[petalas]

Result

Claim type: finite certificate with a short scoped proof of its prime consequence. No closed form, call count, tariff or κ is claimed.

For the unsigned depth-three dirty chain of research/chain-tariff (#4257), let W(m,c) count balanced correct words (m copies of each call) for the target y += c B3 B2 B1 x. Correctness is the cited interval-count criterion: N12 = N23 = 0 and N13 = c modulo m.

• Exact counts W(m,c) for every m = 2..24 and every c are in RESULT.json.
• For every 3 ≤ m ≤ 24 the m−1 nonzero counts are pairwise distinct. So no nontrivial unit u has W(m,uc) = W(m,c) for all c. In this range no gcd-only, sign-invariant or Ramanujan-sum formula can give these multiplicities. The one-source depth-two law of Count optimal unsigned one-source connector words in every characteristic #4236 (equal nonzero counts at prime m) does not extend to depth three. Example: m = 5 gives 5340, 6390, 3000, 2970 for c = 1..4.
• Scoped proof: at prime p, with F(g) the sum of ζ^(g·N13) over pair-valid balanced words, Galois invariance of F(1) under any nontrivial s_u would force W(p,uc) = W(p,c). The certificate therefore shows that F(1) generates Q(ζ_p) for every prime 3 ≤ p ≤ 23.

This answers the chain-tariff open question only partially: it rules out unit-invariant closed-form shapes and supplies exact data; it does not give a closed form.

Evidence and reproduction

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/chain-balanced-multiplicities/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/chain-balanced-multiplicities/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/chain-balanced-multiplicities
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/chain-balanced-multiplicities --check-source-history

Expected marker: PASS_CHAIN_BALANCED_MULTIPLICITIES max_m=24. Each direct mode took about 80–90 s locally (CPython 3.12, macOS arm64), under the 300 s registered timeout.

The chronological DP over letter counts and (N12, N23, N13) residues is checked against an independent gap model (ones/threes before each 2, binomial interleavings) on the full residue distribution for m ≤ 7, and against literal enumeration of every balanced word for m ≤ 5. The total mass (3m)!/(m!)^3 is checked for every m. A positive control reproduces #4236's equal depth-two counts (binom(2p,p)−2)/p at p = 2, 3, 5, 7, and a forged equal-count table is rejected. The zero counts occur exactly at the predecessor's exceptional targets (2,1), (3,2), (4,3); this is agreement, not an audit.

Limitations

Finite range m ≤ 24; no all-m distinctness, asymmetry or closed form is proved. The correctness criterion is cited from chain-tariff; no dirty-register execution is repeated. Only balanced words are counted. No supplier caller, routing, admission, precision or outer multiplication transfer is addressed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Does the unit stabilizer stay trivial (or the nonzero counts stay pairwise distinct) for every m ≥ 3? At prime p: is F(1) never fixed by a nontrivial Galois automorphism?
2. Are the nonzero counts asymptotically equidistributed at prime m? The relative spread is about 1.2e-7 at p = 23 but about 0.4% at m = 24.
3. A closed form or holonomic recurrence for W(m,c).

Related open work

• Prove N(c) ≥ ⌊5c/2⌋ for the unsigned connector via five-source depth-two blocks in every c ≥ 5 (no new κ) #6035 proves a call-count floor N(c) ≥ ⌊5c/2⌋ for a five-source unsigned connector; it studies call counts, not word multiplicities of this chain, so no number is compared.
• Prove the conjugate-pair target needs four calls in every group (no new κ) #6038 (conjugate-pair four-call target) and Classify mixed Q8 call-space shear identities; no pair of shear images separates (no new κ) #6039 (mixed Q8 shear identities) study group call counts, not this chain or its word multiplicities; no number is compared.
• No other open PR studies the depth-three chain multiplicities.

Claim: #1166 (comment)

CI notes

Commit 88331a098ad9f0299e205bb103acc55b2dc087df: both hosted checks passed, plan (4m43s) and replay (research/chain-balanced-multiplicities) (7m38s), in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40608876855. The run head matches the PR head.