[petalas]

Summary

Finite certificate over F2: eight rank-one thin queries are necessary and sufficient for the saved PR #5699 schedule. The six full calls stay in order (1,3,2,4,0,5) and thin queries occur only after two and four full calls.

two aggregate matrices, 128 binary entries
  -> all 1,728 endpoint coefficient equations
  -> rank 104, 24 free coordinates
  -> all 16,777,216 solutions
  -> minimum aggregate rank sum 8
  -> at least 8 rank-one queries

This answers the predecessor's first Open next question for its saved gap pair. The new contribution is the lower bound. Credit theon199 for PR #5699's attaining eight-query square-zero construction and explicit question, Ali Romman for PR #5673's full matrices and normalized model, and Cinna Davis with the retained controller/native/helper/compiler lineage. No new kappa or selected-frontier change.

Evidence

Before: eight queries were an upper bound; the source explicitly left fewer thin queries open. After: every feasible aggregate pair in this fixed binary schedule has rank sum at least eight. Rank subadditivity proves the query lower bound, and the cited construction attains it.

Local CPython 3.12.15, macOS 27.0.1 arm64, standard library only. The direct warning-strict verifier and both registered normal/optimized exhaustive replays pass. Each includes all 16,777,216 pairs, 512 independent kernel/rank controls and nine boundary, rejection or enumeration controls. The complete rank-pair histogram also agrees with the discovery calculation. The verifier imports no predecessor code and prints a generated receipt bound to package and input SHA256 hashes.

Exact reproduction from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-burst-thin-rank/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/two-burst-thin-rank/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-burst-thin-rank
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-burst-thin-rank --check-source-history --check-modes

Use the corresponding CPython 3.12 path on another host. Expected PASS_TWO_BURST_THIN_RANK, solutions=16777216, minimum_rank_sum=8, and PASS_REGISTERED_RESEARCH_PACKAGE.

Both registered checks were also individually timed at 33.699s and 34.327s. Metadata/pins, committed bytes/executable modes, full declared-source ancestry and the official ten-file export with all three declared inputs pass. A read-only Git-free optimized export replay with an empty PATH also passes in 33.13s and preserves all 11 exported files.

Limitations

The certificate covers F2, these exact full matrices/order and these two gaps. Extension fields, changed schedules, extra full calls, stage-dependent thin entries, native lifts, physical restoration and complete supplier/outer admission remain outside it. No Lean/kernel proof or independent human review is claimed. The predecessor's optimistic invoice is cited, not replayed. All five framework assumptions retain their status. All new files are confined to research/two-burst-thin-rank/.

Open next questions

• Can a different gap pair, full-call order or more occupied gaps admit seven binary queries? Start from the changed complete coefficient system. Require a favorable complete invoice before physical expansion.
• Does the rank-sum bound extend to nonbinary characteristic-two fields? The same 24-parameter affine system is a symbolic starting point; no extension-field claim follows from the binary census.

Related open work

Refreshed all account open PRs created since the CLAIM at 2026-10-27T11:59:19Z. The nine other PRs are #5897 through #5905; none studies the fixed PR #5699 full order and two-gap thin-query count or its aggregate rank-sum quantity. No same-object open contribution or conflicting number was found. PR #5699 and PR #5673 remain merged predecessors. This supplements the earlier live 448-open-PR body scan and handoff check; unpublished work remains outside coverage.

CI notes

At commit 8448d0552496963f4cd02ee7abec89c6fb649690, CI run 40502000289 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'. Assertion failures concern frozen-card/current-route expectations, missing-reference handling and sparse omission lists. All ten failure/error headers match PR #5905's failed plan log exactly. This is the reported shared main de5ec2e7a onboarding failure. Maintainers will address it; no shared source or tests were changed.

Hosted replay has not certified this package; only local scientific and publication checks have run. Local normal/optimized exhaustive replays, metadata/pins, committed-byte/mode/source-history checks, official export integrity and read-only Git-free replay pass. The CI result is not mathematical acceptance.

Merge danger

Door: two-way. Blast radius: isolated research package. No shared implementation, frozen source, selected result or predecessor credit is changed.