[petalas]

Summary

Exact finite certificate with a written model reduction. Answer unpaidmoon's #4137 Open next question 1: no restored ten-step five-source walk with exactly one triple and nine singleton steps covers all 32 prefix-difference masks. The sharp maximum is 30.

one triple + nine singletons
  -> 120960 complete words
  -> 0 universal covers, sharp maximum 30/32
  -> any covering ten-step walk with arity >= 3 costs at least 122 source leaves
     at prices (1,4,10,16,28), versus the cited pair-only bill 120

The 122 lower bound is strict over 120; attainment of 122 is unclaimed. No new kappa, supplier admission or selected-frontier change. Only research/one-triple-ten-step-obstruction is added.

This contribution was produced by Nick Petalas. Credit unpaidmoon #4137 for the explicit question, charged triple model and earlier lower bounds; Aaron Wang for the pair-source word; Ross Richardson #943 for the paired-readout model and prefix criterion; Adam Gall for the credited fixed-alphabet tariff work. Predecessor results are cited, not audited or replayed as the deliverable. Advance CLAIM.

Evidence

Before, #4137 leaves bill-120 attainment by a one-triple ten-step walk open. After, complete multiset enumeration excludes it. Rotation and source permutation reduce every active-source case to the enumerated incidence profile (4,2,2,2,2).

• The primary lexicographic census checks all 120960 words across all ten triples, without pruning. A separate position-placement census checks 22680 canonical words with a bitset-translation mask oracle. Their complete histograms agree. Sharp maxima are 30 when the triple contains the four-use source and 26 otherwise.
• Seven invalid-word controls, all ten rotations of an attaining word, a positive full Gray-cycle cover, repeated-prefix controls and exact source-leaf arithmetic pass. The sharp word misses precisely masks 5 and 25.
• Local CPython 3.12.15 warning-strict normal and optimized direct runs pass in 1.58 and 1.53 seconds. Both registered commands pass and are below their separate 120-second limits. Metadata/source pins pass. Committed-byte/source-history preflight passes for all six files at commit 101b8614408dff629a9b4a2d6159ab02427d6eca, with canonical-main ancestor de5ec2e7a6074068452f45b1a191b5c72832df4a. A read-only six-file Git-free export passes optimized replay with empty PATH, produces an identical receipt, and preserves every file hash.
• The prescribed 945-pairing card command and current selected-baseline binding check were run once as orientation. They are not new scientific evidence or the contribution.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/one-triple-ten-step-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/one-triple-ten-step-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/one-triple-ten-step-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/one-triple-ten-step-obstruction --check-source-history

Replace the interpreter path with another CPython 3.12 installation elsewhere. Expected markers are PASS_ONE_TRIPLE_TEN_STEP_OBSTRUCTION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Receipts are generated from the actual census and bind package-file SHA256 hashes; the verifier writes no files. Exact historical blobs and SHA256 provenance are in SOURCES.json. There are no live external mathematical inputs.

Limitations and unrun obligations

This is a finite source-word exclusion and a price corollary in the fixed paired-readout interface. It is not a claim of infeasibility for all connector programs. Extra connectors, specialized operator relations, changed output readouts and other pricing contracts are outside scope. No 122 attaining word is supplied.

No native array execution, physical routing/frame/coefficient/scratch/hosting/precision/tape/stock invoice, recursive supplier or complete multiplication theorem is established. Lean, Docker and the selected scientific cold replay were not run. Independent human and formal proof-kernel review are unclaimed. All registered package commands ran locally.

Open next questions

• Determine the exact minimum bill for a covering ten-step walk with a step of arity at least three. Start with incidence total 14; the new lower bound does not supply a witness.
• Replace the sharp 30-mask finite ceiling with a short structural proof, starting from the five even and five odd prefixes and the two triple-incidence classes.

No follow-on search is included.

Related open work

• Add five-source arity-four connector cost bound #3909 studies the same five-source connector with an arity-four operation and a uniform dispatch-plus-addition ledger. This result excludes the one-triple incidence-12 case and proves a nonuniform source-leaf gap. The distinct 149-unit bill and 23184-word branch are not recomputed or compared numerically.
• Research: high-arity connector toggle cost floor #2249 prices arity-four-through-six calls on six sources with arbitrary output readout counts. Its source count, scope and tariff differ; it does not determine this five-source one-triple class.
• Research: seven-source weight-at-most-four connector boundary (L<=18 excluded, 216-call floor) (no new kappa) #4079 studies the seven-source weight-at-most-four boundary. This package addresses five sources and a different exact ten-step domain; no predecessor numerical result is recomputed.
• research: audit minimal joint-output connector charge #3897 studies a joint output operation on one source and two targets. This package changes source-step arity and retains paired individual readouts.
• Count every finite-field connector chart repair rank (no new κ) #5867 counts finite-field chart-repair ranks for two connector maps. The statistic and alphabet differ from prefix-mask coverage and source-leaf pricing; no common number is compared.

Live open PRs, recent account work and handoffs, local reservations, all-main distinctive README terms and merged searches were repeated before committing. This is bounded overlap evidence, not a claim about unpublished work.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-27T10:38:28Z were refreshed and their titles/bodies inspected:

• Prove exact additive cost for two anticommuting query types (no new κ) #5877 studies two anticommuting query types and their additive tariff.
• Prove the sharp codimension bound for partial query acquisition #5878 studies approximation for partial subspace acquisition.
• Prove the binary-affine call floor with clean erasure (no new κ) #5879 studies an affine-call floor with clean erasure.
• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880 studies clean storage for reversible Gaussian overwrite.
• Construct a 22-channel monotone delivery counterexample (no new κ) #5881 gives a monotone-tariff immutable-channel quotient witness.
• Prove sharp tensor composition of dyadic helper capacity (no new κ) #5882 studies tensor composition of dyadic helper capacity.
• Prove odd-axis Clifford tensor Gram lower bound above one half #5883 strengthens an odd-axis Clifford tensor Gram bound.
• Prove exact tariffs for consecutive queries sharing a detour #5884 studies consecutive native queries sharing a detour path.

None studies this five-source one-triple prefix-mask class or its source-leaf floor. No shared numerical quantity or disagreement was identified, and no extra computations were run to check those results.

CI notes

At contribution head 101b8614408dff629a9b4a2d6159ab02427d6eca, run 40491011484 failed plan solely in the Onboarding helper tests step. All ten failure/error headers name test_research_current_assignment; their source traces point to tests/test_research_current_assignment.py. The exact summary is FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures consist of 0 != 16, five instances of ValueError not raised for missing-helper/reference controls, and two empty-versus-expected sparse omission-list comparisons. This matches the failed step, test file and eight-failure/two-error signature reported on recent canonical-main de5ec2e7a PRs, including #5867's run 40486502162. Shared tests and implementations were not changed; maintainers will handle that failure.

The replay job was skipped. Hosted replay has not certified this package; only local scientific checks have run. Local warning-strict normal/optimized direct checks, both registered commands, metadata/source pins, committed-byte/source-history preflight and the read-only Git-free optimized replay passed as recorded above. The six-file package and mathematical result are unchanged after publication.

Merge danger

Door: two-way. Blast radius: isolated.

Six files in one new opt-in research package. Shared code, frozen predecessors, selected frontier and review ledgers are unchanged.