[petalas]

Summary

Prove that the known twelve-ADD upper bound is sharp for a single source insertion serving four nonzero pairwise orthogonal Boolean demands on exactly four arbitrary dirty rows.

source-column transfer: exactly 6 ADDs
complete dirty restoration: at least 6 ADDs + one insertion + at least 6 ADDs
common ADD price c, insertion prices mu_i: exact cost min_i(mu_i) + 12c

Claim type: exact finite certificate with a scoped Boolean-ring lifting proof. The new lower bound holds for every finite word, every insertion root and every commutative unital Boolean ring under the declared four-row, one-insertion contract. A complete finite-component quotient reduces the lower bound to F2^4. The written upper identity retains arbitrary dirt on the unused complement atom.

This contribution was produced by Nick Petalas. Credit lee-phys #5514 for the triple contract, ideal invariant and prefix/suffix method, retaining #5304/#5259, Ali Romman, Krzysztof Wojt, Ross Richardson and the earlier fanout lineage. Ali Romman's #5310 already supplies the general 4(k-1) restoring construction, hence the twelve-ADD upper bound. The new contribution proves sharpness in the stated four-demand case. The elementary upper identity is credited prior work. Nick Petalas's older #5948/#6663 concern three-row questions and are background; their construction and code are not the main step here.

Only research/four-orthogonal-one-insertion/ is added. The declared card changes scope to an abstract helper-restoration side result. Advance claim.

Evidence

Before, the cited restoring construction supplies an upper bound without proving this four-demand minimum. After, a complete edge-potential certificate supplies the matching lower bound:

• All 50,625 primitive source-column states and 2,750,592 nontrivial discovery transitions are enumerated. A separately encoded binary-component checker covers all 9,720,000 gate transitions, including no-ops. Its start/target potentials are zero/six, and every legal edge changes potential by at most one.
• Four complete attaining words and their inverses are checked over the universal five-atom Boolean ring, including the unused complement. A separate binary interpreter checks all 1,280 atom-input executions. All 52 single-call omissions, four missing-prefix cases, four complement leaks and three corrupted-certificate/boundary controls reject.
• CPython 3.12.15 on macOS arm64 passes warning-strict normal and optimized direct checks with identical computed source-hash receipts. Each registered command was timed locally, at 1.70 and 1.82 seconds. The registered runner passes in 3.65 seconds. These are timeout observations, not performance comparisons.
• Metadata/source pins, AST compilation, exact committed bytes, executable modes and source-history ancestry pass. The official six-file export passes integrity checks, and Git-free read-only normal/optimized exported replays produce identical receipts. No local mathematical obligation is deferred to CI.

Reproduce from the repository root. Another CPython 3.12 installation may replace the interpreter path:

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

The direct commands print PASS_FOUR_ORTHOGONAL_ONE_INSERTION; the registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE. Receipts and distance labels are generated, never handwritten. Both registered commands have separate 120-second limits and use the standard library only. Scientific replay imports no predecessor executable.

Limitations

Exactly four dirty rows and exactly one nonzero insertion are fixed hypotheses. Additional workspace, multiple insertions, arbitrary row/edge prices, restricted coefficients and non-Boolean rings remain outside the result. The lower bound uses a finite certificate and a written lifting proof; independent human and formal proof-kernel review are unclaimed.

Coefficient preparation, routing, frame changes, owned storage, stock, precision, native realization and the complete recursive invoice remain unknown. Scalar ADD counts are not recursive child ranks. All five selected framework assumptions remain. No new kappa, supplier admission, selected-frontier update or complete multiplication theorem is claimed. No predecessor scientific suite, selected cold replay, Lean or Docker run was performed.

Open next questions

• Can a structural invariant replace the finite six-ADD column certificate? First classify five-gate words by ordered destination multiplicities. The finite result supplies exclusion without that classification.
• Can one additional restored dirty spectator shorten the column transfer? First specify its source coefficient and examine the five-row state domain. The current certificate does not cover it.
• Can a native producer and consumer realize the abstract bill? First bind one complete interface and price coefficient access, routing, frames, restoration and target debits. No physical realization is supplied here.

Related open work

• #7183 studies a three-row, non-Boolean radical demand with two insertions and directed-cycle prices. The row count, coefficient ring, insertion contract and ADD alphabet differ. This contribution computes no common numerical optimum.

The own-open-PR check since the CLAIM at 2026-11-11T13:23:16Z covered #7261, #7262, #7263, #7264, #7266, #7267, #7268, #7269 and #7270. None studies this four-row Boolean single-insertion minimum. No new same-object numerical comparison or disagreement was found.

CI notes

CI did not certify this package. At unchanged head 6d3af7bd7eb49b0adf34b7a144df8d17095b02c4, run 43029932349, plan job 130554063786, failed in Onboarding helper tests. The log reports 760 tests, exactly one failure and 24 skips. The failing test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with AssertionError that `make frontier7000-verify` was not found in the CI verification doc. The package replay job was skipped because planning failed.

The mismatch is in unchanged main files. Reading Git objects at canonical base e54523517d0f03e58352e647c403a5af3e525a57 confirms that certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still has Current selected-family replay: #6030 and make frontier6030-verify, without the expected current command. This PR adds only its six isolated package files.

The failed-job logs for run 42905094314, run 42903313790 and run 43029993808 were independently read. Each reports the same exact test and missing command, 760 tests, one failure and 24 skips.

All mathematical evidence above remains local at the unchanged head: normal/optimized direct and registered commands, generated receipt comparisons, metadata/source pins, committed bytes/modes/source history and read-only exported replays passed. No push, retrigger commit, workflow rerun or shared-file edit was made in response to this repository-level failure. Maintainer repair of main remains separate from this contribution.

Merge Danger

Door: two-way. Six isolated research-package files can be reverted together.

Blast Radius: research. Shared code, frozen sources and selected certificates are unchanged.