[petalas]

Result

Answers Open next question 2 of SHIN09-git's #6080 (research/chain-cyclic-schedules): the exact repetition law for balanced depth-four chain words.

For v with n copies of each letter and m = dn, the word v^d is correct for target c iff

(d odd or n even)  and  (3 ∤ d or 3 | n)  and  d | c,

and v is correct modulo n for target c/d. The cubic shift C(d,3)n^3 (kept at depth three) and the new quartic shift C(d,4)n^4 both vanish modulo m under the gates. A new triple gate replaces the depth-three cubic target shift.

Corollary: if gcd(c,m) = 1 (in particular c = 1), no correct balanced depth-four word has a nontrivial rotation symmetry, so 4m | W4(m,c). At depth three a unit-target word can repeat three times. The corollary is vacuous for m ≤ 5 (#4279) and nonvacuous for 6 ≤ m ≤ 64 (#4263, #4280).

Claim type: scoped proof with bounded finite corroboration. No κ, supplier, paid-profile saving or physical speedup. Same alphabet; scope_status=changed is recorded in TASK.json.

Evidence

research/chain-depth-four-repetition-law/verify.py, standard library only:

• composition formula vs literal interval counts of v^d: 1,508,928 (block, d) pairs (all blocks n ≤ 2 with d ≤ 12; all 369,600 blocks n = 3 with d ≤ 4)
• gate law vs literal correctness: 11,482,992 word-target checks, m ≤ 24; fixed-point counts match the F_d formula
• full census m ≤ 3: W4(2,0) = 816 (102 classes), W4(3,0) = 24,984 (2,082 classes), nonzero targets empty; least periods, rotation closure and Burnside agree
• coefficient shift lemma for d ≤ 240, n ≤ 60
• five false-claim controls rejected by explicit counterexamples

Reproduce from the repository root (about 25 s locally, CPython 3.12.10 at ~/git/integer-mult-bounds/.venv/bin/python3.12):

python3.12 -B research/chain-depth-four-repetition-law/verify.py
python3.12 -B scripts/research_package.py --package research/chain-depth-four-repetition-law
python3.12 -B scripts/check_research_commit.py --package research/chain-depth-four-repetition-law --check-source-history

Local results: PASS_CHAIN_DEPTH_FOUR_REPETITION_LAW, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, PASS_RESEARCH_COMMIT_BYTES.

Limitations

• The proof is written; finite checks corroborate it up to the stated bounds. No proof-kernel or independent human review.
• Only m ≤ 3 is censused. The nonvacuity of the corollary for 6 ≤ m ≤ 64 is cited from Attain the depth-four chain connector floor 4m for every 6 ≤ m ≤ 32 (no new κ) #4263/Attain the depth-four chain connector floor 4m for every 33 ≤ m ≤ 64, via an exact row model (no new κ) #4280, not rechecked.
• The correctness criterion is the depth-four form of the chain-tariff interval identity (Prove the shortest unsigned depth-three chain connector in every characteristic (3m; 4m at c ≡ -1 for m ≤ 4; no new κ) #4249); its induction is restated, not formalized.

Open next questions

1. Repetition law at general depth L: which shifts survive modulo dn under the lower gates? Start at depth five.
2. Exact W4(m,c) and cyclic class counts for m ≥ 4, via a residue DP or Attain the depth-four chain connector floor 4m for every 33 ≤ m ≤ 64, via an exact row model (no new κ) #4280's row model.

Related open work

• No open PR studies balanced depth-four chain words or their rotation classes (searched open PRs for chain/cyclic/depth terms, and this account's open PRs Prove exact vector-label Heisenberg central tariffs (no κ) #6648–Settle eight three-coset Hermitian bootstrap families at 784 (no κ) #6662 opened since the claim).

Credit

Builds on SHIN09-git's #6080 (pair-gate argument, fixed-point reduction, Burnside/Möbius transform) and Nick Petalas's #4249, #6037, #4263, #4279, #4280.

CI notes

All checks passed at head e364eeb19e5268967189ed12aa4813ee73edf3dc: plan and replay (research/chain-depth-four-repetition-law) in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/41661221919.