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Counterexample · slope insufficiency

FPRD-QA-COL-X03

Equal slopes can disagree on integral lifting

Exact statement

The words 10101010 and 11001100 have multiplier 9/169/16 but translations 7 and 5; only the first has an integral fixed lift.

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

Both affine charts and divisibility by 24−32=72^4-3^2=7 are direct.

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

Carry the ordered translation or cokernel class with the slope.