[petalas]

Summary

Scoped proof with exact finite corroboration. In ADD-once bipartite dyadic shears, every fully restored source gauge on a source used at most twice can be erased without increasing scalar allowance, regardless of simultaneous target gauges or ADD order. The coupled optimum therefore equals Dugongue's target-only interval formula from #4518. This answers the low-use part of its second Open next question.

Sources with at most two uses: coupled optimum = target-only optimum
Equal-coefficient d-leaf star: d c(o) + 2 min(d,2)|v|
First strict example, three half-weight ADDs: 30 -> 28 scalar units

The threshold is sharp. The proof covers arbitrary finite integer gauge paths and arbitrary dirty initial data, charges every entry/return SCALE, and supplies the matching inverse-dual interface. The elementary shared-factor star construction and classical Levi identity are credited; the new result is the source-elimination theorem and exact threshold under this schedule contract. No new kappa is claimed.

This contribution was produced by Nick Petalas. Credit David Ganz / Dugongue's #4518 tariff, target-only theorem and question, its #4502/#4505/native lineage, Ali Romman's #5783/#5824/#5830 fixed-graph optimization, and Richard Hoekstra's #1235 phase-gauge tour work. No predecessor scientific suite was replayed or audited.

Evidence

• Before: Levi-adaptive gauges: 666 → 616 scalar allowance; lower suffix error #4518's interval optimum restricts all source gauges to zero and leaves coupled paths open. After: the same optimum is proved for the enlarged class whenever source use is at most two; a fully restored degree-three star attains the exact smaller cost.
• A separate bounded Dijkstra implementation checks all 256 two-source/two-target absent-or-valuation lists, every ADD order and gauge states in {-1,0,1}. Both coupled and source-fixed minima match the interval formula. The written proof, rather than the finite gauge box, establishes the unbounded conclusion.
• 81 prescribed gauge-path fixtures, 18 star and 18 reversed-star shortest-path problems, 51 further exact star witnesses, full arbitrary-input matrix/inverse-dual endpoints, and eight rejecting controls pass in normal and optimized modes. Reversed stars exercise profitable target movement while every source is low-use.
• CPython 3.12.15, standard library only. Warning-strict direct checks pass in 1.007 s and 0.953 s; the registered wrapper runs both commands and passes in 2.099 s. Each registered command is below its 120 s timeout. These are check durations, not a speed comparison.
• Metadata/pins, committed bytes, source ancestry and executable modes pass at 00675f5c3e011bc0627d5e1ea381ed30f922eb0c. The official eight-file export passes integrity and declared-input coverage; normal and optimized replays pass outside Git with an empty executable PATH. A generated stdout receipt binds the six package files by SHA256.

From the worktree/repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -X dev -I -B research/low-use-source-gauges/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -X dev -I -O -B research/low-use-source-gauges/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/low-use-source-gauges
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/low-use-source-gauges --check-source-history --check-modes

Any CPython 3.12 executable can replace that path. Direct replay prints PASS_LOW_USE_SOURCE_GAUGES no_new_kappa; registered replay prints PASS_REGISTERED_RESEARCH_PACKAGE. Sparse checkouts should materialize research/levi-adaptive-gauges first. SOURCES.json pins the cited mathematical inputs and historical task/selected provenance. The cheap selected-baseline binding orientation also passes after materializing its closure.

Limitations

The theorem fixes the list of ADD contributions, allows only disjoint pure source and target roles and positive power-of-two gauge scales, uses #4518's scalar tariff, and restores every gauge. It does not optimize general circuits, coalescing, borrowing, source writes, target reads, non-diagonal changes or unreturned gauges. No actual P104 word or paid/target histogram is changed.

Native implementation, physical precision/tape costs, rounded arithmetic, caller integration and complete multiplication remain outside this result. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. Selected cold replay, predecessor scientific suites, Lean and Docker were not run. Ruff is unavailable locally; warning-strict Python checks pass. The README records the bounded CPython allocation deviation under Power of Ten.

Open next questions

Which sources with at least three uses in one named complete native window admit a profitable restored gauge after the surrounding interfaces are charged? Extract one ADD-once bipartite window, retain its target-only optimum and prune its low-use sources by this theorem. The emitted-window, caller, native and precision/tape contracts are the blockers to adoption. This contribution stops at the first sharp threshold result.

Related open work

• Optimize THREE-base logical defect guards: save 2336 per direction (no κ) #6121 optimizes ADD/SCALE logical defect guards on the historical THREE-base suffix ledger. This contribution optimizes a different quantity, scalar allowance in bipartite gauge schedules, and supplies no new defect budget or native-word installation.
• Resynthesize P104 scalar prefix: 3,577 fewer events (No new κ) #6116 resynthesizes the selected P104 scalar prefix by changing its arithmetic word. This theorem fixes the ADD list and prices source/target representation paths; its 28/30 star is not a cost claim for that complete prefix.

Main, merged-work, open-PR, recent account, Discussion and local-claim searches were refreshed before the commit. No competing claim on this source-use threshold was found. Cached local-note coverage is sparse and unpublished work is not excluded. No global literature-priority claim is made.

Completion refresh inspected all eleven other open account PRs created since the advance CLAIM: #6121, #6122, #6123, #6124, #6125, #6126, #6127, #6128, #6129, #6131 and #6132. #6121 concerns the same historical native family and is distinguished above. The others concern query capacity, centre kernels, sampler tables and reversal, leaf tariffs and tails, Gaussian conditioning/concentration, rounded Walsh error and ADDRESS characters. None supplies a competing coupled dyadic source-gauge optimum or a value for this star. No extra computations were run to compare their results.

Advance claim: #1166 (comment)

CI notes

CI uncertified at commit 00675f5c3e011bc0627d5e1ea381ed30f922eb0c. In run 40693369435, plan failed in Onboarding helper tests and replay was skipped. The package replay did not run in CI. No package check was certified by that run; the scientific checks above are local evidence only.

The failure is inherited from canonical main 721c03822c421b7f7fcbb2598993b947c4d88419, the PR6030 kappa-admission commit pushed with [skip ci]. It updated the selected record without matching the required documentation format/navigation. The failed-job log confirms exactly these two onboarding failures:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects **0.003125804** in docs/research/current-status.md. The document instead bolds the longer expression 0.003125804 = 781451/250000000, so the exact marker is absent.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes selected PR5599 and make frontier5599-verify.

The log reports 678 tests, two failures and 23 skips. PR #6120's run 40691776027 fails the same two tests, confirmed from its failed-job log. This package did not cause either failure. Incidental negative-fixture diagnostic text is not counted as another test failure.

A maintainer re-run is required after main's documentation is fixed. No branch push, documentation fix or CI re-run was performed for this blocker. Local normal/optimized, registered, committed-source and portable-export checks remain passing evidence; hosted replay remains uncertified.

Merge danger

Door: two-way. Blast radius: isolated.

Adds one standalone research package and opt-in replay manifest. No shared implementation, frozen source, existing attribution or selected record changes.