[petalas]

Summary

Scoped proof with finite corroboration, no new κ. In characteristic two, with a central formal involution Q, arbitrary free constant invertible bases, no auxiliary workspace, and paid shears I+Q N with N²=0 and rank N≤r:

At most four calls attain Q I_n
    exactly when n is even and n ≤ 2 floor(4r/3).
Minimum maximum paid rank = ceil(3n/8).
Minimum total paid rank = 3n/2.

The new Gram-complement argument strengthens the elementary n≤3r capacity bound. A balanced omission schedule attains the bound: four rank-three queries twist eight independent coordinates, while four rank-two queries cannot twist six. The proof allows arbitrary scalar mixing across the coordinates. This is a sharp abstract specialization of the higher-rank primitive question in Matarener's #6114, not a physical supplier or an extension of its rational Laurent theorem.

The package is research/four-block-query-capacity/. This contribution was produced by Nick Petalas. Credit Cinna Davis / 1337isnot1337 for the minimum-query simplex and controller lineage, Ali Romman #5617 for the coupled-port Gram framework, and Matarener #6114 for the source question. The predecessor proof documents are pinned as provenance. Their verifiers and reported numbers were not replayed.

Advance CLAIM. The delivered theorem matches its predicted scope.

Evidence

• Before: the complement-dimension relaxation gives n≤3r; at r=2 it leaves n=6 open.
• After: the complementary-idempotent rank inequality gives 3n≤2Σr_i≤8r, excludes n=6 at r=2, and an explicit schedule attains every feasible even dimension.
• Exact quotient-ring and independent lifted-bit interpreters pass 392 construction fixtures for caps 1 through 24, up to dimension 64, checking 17,696 original coefficient columns. The verifier also checks 170 Gram factorizations, four mixed-basis words, and ten rejecting controls.
• Complete binary-matrix enumeration in dimensions 2, 3 and 4 finds square-zero counts by rank [1,3], [1,21] and [1,105,210]. Every pair sum is included in the four-summand feasibility decision for each cap.
• Warning-strict normal and optimized direct checks pass in approximately 0.22 seconds each. Both VERIFY.json commands pass in the registered package replay, approximately 0.71 seconds total. These are local check durations, not a speedup claim.
• Committed-byte/source-history preflight passes all six files at e4fb5699c1c390ada0e6ac738d48548c462448b1, with base 721c03822c421b7f7fcbb2598993b947c4d88419 present in publication ancestry. Package metadata and pins pass. Each direct run emits a generated receipt bound to package SHA256 hashes and writes no files.

Reproduce from the repository root with the interpreter used locally:

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

Elsewhere use a CPython 3.12 path. Standard library only; no runtime predecessor inputs. Expected markers are PASS_FOUR_BLOCK_QUERY_CAPACITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

This is a scoped informal algebraic proof, not a complete multiplication theorem. A physical multi-bank query, native alignment, arbitrary-input helpers, restoration, paid basis maps and the complete signed invoice remain missing. Free constant bases are an explicit model hypothesis. Extra workspace, characteristic zero, infinite-order Q, non-square-zero paid operations, independent stage variables and nonconstant free maps are outside the theorem.

No selected supplier cold replay, full framework proof, outer-ledger change, physical implementation or proof-kernel verification was run. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

Open next questions

• Can a legal physical multi-bank job implement the rank-three query in the eight-dimensional witness? Start from its four shear blocks and specify all returned helpers and actual paid ranks; a native realization and complete invoice are missing.
• What is the exact capacity at six or more calls in the same model? Start from the Gram complement and feasible omission schedules; this package stops at four calls.

Related open work

• Controller income is capped at full calls minus one: even-part rank bound in the p16 Laurent alphabet (scoped negative result; no new kappa) #6114, Matarener: proves the rank-one rational Laurent income obstruction and asks about multi-bank primitives. This package answers an abstract characteristic-two four-call specialization; it does not compare the Laurent income numbers or supply a physical primitive.
• Prove paired-stage loss at m = 4 under the chronology cut is exactly 10 (no new κ) #6041, Nick Petalas: studies chronological paired-stage rank-two projectors on a fixed binary caller. This package changes the query rank and removes stage/chronology claims; no counts or paid invoices are compared.

Overlap was refreshed after the hosted run; no additional open PR studies this four-query capacity.

CI notes

Uncertified by CI. At head e4fb5699c1c390ada0e6ac738d48548c462448b1, run 40692036317 failed plan in the onboarding helper tests. The package replay job was skipped and did not run in CI. No hosted mathematical replay is claimed. A maintainer re-run is needed after main is fixed.

The exact unrelated failures on canonical main 721c03822c421b7f7fcbb2598993b947c4d88419 are:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal **0.003125804** in docs/research/current-status.md; the document instead bolds the longer expression 0.003125804 = 781451/250000000 as a whole.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

Main commit 721c03822 admitted the PR6030 κ record with [skip ci] without matching both onboarding document expectations. This PR adds only its isolated research package and does not change either document or the selected record. The logs for PR #6120's run 40691776027 show the same two test failures. Local normal/optimized registered replay, package metadata and committed-byte/source-history checks passed as recorded above. Nothing was pushed to address these unrelated failures.

Merge danger

Two-way door; blast radius is the isolated research package. No shared code, frozen predecessor evidence or selected record changes.