[petalas]

Summary

Scoped proof with exact finite corroboration. For six independent main fields with overlapping stage demands, the exact signed whole-idle inventory minimum is 2H(24)+2H(8) for every nondecreasing concave tariff H with H(0)=0. Inverse calls and arbitrary constant invertible mixing cannot lower it. Strict concavity makes the optimal signed subset multiset unique.

target stage sets:  ABC ABC AB AB BC BC
native stage uses:  one full query pair per stage
idle degree debt:   A:2 B:4 C:2
optimal inventory:  ABC ABC B B, all forward

The resulting target-minus-idle reward strictly exceeds uniform six-ABC completion for every H(r)=r^s, 0<s<1. Its exact gain is 4H(17)+2H(24)-2H(8)-4H(25)>0, proved by four balancing transfers. This is a sharper optimistic caller budget. No six-field operator word, new kappa, supplier admission or selected-frontier change is claimed.

The proof and verifier were produced by Nick Petalas. Credit Ali Romman for the released overlapping AB/BC proposal in Discussion5593 and PR6348's full-query, fresh-axis and whole-idle conventions; Cinna Davis for thin-query/controller interfaces; DreamingOfClouds and the retained P104/helper/native/compiler/outer lineage. Determinant multiplicativity, submodular greedy duality and concavity are classical. The new step is this exact signed completion and strict comparison, not the predecessor module or tree.

Advance CLAIM. Base canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, refreshed before submission. Live claims, local reservations, open PRs, merged keyword searches and main-tree notes were checked. Sparse cached notes and unpublished work remain coverage limits.

Evidence

Before, the overlapping proposal lacked an exact signed idle-inventory optimum. After, a dual proves the all-length minimum and equality classification. Ten independent bounded signed-lattice shortest paths corroborate the optimum for five exact tariffs and both the overlapping and uniform deficits. They do not replace the unbounded written proof.

Each warning-strict normal/optimized replay reports:

PASS_SIGNED_OVERLAP_IDLE_COMPLETION
signed_dual_inequalities = 42
concave_transfer_checks = 100
finite_shortest_paths = 10
rejecting_controls = 6

The registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE, outcome partial. Both commands passed locally on CPython 3.12.15, macOS arm64, in approximately 0.08 seconds each. These times establish the CI budget and are not a speedup claim. Package metadata, committed bytes/modes and full declared source history pass. The generated receipt binds all package proof/source/metadata hashes.

The official committed export passes integrity checking. Normal and optimized direct replays pass with read-only files, no Git on PATH and identical stdout hashes. All eight exported files retain bytes and modes. Actual proof-file corruption rejects in both modes. No predecessor scientific suite was replayed as this contribution.

From the repository root, reproduce with the actual local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/signed-overlap-idle-completion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/signed-overlap-idle-completion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/signed-overlap-idle-completion
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/signed-overlap-idle-completion --check-source-history

Another host may substitute its CPython 3.12 executable. The standard library suffices. The source-history pins are provenance only; there are no external scientific runtime inputs.

Limitations and unrun obligations

This is a formal Laurent-stage determinant relaxation. It fixes one native query-pair occurrence per stage, six arbitrary main inputs, autonomous helpers and whole-idle subset atoms of rank eight per fresh stage. A finite-order native specialization may have additional relations. No impossibility conclusion for every specialized native program follows.

An optimal inventory need not have a legal chronological placement or correct query coefficients. The comparison cancels identical full-call/helper charges only. Thin-query counts, ordinary work, storage and all changed charges must be priced separately. Native frames/phases, helper preservation, owned erasure, precision, stock, tape, uniform compiler and outer funding remain open. The five selected assumptions retain their status.

The determinant, strict-concavity and equality arguments are written proofs outside a formal kernel. Independent human and formal proof-kernel review are unclaimed. No selected cold replay, native execution, Lean, Docker or complete rounded multiplier was run.

Open next questions

• Can a six-field chronological word attain the overlapping query/Gaussian target at the exact four-atom idle minimum? Start with every independent input and the exact Laurent endpoint, then charge all thin queries.
• If minimum-idle placement fails, what extra paid inventory is necessary? Analyze the two ABC and two B atoms around the three full-query stages; determinant equality alone does not decide it.

Related open work

• Prove the exact nineteen-leaf query-tree topology optimum (no κ) #6380 optimizes a fixed nineteen-leaf binary tree; this changes to an overlapping six-field determinant contract and does not optimize any tree.

• Construct transported-query substitution for noncommuting coded jobs (no κ) #6382 constructs transported noncommuting child substitution; this studies commuting independent stage degrees and does not supply a child substitution.

• Compose native input lifts with the query tree at conditional κ 0.00314980 #6350 changes native input lifts while retaining the balanced caller; its local-word gain is separate from this idle-completion relaxation.

• Contract nine native helper outputs at conditional κ 0.00315057 #6353 changes native helper outputs with that caller; its native boundary and invoice are not rechecked here.

• Conditional κ = 0.00315087 (+3.0e-7 over #6353) #6372 changes a two-anchor native boundary with the same caller; this result constructs no native boundary.

• Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375 shares native input carriers while retaining the caller; its input-carrier transport differs from the stage-idle inventory studied here.

• Independently audit PR6353 rational joint basis and public boundary #6355 studies Contract nine native helper outputs at conditional κ 0.00315057 #6353's rational boundary; those scalar coefficients are not computed here.

• Independently recount PR6353 recursive invoices and fixed-saving margin #6359 studies Contract nine native helper outputs at conditional κ 0.00315057 #6353's frozen formal invoice; this proves a changed-contract inventory optimum rather than recounting that invoice.

• Independently check four frame1094 native matrices #6360 studies fixed native matrices; native realization remains an open obligation here.

• Independently verify the fixed balanced caller Laurent endpoint #6361 studies the fixed balanced caller endpoint; the present package supplies no six-field endpoint.

• Independently audit PR6372 two-anchor scalar boundary and endpoint #6376 studies Conditional κ = 0.00315087 (+3.0e-7 over #6353) #6372's scalar boundary/restoration; those identities are outside this completion proof.

• Certify PR6375 matched profiles and unique relative-tariff threshold #6378 studies Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375's relative tariff switch; this compares different main stage demands and computes no common crossing value.

• Prove exact tariffs for carriers visiting distinct entrance lines (no κ) #6388 proves a multi-entrance frame-transport tariff; its carrier itinerary differs from the signed stage-subset inventory here, with no common computed optimum.

• Prove fixed-prefix P104 tails are optimal for every concave tariff #6394 proves fixed-prefix P104 delivery-tail optimality for concave tariffs; those supplied-copy chains differ from the six-field stage demand here.

The completion refresh read the account's open PRs created since the advance CLAIM. No disagreement on a common computed quantity was identified; no additional computation checked another contribution's numbers.

CI notes

Both plan and replay (research/signed-overlap-idle-completion) passed at head e384ae7181f085f92d21e3578976e3231c76e26d in run 41512073720. The completed checks were confirmed once with gh pr checks 6391 --repo CrocSwap/integer-mult-bounds. Hosted success covers the registered bounded replay and controls; it is not independent mathematical or formal-kernel acceptance. All registered commands and committed/export checks also passed locally.

Merge Danger

Door: two-way. Only one isolated research package is added.

Blast Radius: isolated. Shared code, frozen predecessor evidence and the selected record are unchanged.