[petalas]

Summary

Scoped proof with exact finite corroboration: pruning any cheapest full-span query basis to a dimension-d target space in a rank-r query pool has the sharp acquisition approximation factor r-d+1. The theorem holds over every field with nonnegative costs. A binary family approaches the factor with positive integer costs and a unique cheapest full-span basis.

B = any minimum-cost basis of the entire query span
C = union of B-coordinate supports of the target subspace
cost(C) <= (r-d+1) OPT(target)

The new step uses target dimension in a price-threshold argument and supplies a matching family. Ali Romman #5791 supplies the partial-target question and whole-span acquisition contract; the cheap-prefix property and weighted basis exchange are classical. No predecessor implementation or result is audited or replayed. No literature-wide priority for approximation theory is claimed.

Evidence

Before, whole-span optimality supplied no stated partial-target guarantee. After, the written proof gives the sharp codimension factor, including all tied minimum bases and zero costs.

Local CPython 3.12.15 checks pass at 79965dcf726c077859afe8901eb8e32a236e642f:

• 123,481 minimum-basis/target comparisons across 256 weighted pools and all 67 subspaces of F2^4, with exhaustive partial-acquisition optima.
• 243,947 pointwise price-threshold checks, 30 unique-basis sharp fixtures, three hypothesis controls and four malformed-input rejections.
• Warning-strict normal and optimized direct replays, both registered commands, package metadata, committed-byte/source-history preflight, official export integrity and a separate optimized replay of the Git-free export.
• PASS_PARTIAL_QUERY_BASIS_GAP, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES and PASS_RESEARCH_EXPORT. The two registered checks took 1.64 seconds together; direct normal and optimized checks took 0.63 and 0.61 seconds. Generated receipts bound all five package files; temporary receipts and research scratch are removed after publication.

From the worktree root, reproduce with:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/partial-query-basis-gap
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/partial-query-basis-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/partial-query-basis-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/partial-query-basis-gap --check-source-history

Limitations

Each acquired query is paid once; linear reconstruction is free only in this declared relaxation. Physical delivery, XOR/scalar reconstruction, restoration, owned storage, native legality and supplier admission require separate costs and proofs. Binary finite checks corroborate the all-field written argument. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker, selected cold replay, rounded caller or complete multiplication theorem was run. The selected frontier and its five framework assumptions are unchanged.

The formal-interface card is explicitly changed to this algebraic side result. The advance CLAIM, exact source citation, scope and overlap limits are recorded in the package. Refreshed canonical main remains de5ec2e7a6074068452f45b1a191b5c72832df4a.

Open next questions

• Can target-aware acquisition improve the r-d+1 factor with a small checkable certificate? A full-span basis tie rule cannot remove this obstruction, since the sharp family has a unique minimum basis.
• Which actual source caller satisfies the additive acquisition model after reconstruction, delivery, restoration and storage are priced?

Related open work

• Construct minimum-cost binary query bases with exchange certificates #5791 optimizes whole-span binary query acquisition and explicitly leaves partial-span targets separate. This contribution proves the sharp partial-target loss bound for pruning any of its valid whole-span optima, without depending on its pending code.

Related open work was refreshed after CI against this account's open PRs created since the advance CLAIM. No additional PR on the same partial-target acquisition quantity was found.

CI notes

At commit 79965dcf726c077859afe8901eb8e32a236e642f, the registered-package run failed in plan, solely at Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py; the log ends FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. This is the shared onboarding failure on base de5ec2e7a6074068452f45b1a191b5c72832df4a, outside this package. The ten failure/error labels and final failure summary match PR #5879's failed run exactly; both logs were inspected.

The package replay was skipped. Hosted replay has not certified this package; only local verification has run. The local checks listed above passed. The shared failure is left for maintainers; no shared file or onboarding test was changed.

Merge Danger

Door: two-way. Five new files in one isolated research package.

Blast Radius: research. No shared implementation, frozen predecessor or selected-result change.