[petalas]

Summary

Scoped written proof with exact finite corroboration. One lower-only phase in a rational shear superposition word can be replaced by its binary mixing, preserving every required upper-supported endpoint entry and the exact paid profile.

X = L F B
L,B over the upper-shear algebra; F over the lower-shear algebra
P([u_A v_K u_B]) = [u_(A+B)]
P(X) = L epsilon(F) B
upper-supported required entry => deletion preserves that entry

The new label retraction is well-defined for singular K and arbitrary rational or dyadic coefficients. It eliminates a proposed single-phase nonabelian escape before further word search. No new kappa, supplier or complete multiplication theorem is claimed.

Task and overlap

Card bit-helper-contract, scope changed from G20 helper construction to the explicit first open question of Sagar Thakrar's PR2613. Base main is 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8. TASK.json retains the full frozen card, changed scope and exact source pins.

The theorem is an algebraic deletion reduction, distinct from ywz65's #2671 and Nick Petalas's #6027 invariant-lattice filters. dcharb78's #3183 separately reports a finite width-two block-Borel screen; its result is neither replayed nor claimed here. Exact main READMEs and the source proof are provenance pins, with no predecessor executable dependency.

Advance claim. The delivered factorization condition clarifies the claim's initial contiguity wording, as recorded in the scope update.

Evidence

Before, the source handoff proposed one lower-left insertion without this deletion reduction. After, PROOF.md constructs an upper-shear bimodule retraction and proves deletion for every width and finite register count in the stated factorization class.

Local CPython 3.12.15, Darwin arm64, standard library only:

• Normal and optimized direct checks print PASS_SINGLE_LOWER_PHASE_DELETION. Each checks 4,256 complete finite-field triples, including 1,420 repeated-label comparisons; 144 rational triples, including 92 singular K; 144 bimodule identities; 24 full formal factorizations with 112 entries, including 42 upper-supported entries; six hypothesis/mutation controls and three dimension rejections.
• Both registered commands pass, with each direct command below one second locally against a 60-second limit. The final complete registered replay took 0.66 seconds. These timings establish replay budget only, not a performance improvement.
• Metadata and provenance checks pass. PASS_RESEARCH_COMMIT_BYTES authenticates all six package files and source ancestry to the pinned base.
• Official committed export and declared-input coverage pass. A fresh optimized Git-free replay with empty PATH and read-only exported files passes, preserving all seven exported files, including the export manifest. Receipts are generated and bound to package hashes.
• The selected binding-only orientation prints PASS_RESEARCH_BASELINE_BINDINGS_ONLY; it was not a scientific selected replay.

Reproduce from the repository root:

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

Any CPython 3.12 executable may replace the displayed path. The universal theorem follows from the written proof, not finite extrapolation.

Limitations and unrun obligations

The final product must have the factorization L F B. Ordinary one-sided circuits with a contiguous lower-only phase qualify. Chronological contiguity in a model permitting both left and right actions does not suffice by itself. Internal register-dependent Levi maps, separated lower phases and non-upper required entries are excluded. Clean auxiliary garbage is allowed; dirty-input independence requires including the relevant data-from-auxiliary zero entries among the preserved entries.

No general full-Borel impossibility or new three-call census is claimed. Physical legality, native operators, ownership/restoration beyond the declared entries, stock, precision/tape, uniform compiler, complete signed invoice and outer admission remain separate. No selected scientific replay, native replay, Lean, Docker or complete multiplier ran. Independent human and formal proof-kernel review are unclaimed.

Credit

This contribution was produced by Nick Petalas. Credit Sagar Thakrar's PR2613 for the question and model, eumemic's PR288 for the register framework and existing upper-only obstruction, Swapnil Jain's related rank framework, ywz65's distinct gauge work, and the retained BIT lineage. General literature priority for elementary block multiplication and bimodule methods is unclaimed. Apache-2.0; predecessor notices remain unchanged.

Open next questions

• Can separated lower phases yield a favorable completed word? Start with a width-two word, exact data endpoint and full paid/target invoice. No such saving or physical caller is supplied here.
• What deletion criterion survives internal Levi maps? First specify the enlarged middle alphabet and test its label collisions. The supplied Levi identity already rejects naive deletion.

Related open work

Completion refresh inspected the titles and stated scopes of all 14 other open account PRs created since the advance CLAIM, #6149 through #6163 excluding this PR. None studies rational upper/lower single-phase deletion or supplies a competing value. No additional mathematical replays or comparisons were run.

No other open PR found in the live overlap corpus studies this exact single-lower-phase deletion question. Adjacent rational formal-register work has different endpoint/alphabet contracts:

• Prove eight-call minimum for one arbitrary scaled rank-one frame #6079 proves a minimum for one added scaled rank-one frame; this contribution changes the lower-update phase contract and computes no competing minimum.
• Classify usable positive rank-one frames and their exact call counts #6089 classifies positive extra-frame use and attainable lengths; this contribution supplies a deletion map, without a positive-frame classification or length spectrum.
• Two added positive rank-one BIT frames cannot give a seven-call word: answers #6104 Q1 (no new κ) #6111 excludes two added positive frames at seven calls; this contribution does not evaluate those profiles or claim their floor.
• Exclude PR6111's eight-call half-frame candidate (No new κ) #6148 excludes one half-frame candidate by two-use lattice descent; this contribution requires no such lattice descent and computes no candidate call count.

No extra computations were run to compare predecessor numbers. The complete open-PR/Discussion overlap snapshot, targeted merged searches and current-main text searches are coordination evidence, not a novelty certificate; sparse local-note coverage remains incomplete.

CI notes

CI is uncertified at contribution commit e88627a4a85d57d204e950e9b35c29736490bbae. In run 40726511772, plan was cancelled after running from 16:29:48 to 16:40:03 UTC. The job annotation says "The job has exceeded the maximum execution time of 10m0s". The workflow sets timeout-minutes: 10; cancellation occurred during "Onboarding helper tests". The package replay was skipped and did not run in CI. This timeout occurred in shared planning before the package replay, not in this contribution's verifier.

Separately, completing planning would still fail on two documentation mismatches inherited unchanged from main 721c03822c421b7f7fcbb2598993b947c4d88419, whose PR6030 admission commit includes [skip ci]:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects literal **0.003125804** in docs/research/current-status.md; the existing bold span also includes the rational expression.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md; the current section still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and its replay command.

I read the comparison logs: PR6150 run 40724809789 and PR6120 run 40691776027 both finish with exactly those two failures. Open PR6050 targets the documentation repair. This isolated package changes neither affected documentation nor their tests.

All local checks listed above passed, but they are not hosted certification. A maintainer must re-run CI after main is fixed, with planning able to finish within its time budget. No code was pushed, no CI re-run was requested, and main's docs were not changed for this completion.

Merge danger

Door: two-way. Blast radius: package.

Only six new files under research/single-lower-phase-deletion/, with opt-in replay. Shared code, frozen inputs and the selected frontier are unchanged.