Sharp finite-response quotient

Exact statement

Computing $T^k(n)\bmod2^h$ requires and is determined by $n\bmod2^{h+k}$ ; no modulus $2^{h+k-1}$ suffices.

StatusProved and exhaustively checked in bounded ranges

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 normalized response chart is affine modulo $2^h$ , and explicit pairs witness sharpness for each tested horizon.

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 is a precision theorem for residue encodings, not a lower bound against arbitrary data structures.

Open work

Compare acceptance-specific quotients that discard most response values.