composing affine lifts with “deck translations” so that the induced permutations act transitively on fibers and uniformly gap the fiber component. This is generally false: for a non‑regular branched cover the deck group need not act transitively (and may be trivial), and the ‘induced permutation of m sheets’ is not a single global permutation independent of the basepoint. Thus the model’s Step 2 (and the resulting uniform fiber gap claim) fails, while the paper’s extension argument does not require such an assumption.