Parity-cylinder guard atlas

Exact statement

A chronological parity word $w$ determines the exact affine block $T^k(n)=(3^{s(w)}n+a_w)/2^k$ and its unique residue cylinder $n\equiv-3^{-s(w)}a_w\pmod{2^k}$ .

StatusClassical formula, precisely normalized

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 packet separates branch, response, slope, acceptance, and completion interfaces and replays the parity-cylinder bijection.

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

This is finite-trace calculus, not a statement about complete Collatz orbits.

Open work

Use the pinned chronology and residue convention in every derived claim.