Finite positive prefixes sew to minus one

Exact statement

Every finite prefix of $1^\infty$ has positive integer realizations, but the coherent least residues $2^k-1$ sew to $-1\in\mathbb Z_2\setminus\mathbb N_0$ .

StatusExact

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 residue tower is explicit and unbounded.

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

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

Open work

Require bounded least representatives when the target doctrine is the nonnegative integers.