Exact guard seam

Exact statement

For $w=uv$ , $a_{uv}=3^{s(v)}a_u+2^{|u|}a_v$ , and a unique tail coordinate gives $r_{uv}=r_u+2^{|u|}c_{u\mid v}$ .

StatusProved and 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

Block composition and 4,284 seam instances pass exact integer replay.

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

Treat the seam coordinate as part of the chart, not informal compatibility.