[petalas]

Summary

Scoped proof with exact finite corroboration: at exponent 99/100, a proper positive-integer target partition charged one additional paid rank-one child per resulting target improves the complete signed bill exactly when the original rank is at least 288.

profit = sum_i r_i^(99/100) - R^(99/100) - k
R <= 287: every proper partition has profit < -0.004077
R  = 288: 96+96+96 gives 0.006289 < profit < 0.006290
R >= 288: balanced three-part witnesses remain strictly profitable

This specializes Ali Romman's #5769 fee-charged target-partition contract. Its rank-256 menu cap contains no profitable partition under this explicit fee/exponent; the first opportunity is 288. Classical concavity, Jensen and integer balancing retain their credit. This contribution is produced by Nick Petalas. No new kappa, physical supplier or selected-frontier change.

Evidence

Before: the finite fee-charged optimizer supplies no all-rank threshold for this specialization. After: the written proof reduces every smaller rank and arbitrary piece count to 286 exact Jensen inequalities, with a separately certified tighter integer margin, and supplies the rank-288 witness.

Local CPython 3.12.15 checks pass at 53b61f87e3f2f358a38b4868831ef7fa0ac07603:

• All 286 Jensen and balanced-integer rejection bounds, the positive witness and rational margin checks pass.
• Every one of 7,313 proper sorted integer partitions through rank 24 passes the strict-balancing and losing-bill checks; eleven independently implemented bisection power checks agree. Eight forged certificates and five invalid partitions reject.
• Warning-strict normal and optimized registered commands pass, locally timed at 4.47 and 4.50 seconds. The final registered package run passes in 9.22 seconds total. These are check durations, not speedup claims.
• Package metadata/pins, six-file committed-byte and declared-source-history checks, official committed-export integrity and optimized read-only Git-free replay with empty PATH pass. The cold receipt equals the fresh local receipt. Receipts are generated on stdout and bind all six package files by SHA256.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/unit-fee-target-threshold
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/unit-fee-target-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/unit-fee-target-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/unit-fee-target-threshold --check-source-history

Another CPython 3.12 installation can replace the absolute interpreter path. Expected JSON marker: PASS_UNIT_FEE_TARGET_THRESHOLD. --check-receipt PATH recomputes the certificate and source hashes. No unmerged predecessor object, network or third-party package is needed for the verifier.

Advance CLAIM. Canonical main remained de5ec2e7a6074068452f45b1a191b5c72832df4a, priority 8, at the pre-commit refresh. Live complete overlap snapshots covered 459 open PR bodies and 5,400 Discussion records; the sparse local helper omitted 7,663 HEAD note paths. Distinctive searches across all main research README blobs and live merged-PR searches supplemented that incomplete local coverage. No keyword search proves novelty.

Limitations and unrun obligations

The written all-rank proof is not proof-kernel verification. The exponent is fixed at 99/100, a one-percent raw power screen rather than the selected supplier exponent or nine-row outer budget. The fee is one recursive child per piece; scalar ADDs, shared fees, paid removals and other credits are different contracts. Extra nonnegative fees preserve the rejection below 288 but may erase the positive witness.

Independent useful target realization, literal source maps, native correctness, chronological delivery, dirty restoration, owned lifetimes, compiler/precision/tape/outer admission and the full multiplication theorem remain unproved. No predecessor replay, selected cold replay, Lean or physical supplier ran. Independent human and formal proof-kernel review are unclaimed. The five selected framework assumptions remain.

Open next questions

• Pin an actual supplier exponent and determine the corresponding adjacent-rank threshold under the same fee. Balanced partitions give the first bounded calculation; a matched complete caller tariff is required.
• Find one literal helper/cut where paid routes can be shared across independent targets. Start with its complete invoice; target realization and dirty restoration remain blockers.

Related open work

• Optimize complete target partitions with explicit piece fees #5769 optimizes bounded fee-charged target menus; this gives the unrestricted all-rank threshold for the explicit uniform unit fee at one exponent, without replaying that optimizer.

• Compute exact two-fee tradeoff regions for concave invoice gains #5762 computes two-fee regions valid for all increasing concave tariffs. This fixes one power tariff and varies original target rank; no common numerical optimum is claimed.

• Substitute recursive jobs with matched-target and fee accounting #5754 substitutes matched recursive jobs with complete target debits and fees. This studies partition credit instead of that composition operation.

• Prove exponential invoice frontiers at fixed count and rank mass #5901 constructs incomparable invoice pools at fixed count and rank mass. It does not determine existence of a profitable unit-fee partition and supplies no premise here.

• Prove the exact paired-color batch frontier and sharp 10/9 penalty #5911 optimizes capacity-three batches and occupied-color incidence for fixed targets. That grouping objective differs from changing target ranks under a per-result recursive-child fee; no common numerical threshold is computed.

This account's open PRs created since the CLAIM at 2026-10-27 12:32:14 UTC were refreshed before DONE. The new #5910 sector-workspace result is unrelated to this partition tariff. No new same-threshold result or numerical disagreement was found in that snapshot.

CI notes

At commit 53b61f87e3f2f358a38b4868831ef7fa0ac07603, run 40505454496 finished with plan failed; replay skipped. The only failed step was Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The assertion failures concern frozen-card/current-route expectations, missing-reference handling and sparse omission lists. All ten failure/error headers match PR #5906's failed plan log exactly. This confirms the reported shared main de5ec2e7a onboarding failure. Maintainers will address it; no shared source or tests were changed.

Hosted replay has not certified this package; only local scientific and publication checks have run. Local normal/optimized checks, metadata/pins, committed-byte/source-history checks, official export integrity and read-only Git-free replay pass. This CI result is not mathematical acceptance.

Merge Danger

Door: two-way. One isolated opt-in research package can be removed independently.

Blast Radius: research. Six new package files; no shared implementation, frozen source or selected record changes.