Rotation and repetition transport receipts

Exact statement

Cyclic rotation transports the integral cokernel receipt by a unit, while repetition injects the original cokernel and preserves integrality and positivity.

StatusProved and extensively checked

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

Context

Parity words are exact affine charts. Their seams, response quotients, integral cokernels, and 2-adic inverse limits show precisely where finite compatibility differs from membership in the ordinary nonnegative integers.

Proof or evidence

Exact arithmetic verifies 81,924 rotations and 10,230 repetitions.

Collatz guard-chamber review and independent audit →

Verification notes

The affine word, residue-cylinder, seam, response, integral, positive, 2-adic, selector, rotation, and fixed-weight arguments were reconstructed. An independent implementation checked 131,070 parity words, 98,304 seams, 98,305 swaps, 8,800 primitive necklaces, and all selectors through depth four, including the 001 and all-ones hostile examples.

Limitations

Rotation changes based phase; repetition is a nonprimitive spelling of the same fixed point.

Open work

Use primitive cyclic classes when counting distinct periodic presentations.