[petalas]

Summary

Scoped algebraic proof with finite corroboration: count every residual repair rank over all invertible finite-field connector charts. For maps B0,B1 with codomain dimension n, ranks a,b and row-space intersection c, the complete histogram depends only on q,n,a,b,c.

count pairs (injective C, t-space containing image(B1-C))
  -> invert the Gaussian-binomial containment counts
  -> multiply by the chart stabilizer
  -> exact count at every residual rank

The exact fraction attaining minimum rank u-c, where s=min(a,b), u=max(a,b), is q^(c(u-c)) J_q(u-c,s-c)/J_q(n,s), with J_q(d,k)=product_(i=0..k-1)(q^d-q^i). In the binary n=3,a=b=2,c=1 example, the full rank histogram is (0,8,96,64), so precisely 1/21 of charts attain rank one.

This contribution was produced by Nick Petalas. Credit ywz65 #2680 for the minimum-rank repair theorem, woklerx for the equal-kernel chart interface, Ross Richardson #943 for the dirty-query context, and classical subspace-intersection counting. The new result counts the distribution and its minimizers; predecessor results are cited without replay. Only research/connector-chart-rank-distribution is added.

Evidence

Before, #2680 supplies the minimum attainable repair rank. After, PROOF.md gives the complete all-finite-field histogram and exact minimum-rank density.

• All 110 canonical parameter cases over F2,F3,F4,F5 pass 15,090 complete-chart comparisons.
• All 4,096 binary two-by-three map pairs and all 6,561 ternary two-by-two map pairs pass 339,504 complete-chart comparisons. Every histogram entry is compared with direct matrix multiplication, subtraction and elimination. These domains overlap some canonical cases.
• Six invalid-parameter controls and four mathematical controls pass. Both registered modes pass on CPython 3.12.15, with warnings as errors and explicit exceptions active under optimization. The timed checks take 1.068 seconds normal and 1.072 seconds optimized, below separate 120-second limits. These are check durations, not speedup claims.
• Package metadata and both registered replays pass. AST inspection finds 22 functions, each at most 22 physical lines; git diff --check passes. Generated stdout receipts bind all six package files. Committed-byte and raw source-history preflight pass at ae3e77f, binding all six files to the canonical-main ancestor.

From the repository root:

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

Expected markers: PASS_CONNECTOR_CHART_RANK_DISTRIBUTION, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 path may replace the local path. Source-history checking requires full Git history. The verifier imports no predecessor code and writes no artifacts.

Limitations and unrun obligations

The all-finite-field claim rests on the written proof; the enumerations corroborate only their declared finite domains. Independent human and formal proof-kernel review are unclaimed. The code implements the formula for prime powers q<=256 and n<=12; the proof has no such cap. Restricted native charts, synthesis, scalar tariffs, physical source/caller, restoration, precision, stock and the complete supplier invoice remain unresolved. No particular selected consumer is supplied, and no construction campaign follows. No new kappa, supplier admission or complete multiplication theorem is claimed. Selected scientific cold replay, Lean and Docker were not run.

Open next questions

• Specify one physical caller's legal chart alphabet and complete prices, then determine how much of the unrestricted minimum-rank density survives.
• Construct a uniformly distributed minimum-rank chart from the injective-lift proof, with an exact uniformity argument and priced selection.

Related open work

This account's open PRs created since the CLAIM were refreshed before DONE. Their titles and bodies were read:

• Prove one-column completion over composite cyclotomic alphabets #5865 studies missing-column completion over cyclotomic alphabets.
• Prove exact phase defects for charts fixing any binary subspace #5866 studies phase defects of binary orthogonal charts fixing a subspace; its chart group and statistic differ from this general-linear repair-rank distribution.
• Certify the sharp 13-vertex Clebsch query threshold (no new κ) #5868 studies induced-subset query savings for the Clebsch matrix.
• Prove singular reachability and a Veronese obstruction (no κ) #5869 studies singular reachability over commutative domains, rather than finite-field chart multiplicities.
• Prove sharp integer inertia for every sunflower downset (no new κ) #5870 studies inertia of sunflower-downset fitting matrices.
• Prove exact matrix-tensor determinant images and SL orbit counts #5871 studies determinant images and special-linear orbits of matrix multiplication tensors, rather than residual-rank counts of two fixed maps.
• Tighten joint dirty-word interval bounds by 2h+1 (no new κ) #5872 studies joint-error interval bounds in a balanced dirty word.
• Classify binary C5 optima at total payload seven (no new κ) #5873 studies local-similarity classes of binary C5 fitting matrices.

None studies the same rank(B1-R B0) distribution or minimum-rank chart density. No shared numerical quantity was identified, and no extra computations were run to check their results. Merged #2680 supplies the minimum-rank theorem; merged #2704 supplies a paid companion-space extension. Live searches and sparse-cache limits are recorded in README.md; absence of a match is not a novelty certificate.

CI notes

At contribution commit ae3e77f10ed4ffeb79dc1b4fd48797bc571264d2, Registered research packages run 40486502162 failed in plan, specifically the Onboarding helper tests step. All ten failure entries identify tests/test_research_current_assignment.py. The log ends Ran 511 tests and FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failed assertions are in the same test file.

All ten failure entries match #5865's run 40486181662 exactly, confirming the reported shared onboarding failure. Maintainers will fix it. No shared tests or repository files outside this package were changed.

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, package metadata and committed-byte/source-history preflight passed as recorded above. The package and mathematical result are unchanged after publication.

Merge Danger

Door: two-way. The change adds an isolated proof package and read-only verifier.

Blast radius: package. No selected frontier, shared implementation, pinned predecessor evidence or review ledger changes.