[petalas]

Summary

Prove an explicit exponential-size obstruction to increasing-concave invoice pruning, even with a common call count, paid rank mass, target and paid rank bound. Claim type: scoped mathematical proof with exact finite corroboration.

P_S = sum_(i not in S) 2 H_(2i) + sum_(i in S) (H_(2i-1)+H_(2i+1))
I_S = P_S - H_(2m(m+1))
I_S <=_c I_T  iff  S contains T
middle layer: binom(m,floor(m/2)) >= 2^m/(m+1) incomparable invoices

At m=11, all 462 middle-layer candidates survive concave-order pruning. Each has 22 paid calls, rank mass 264, paid ranks at most 23, and the same target H264. This exceeds PR #5778's 256-candidate cap with small integer ranks. The proof gives an explicit hinge tariff with exact gap one in each direction for every incomparable pair.

The explicit invoice family, proof application and verifier were produced by Nick Petalas. Credit Ali Romman #5778 for the frontier contract, zxv for complete signed invoice accounting, mightyOil and retained predecessors for moment-order context, and Dugongue for the distinction between concave and power order. Mean-preserving spreads, hinge tariffs and binomial identities are classical. No predecessor is audited or replayed, and no general literature priority is claimed.

Evidence

Before: the #5778 contract retains every incomparable pool member. Equal count and rank mass alone leave the possible surviving pool size unspecified.

After: the written proof gives the exact reverse-subset order for every m>=1. Local CPython 3.12.15 warning-strict normal and optimized checks pass 254 complete cube members, all 21,844 ordered cube pairs, 714 middle-layer members, all 138,117 unordered layer pairs, 276,234 direct counter-tariff evaluations, and 17 rejection controls. The two registered verifier commands took 1.058 and 1.122 seconds locally, below their 120-second limits. These are replay timings, not a performance improvement claim.

Exact reproduction from the repository root:

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

Direct replay prints PASS_CONCAVE_INVOICE_ANTICHAINS and a generated receipt bound to all five package hashes. Registered normal/optimized replay and exact committed-byte/source-history checks pass at 3ee3c8ebd2c1afe945fe906cbb9b77186bc070bf. Standard library only; no pending source commit is needed. Full provenance is pinned to canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8.

Live overlap covered 448 open PRs and 5,379 Discussion records with complete pagination, followed by fresh open-PR and handoff checks before commit. Distinctive all-main README and merged-PR searches supplement the sparse local-note search, whose 7,631 omitted note paths were explicitly reported. This coverage is not proof of novelty. Advance CLAIM.

Limitations and unrun obligations

These are synthetic complete signed accounting objects. Their common boundary is a declaration, not a physical construction. No supplier, native implementation, target independence, ownership/restoration, compiler admission or new kappa is claimed. The five framework assumptions remain. Selected cold replay, Lean, independent human review and formal proof-kernel review were not run.

The obstruction concerns replacement by supplied pool members under every increasing-concave tariff. It does not rule out useful typical-case pruning, approximate methods, extra representatives or stronger power-only comparisons. The fully spread cube member dominates all other cube members if it is added to the pool; the antichain assertion concerns the supplied middle layer. Hinge incomparability does not establish a power-curve crossing. The proof covers every m; the finite verifier enforces m<=11 and streams certificates.

Open next questions

• Realize two independent spread blocks in one legal source, with complete target, restoration, native and compiler charges. This is the first step toward a physical large-frontier example; no realization is supplied here.
• Certify power-only order for the displayed m=2 invoices on a declared exponent interval, then seek a family theorem. The present hinge witnesses do not settle that narrower order.

Related open work

• Prune invoice search pools with exact retained dominance witnesses #5778 supplies the exact retained-incomparable frontier contract. This PR adds a worst-case family; its implementation and tests are unchanged.
• Construct and independently replay concave rank-transfer plans #5760 constructs equal-mass concave rank-transfer plans. This family uses classical mean-preserving spreads to prove an exact subset order and antichain obstruction.
• Compute exact two-fee tradeoff regions for concave invoice gains #5762 computes fee regions in the same hinge order. Here every candidate shares its target and has no varying fee.
• Certify best-invoice regions across common-base power costs #5768 certifies common-base power winner regions. This result concerns all concave tariffs and leaves power-only pruning open.
• Certify arbitrary-rank invoice order across exponent ranges #5753 certifies arbitrary-rank power intervals. It can study the narrower power question; its numerical examples were not recomputed here.
• Certify all-exponent order for general-rank recursive invoices #5788 certifies general-rank all-exponent power order. The hinge obstruction is a different statement and does not assert a power crossing.
• Prove sharp mixed-rank invoice envelopes with touching roots (no κ) #5887 proves synchronized touching-root power envelopes for synthetic invoices. This family instead embeds subset order at fixed call count and rank mass; no shared numerical fixture is computed.
• Select compatible invoice edits with an exhaustive optimum certificate #5759 selects compatible invoice edits under capacities. This PR gives a candidate-pool pruning obstruction, not a resource-constrained optimum.
• Optimize complete target partitions with explicit piece fees #5769 optimizes fee-charged partitions of a declared target. These synthetic invoices retain one identical target and claim no legal partition construction.
• Assemble complete invoices with source identity and coverage checks #5770 binds invoice assembly to source identities. This result remains arithmetic and supplies no source-realization receipt.

Related work was refreshed against this account's eight other open PRs created since the advance CLAIM: #5897, #5898, #5899, #5900, #5902, #5903, #5904 and #5905. They study other objects and do not compute this concave-invoice frontier or a conflicting same-instance value. No extra mathematical computations were run to check those contributions.

CI notes

At head 3ee3c8ebd2c1afe945fe906cbb9b77186bc070bf, run 40501075176 finished with plan failed; replay skipped. The only failed step is 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' in test_cli_distinguishes_current_orientation_from_historical_control and KeyError: 'catalog_first_step' in test_cli_preserves_legacy_output_and_emits_both_commands. The eight failed assertions concern the frozen-card count, missing-helper/reference rejection, and sparse current/historical command receipts, including four missing-historical-reference subtests.

The ten failure/error headers, both KeyErrors and the final summary match the inspected PR #5897 plan log exactly. This confirms the reported shared onboarding failure on canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a. No shared test or implementation file was changed; maintainers will handle that repair.

The package replay job was skipped. Hosted replay has not certified this package. Only the local mathematical and package checks listed above have run.

Merge Danger

Door: two-way. Blast radius: isolated. The diff adds one opt-in research package and changes no shared implementation, predecessor credit, frozen evidence or selected result.