[petalas]

Summary

Scoped algebraic proof: two faithful anticommuting involution query types have an exact additive tariff for the unmixed shear y += (A0+BQ+CR)x over A=product_c Mat_(d_c)(K), with characteristic different from two.

Q calls >= max_c ceil(rank(B_c)/d_c)
R calls >= max_c ceil(rank(C_c)/d_c)
Both bounds are attained together, using one zero-return owned query field.

The lower bounds permit arbitrary linear feedback, retention, overwriting and restored borrowed dirty fields. Granting the opposite query type for free proves each bound. A two-component example needs two calls globally although each component needs one, so maximizing component totals undercounts the actual fixed-type alphabet. Noncommutative spectator blocks still pack same-type demands.

This contribution was produced by Nick Petalas. It answers the anticommuting-involution specialization of David Ganz's RQ09 in #5803, head 0acda5ab421d28b98fdf916968696c8f9453376f. Credit its R03-R06 framework, Ali Romman #5626's faithful-sector compiler, Dugongue #5578 T6/T7, and the native, phase, ownership, controller and compiler contributors retained in the package. Classical rank factorization and Clifford algebra are not claimed as globally new. No new kappa or selected-frontier change.

Evidence

Before this contribution, #5803 left a typed generalization for two noncommuting queries open. The new proof gives exact separate minima, a simultaneous constructor and a sharp sector accounting gap in a declared specialization. The proof is self-contained and cites predecessors without replaying their verifiers or checking their numbers.

Local CPython 3.12.15 checks pass:

• 13,338 complete shear fixtures in formal Clifford coefficients and independently written literal payload matrices, with 149,742 original input-column comparisons and 213,064 circuit instructions across those fixture families.
• All 1,555 feedback words of length at most four in the stated six-instruction alphabet, the explicit packed example and eight rejection controls.
• Warning-strict direct normal and optimized runs, locally timed at 2.947 and 2.924 seconds respectively. Both registered commands pass; the package runner takes 6.123 seconds in the measured run. These are check durations, not performance improvement claims.
• Package metadata and source pins pass. Committed-byte/source-history preflight passes for all six files at 5c72ad1fef28cdb5f07d4585bc4e859dbeb02760, with base de5ec2e7a6074068452f45b1a191b5c72832df4a present in publication ancestry.

Every verifier run emits a generated receipt bound to SHA256 hashes of the six package files. It writes no artifacts. No receipt is handwritten.

From the repository root, reproduce with the interpreter used locally:

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

On another machine use its CPython 3.12 path. Expected outputs include PASS_ANTICOMMUTING_QUERY_ADDITIVITY, PASS_REGISTERED_RESEARCH_PACKAGE and the committed-byte preflight's success report. The package has no runtime inputs outside its directory or third-party dependencies. Bibliographic pending work is not a hidden Git-object dependency.

Limitations and unrun obligations

This is a scoped informal proof with finite corroboration. It excludes a nonzero QR target coefficient, characteristic two, nonfaithful representations, extra ordinary operators, nonlinear instructions, multi-field queries and per-component query-type selection. The finite feedback enumeration does not exhaust arbitrary programs.

No actual native Gaussian realization, physical coefficient words, full movement/precision/stock ledger, productive-target credit or complete signed invoice is supplied. No selected supplier cold replay, all-size framework proof or proof-kernel verification was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Find the exact tariff for mixed QR outputs, beginning with both opposite-type projected-rank bounds and explicit small mixed words.
• Instantiate one lawful native caller, first proving the operator relations, faithful four-coefficient decomposition and ordinary coefficient implementation, then price its complete producer/consumer.

Related open work

• Add a mathematical atlas of shared queries, helper hubs and compiler routes #5803 poses RQ09 and supplies the single-query noncommutative coefficient framework. This PR settles only the faithful anticommuting-involution, unmixed-shear specialization. It does not install the atlas's proposed Gaussian bundles.

• Construct minimum-cost binary query bases with exchange certificates #5791 optimizes binary query-pool bases. Its selectable pool and XOR reconstruction differ from this fixed two-type operator alphabet; no shared numerical optimum is asserted.

• Prove the sharp codimension bound for partial query acquisition #5878 proves a partial-target acquisition guarantee for a selectable scalar query pool, extending Construct minimum-cost binary query bases with exchange certificates #5791. This PR counts two fixed noncommuting operator types over a specified ordinary algebra; no shared numerical optimum is asserted.

• Prove exact tariffs for consecutive queries sharing a detour #5884 proves a signed path tariff for consecutive queries sharing a detour. Its metric-chart invoice differs from this fixed-type shear count; no common numerical quantity is compared.

• Prove the binary-affine call floor with clean erasure (no new κ) #5879 permits clean erasure in a binary-affine paid alphabet. Its characteristic-two address model differs from the faithful odd-characteristic anticommuting algebra here; no shared call-count instance is asserted.

The completion refresh read the titles and bodies of this account's nine other open PRs created since the advance CLAIM at 2026-10-27T10:35:58Z. None supplies the same two-type shear result or a conflicting same-instance number. No additional computations were run to check another contribution.

CI notes

Hosted CI did not certify this package. At commit 5c72ad1fef28cdb5f07d4585bc4e859dbeb02760, run 40491049099 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 eight assertion failures concern frozen-card/current-command handling, missing current/historical references and sparse reference semantics. The complete ten failure headers match PR5876's plan log, checked directly. This confirms the reported shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a; no package failure is reported by this run.

The replay job was skipped. Hosted replay has not certified the package; only local scientific checks have run. No shared implementation, tests, selected record or frozen evidence was changed or rerun to address this failure. Maintainer repair remains outstanding.

Merge danger

Two-way door. Blast radius: isolated. The PR adds one six-file research package and changes no existing implementation or frontier.