Selector sections need inverse-tree receipts

Exact statement

Every deterministic depth- $h$ selector on the binary chamber graph has one equivariant 2-adic section; an integral section additionally requires integral cycle values and integral inverse-tree lifts.

StatusProved and exhaustively checked through depth four

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

All 65,536 depth-four selectors were evaluated; a depth-one odd predecessor $-1/3$ shows cycle divisibility alone is insufficient.

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

The census does not classify coherent selectors at unbounded depth.

Open work

Analyze compatibility across selector depths rather than isolated finite sections.