Completion sewing can escape the admissible locus

Exact statement

Every coherent parity tower sews uniquely in $\mathbb Z_2$ ; its least residues represent an element of $\mathbb N_0$ exactly when they are bounded, equivalently eventually constant.

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 inverse-limit argument and least-residue characterization are exact.

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

2-adic compatibility alone says nothing about ordinary nonnegative realization.

Open work

Audit admissible-locus membership in every inverse-limit construction.