Primitive parity-necklace census

Exact statement

Among 8,800 primitive parity necklaces through period sixteen, exactly five have integral fixed lifts; the positive graph induced on $\{1,\ldots,100000\}$ contains only the $1\leftrightarrow2$ cycle.

StatusReproduced

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 independent guard census reproduces every pinned count.

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 bounded scan excludes neither larger positive cycles nor other representations.

Open work

Do not enlarge the search without a new structural criterion.