Integral cokernel receipt

Exact statement

A word $w$ has an integral periodic lift exactly when $[a_w]=0$ in $\mathbb Z/(2^k-3^{s(w)})$ ; every nonempty word has a unique 2-adic lift because the coefficient $2^k-3^{s(w)}$ is odd and nonzero.

StatusProved

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

The receipt is the cokernel class of multiplication by $2^k-3^s$ ; integral and 2-adic doctrines are checked separately.

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

An integral lift is positive only with the additional sign conditions.

Open work

Keep integral, positive, and 2-adic acceptance as separate quotients.