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Counterexample · completion mismatch

FPRD-QA-COL-X01

A nonzero integral obstruction killed 2-adically

Exact statement

For w=001w=001, the integral receipt is 4 mod 54\bmod5, so no integral lift exists, while the unique 2-adic lift is 4/54/5.

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

Direct substitution verifies both the nonzero cokernel class and 2-adic solvability.

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 concerns one periodic parity word, not global Collatz behavior.

Open work

Retain as the smallest completion-versus-integrality fixture.