[petalas]

Result

Claim type: scoped written proof with exact finite corroboration. No κ change; the selected record stays 781451/250000000.

A productive target of rank R is refined into k independent integer-rank targets, and each piece after the first pays the same fee c. The rank tariff φ is nondecreasing and concave with φ(0)=0, which covers r^t for 0<t<1 and the comparator's cap tariffs. Under these assumptions:

1. For each k, the balanced partition is optimal (Karamata).
2. The balanced credit S(k) is concave in k. It equals k·ψ(R/k), the perspective of the piecewise-linear interpolant ψ of φ.
3. The optimum is S(k*) − (k−1)c, where k is the first k with S(k+1) − S(k) ≤ c.
4. Halving test: some refinement pays if and only if φ(⌊R/2⌋) + φ(⌈R/2⌉) − φ(R) > c.

At t = 1 − 781451/250000000, with one extra paid rank-q job per piece (c = q^t), a refinement can first pay at R = 470, 940, 1880 and 3760 for q = 1, 2, 4 and 8. These numbers use the selected value as the tariff exponent for illustration only. They make no claim about the binding supplier's normalization or about any actual helper target.

Predecessors and the new part

• Ali Romman's Optimize complete target partitions with explicit piece fees #5769 defines the fee-charged target-partition contract and its finite menu optimizer. This PR generalizes that contract's uniform-fee case to a closed form.
• Merged Prove sharp unit-fee target threshold at rank 288 (no new κ) #5909 (Nick Petalas) specializes Optimize complete target partitions with explicit piece fees #5769 to exponent 99/100 with one paid rank-one child per resulting piece. It proves, using concavity, Jensen's inequality and integer balancing, that a profitable partition exists exactly when R ≥ 288. That threshold is Prove sharp unit-fee target threshold at rank 288 (no new κ) #5909's result and is not claimed here.
• New here: the closed form for every nondecreasing concave tariff with φ(0)=0 and every uniform fee; concavity of the balanced credit in the piece count; the halving test; and the certified selected-exponent thresholds for fees of one rank-q job, q ∈ {1, 2, 4, 8}.
• Fee normalization. Prove sharp unit-fee target threshold at rank 288 (no new κ) #5909 charges k fees (one per resulting piece). This PR charges k−1 (one per piece after the first). The credit changes differ by the constant −c, so items 1–3 cover Prove sharp unit-fee target threshold at rank 288 (no new κ) #5909's model with the same optimal partition. The halving test does not: with k fees, a three-part split can pay while halving does not, as Prove sharp unit-fee target threshold at rank 288 (no new κ) #5909's 96+96+96 witness shows.
• No cross-check. This verifier evaluates only the selected exponent, not 99/100, so it computes no number that Prove sharp unit-fee target threshold at rank 288 (no new κ) #5909 also computes. No comparison was made.

Evidence and reproduction

Package: research/uniform-fee-target-refinement/. The written proof is in the README. Interpreter: ~/git/integer-mult-bounds/.venv/bin/python3.12 (CPython 3.12, standard library only).

python3.12 -B research/uniform-fee-target-refinement/verify.py
python3.12 -O -B research/uniform-fee-target-refinement/verify.py
python3.12 -B scripts/research_package.py --package research/uniform-fee-target-refinement
python3.12 -B scripts/check_research_commit.py --package research/uniform-fee-target-refinement --check-source-history

Local results at 7f73f71ad336ab6b16c1f1d2e71ded7ce19444c5. The second commit changes only README credit text and re-passes every check:

• PASS_UNIFORM_FEE_TARGET_REFINEMENT exact_checks=10392 controls=2 thresholds=q1:R>=470 q2:R>=940 q4:R>=1880 q8:R>=3760 under both flags, about 1.6 s each.
• PASS_REGISTERED_RESEARCH_PACKAGE.
• PASS_RESEARCH_COMMIT_BYTES.

The verifier enumerates every integer partition of R ≤ 22 under eight exact rational concave tariffs and 38 fees, and checks items 1–4 against brute force. Two controls show that the hypotheses matter: a non-concave nondecreasing tariff where halving loses but a three-way split gains, and a convex tariff where an unbalanced pair wins. The thresholds use 60-digit correctly rounded Decimal arithmetic with a certified gap of at least 1e-40. Every R up to 7520 is checked on both sides of each threshold.

Limitations

• This is an accounting statement for a uniform per-piece fee. It does not establish independence, chronological availability, native delivery, restoration, compiler closure or caller cost for any split.
• Rank-dependent or shared fees fall outside the theorem.
• The finite checks corroborate the proof. The all-R statement rests on the written argument.
• No selected replay, Lean or Docker run. Independent human and proof-kernel review are unclaimed.

Open next questions

• Do the balanced optimum and a halving test survive rank-dependent fees, for example c + d·φ(r_i)?
• Which selected P104 productive targets have rank above 470, and what is the cheapest legal per-piece fee for each? This is blocked because scripts/export_selected_invoice.py reports the PR6030 source as unsupported.
• What is the joint optimum when several splits share one fixed adapter fee?

Related open work

• No open PR studies uniform-fee target refinement, as far as the searches found. The merged predecessors Optimize complete target partitions with explicit piece fees #5769 and Prove sharp unit-fee target threshold at rank 288 (no new κ) #5909 are described above.
• Prove exact heterogeneous tensor-cut multiplicities (no κ) #6649 (heterogeneous tensor-cut multiplicities) covers a different object: cut-type schedules, not target partitions.

Credit

Produced by Nick Petalas. Predecessors: Ali Romman's #5769 fee-charged target-partition contract and optimizer, and #5909's unit-fee threshold at exponent 99/100. The signed mixed-target invoice model and the concave-order comparator are by zxv. Karamata's inequality and perspective concavity are classical.

CI notes

Both hosted checks passed at head 7f73f71ad336ab6b16c1f1d2e71ded7ce19444c5 in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/41661215990: plan (5m2s) and replay (research/uniform-fee-target-refinement) (2m12s).