[petalas]

Summary

Finite exact certificate with a scoped counting proof: every ordered rank-two idempotent pair on F2^4 has a complete two-guard header exactly when Q != I+P. The sharp minimum positive success probability is 144/20160 = 1/140, attained by exactly two parent conjugacy classes.

The package supplies the complete thirty-row Schur continuation table, adding 23 rows beyond PR4735's seven source-specific parents. Each row retains all seven ordered rank-one successor types, split multiplicities, witnesses and exact guard-depth counts.

rank-two pair
  canonical parent
    seven ordered successor counts
      exact complete two-guard header count

The thirty-state classification is MatShep / #4762's result. The recurrence, terminal invariant and terminal split counts are codyw912 / #4735's; Mihir Srivastava / #4689 retains the seven original states and guard numbers. They are cited dependencies, not audit deliverables. Nick Petalas's older #4915 orbit weights and #4817 first-pivot law are related context. The new result and package were produced by Nick Petalas.

Advance CLAIM. No new kappa or selected-frontier change.

Evidence

Before, the full parent classification leaves split transitions open, and the source-specific continuation table covers seven parents. After, every parent has exact successor multiplicities and a complete-header count.

• Packed intrinsic split enumeration agrees with a separate dense implementation on all thirty rows and 900 distinct admitted representative splits.
• The oracle directly evaluates the two leading minors over all 604800 parent/header pairs. It does not use the terminal g table or recurrence to count full successes. It independently checks all transition cells and witnesses through literal block Schur complements.
• The complete parent cover is checked against both raw binary matrices and the full-header conjugacy domain. Five rejection controls cover false first/full counts, false transitions, false orbit size and singular inversion.
• All four registered warning-strict normal/optimized commands pass on CPython 3.12.15, macOS arm64. Individually timed direct runs took 0.2021, 0.2028, 1.4666 and 1.4270 seconds, rounded upward. These are check durations, not speed comparisons.
• Package metadata/pins and the registered package replay pass. Seven-file committed-byte/source-history preflight passes at 4d72cf3d54c1f4c547e52d7c5cf1c1d5009b1043.
• Official committed export/integrity, both optimized read-only Git-free exported verifiers, unchanged eight-file export inventory, and two actual source-corruption rejection runs pass. In-memory syntax/function-size inspection and whitespace checks pass.

From the worktree root, using the exact local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/binary-rank2-schur-continuations/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B -O research/binary-rank2-schur-continuations/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/binary-rank2-schur-continuations/oracle.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B -O research/binary-rank2-schur-continuations/oracle.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/binary-rank2-schur-continuations
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/binary-rank2-schur-continuations --check-source-history

Any CPython 3.12 installation can replace that interpreter. Expected markers are PASS_BINARY_RANK2_SCHUR_CONTINUATIONS, PASS_BINARY_RANK2_DIRECT_HEADERS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. RESULT.json is generated and bound to all six other package files.

Limitations and unrun obligations

Exactly two rank-two idempotents on F2^4, leading guards of orders one and two, and complete equally weighted headers. Larger dimensions, other fields and restricted or weighted covers are outside scope.

Header selection/emission, actual producer compatibility, physical reads, dirty restoration, native alignment, stock, precision/tape and complete consumer costs remain open. The finite probability does not supply a physical header cover or supplier admission. All five selected framework assumptions remain. No complete multiplication theorem, independent human review or formal proof-kernel acceptance is claimed.

Every registered obligation ran locally. Predecessor scientific suites, the selected cold replay, Lean and Docker were unrun. Bounded Python managed allocation and unavailable separate static analyzers are documented in the README.

Open next questions

• Derive the two sharp 1/140 rows from image/kernel incidence without enumerating binary 4x4 partners, preserving ordered terminal types. This could support a scalable symbolic successor law.
• Bind an actual producer's two independent rank-two requests and payload field to a fully priced header-selection/consumption contract. Physical implementation and odd-field transfer remain missing.

Related open work

• Reduce complete nilpotent-Schur replay time by 9.15% (no κ) #6460 accelerates the complete PR5267 nilpotent-Schur replay while preserving its outputs. This package supplies new binary rank-two continuation counts and does not extend that cache or recompute its performance or predecessor counts.

Fresh open-PR and handoff checks found no other open PR computing these exact continuation rows or the 1/140 minimum. Main/merged searches and the complete cached text snapshot supplement that check; arbitrary historical diffs and unpublished work remain coverage limits. No extra computations were run to audit other PRs.

Before DONE, the completion refresh inspected all 14 other open PRs created by this account since the advance CLAIM. They study other circuit, connector, carry, feature, repair-rank or replay quantities. None computes these ordered binary rank-two Schur continuation rows or the 1/140 minimum. No additional mathematical computations were run to check those contributions.

CI notes

Both hosted checks passed at exact head 4d72cf3d54c1f4c547e52d7c5cf1c1d5009b1043 in run 41609889636: plan in 4m26s and registered package replay in 2m28s. The completed checks were read once; separate run metadata confirms the matching head, completed status and successful conclusion.

All four registered normal/optimized commands, local package metadata/replay, byte/ancestry preflight and export checks passed. Hosted CI certifies execution of this scoped registered package; independent human review, formal proof-kernel acceptance and physical supplier admission remain unclaimed. No check failed.

Merge danger

Door: two-way. Blast radius: isolated package. Seven new files; shared code, frozen predecessors, their credit lines and the selected record are unchanged.