[SHIN09-git]

Change and mathematical scope

Balanced correct-word counts cannot always be divided by the word length to count cyclic schedules. This package proves the exact repetition and rotation-period law for the unsigned depth-three dirty chain, for every modulus. At (m,c)=(3,1), the known 24 length-nine words form four cyclic classes: two of size nine and two of size three.

For a unit target, every nonprimitive balanced correct word repeats exactly three times. Such words exist precisely when m=3n, 3∤n, and c≡n³ (mod 3), except at (6,5) and (12,1). The sufficiency direction uses the credited predecessor balanced-existence theorem. General fixed-point and Möbius formulas give every least-period population and the exact cyclic count from W(m,c).

Only research/chain-cyclic-schedules/ is added. This is a scoped proof with finite exact corroboration, with no new κ, supplier, paid-profile change or measured speedup. Rotation equivalence concerns the universal endpoint; routing, precision and word-dependent physical costs remain separate. This does not prove #6037's all-modulus distinctness conjecture or a closed formula for W.

Proof and boundaries

Task and overlap

Canonical base: 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8. The original connector-alphabet-change card is retained verbatim, with scope_status=changed to a counting side question in the unchanged alphabet. The result supplies exact rotational deduplication and divisibility checks; it does not supply a favorable completed budget.

Nearest work: Nick Petalas's #6037 counts literal words; #4257 classifies depth-three profiles and tariffs; #4281 already observes rotation closure; #4748 classifies one binary depth-six rotation class. The all-modulus depth-three stabilizer law and unit-target existence classification are the addition here.

Original #6037 head: 88331a098ad9f0299e205bb103acc55b2dc087df. Consumed bytes are retained unchanged in publication ancestor ccff0fb7ed8f53a686a3f2ecc2a5585d7bd8a505; source pins resolve to that ancestor, with original path/head/blob and credit preserved separately. Cold reproduction does not require the pending predecessor's cached Git object.

The final overlap report covers 341 open PRs, 5,722 Discussion records and 5,222 closed/merged PRs at the recorded timestamps. The closed refresh timed out; a subsequent updated-order API check found no closed PR updates after the retained cutoff. Latest open submissions through #6072 were checked separately. These metadata searches are not a novelty certificate or a search of all historical diffs.

Validation

CPython 3.12.10, Windows 11 x64, standard library only:

python3.12 -B scripts/research_package.py --package research/chain-cyclic-schedules

• Exhausted 793,182 balanced words for m=1..5, including the formal modulus-one convention. Independent literal fixed points, least periods and canonical rotations agree.
• Checked 536,472 full-matrix rotations and 291,408 repeated-block identities; independently reproduced W by chronological DP through m=8.
• All 179 unit-target cases through m=24 satisfy the new classification. Rows above m=8 are derived from frozen Certify that balanced depth-three chain multiplicities break every unit symmetry for m ≤ 24 (no new κ) #6037 counts, not a new independent high-modulus census.
• Normal and optimized full receipts match; 13 explicit false-claim controls pass. Eleven actual private-copy mutations are rejected in both modes (22 rejections), with two passing unchanged probes.
• Committed-byte and official export checks pass. The exact scientific snapshot was replayed without Git from read-only exported files: normal, optimized and mutation checks all pass, with no exported file changes. This also verifies the repaired Windows temporary-copy permission handling.

The unchanged canonical base still fails two current-status documentation tests; the separate #6050 addresses them. These are distinguished from the passing package checks. On the submitted head abdcee87d8eb0f46db522b16596406f43e890d7e, hosted run 40636404155 ran 678 helper tests and ended with exactly those two failures (23 skipped); the scientific replay job was consequently skipped. This is a shared-plan failure, not a remote failure of the new mathematical checks. Full selected-supplier cold replay, full-history preflight in this shallow checkout, proof-kernel verification and independent human review are not claimed. The validation note records the precise limits.

Credit

Nick Petalas retains credit for the counts, interval reduction, balanced-existence theorem and rotation precedents; Ross Richardson's #943/#934 lineage supplies the complete-call dirty-connector interface. Burnside's lemma and Möbius inversion are classical. Source notices and Apache-2.0 are retained.

Prepared for SHIN09-git with substantial OpenAI Codex assistance in proof, implementation and writing. Both implementations have the same assisted author; no global literature-priority claim.

Open next questions

Can a representative generator attain the cyclic quotient with a fully counted runtime improvement? The smallest useful test is the 356 classes at (5,1). Separately, derive all depth-four repetition constraints before attempting a higher-depth quotient. Neither follow-up is claimed complete here.