Fixed-weight residue filtration

Exact statement

An adjacent rewrite $x01y\to x10y$ changes the least residue by a term of exact 2-adic valuation equal to the swap position; the depth- $h$ receipt count is a truncated binomial sum.

StatusProved and exhaustively checked through length fourteen

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 checker covers 32,766 words, 98,305 swaps, and 65,658 fundamental-cycle sums.

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

On the fine graph the label is an exact coboundary; after slope collapse it is vertical loss, not automatically quotient cohomology.

Open work

Use the receipt count as a loss bound only for the pinned fixed-weight quotient.