The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture

Ramachandra and Natarajan conjectured a bound on the pairwise independent correlation gap in their 2025 Operations Research Letters paper. An explicit counterexample refutes it.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Jun 18, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: preprint. AI contribution: ai-discovered. Imported under CC BY 4.0.

Canonical aliases: The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture · Correlation gap

Confidence: Not scored

Registry verification: lean checked · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

Arjun Ramachandra
human · human collaborator

Karthik Natarajan
human · human collaborator

GPT-5.5 Pro
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture

specialist registry · passed

The refutation is an explicit counterexample, so it is a finite check. Short arXiv note, not peer-reviewed. A Lean 4 / Mathlib formalization of the counterexample was contributed in August 2026 by its author, produced with Codex. Curator source audit: all 579 lines read, with no sorry, admit, native_decide, unsafe declaration, user-declared axiom, implemented_by or partial def anywhere; finite checks go through kernel decide and rational identities through norm_num, and Mathlib is pinned to an exact revision on toolchain v4.33.0-rc2. The curator has not compiled it, and it is the work of the same person who reported the result, so it is not third-party corroboration. What the formalization does and does not settle is worth stating exactly. Its final theorem is a seven-part conjunction certifying the witness and its bounds: the three-atom distribution attains the target marginals with expected coverage 4, no distribution exceeds 4, the product distribution is pairwise feasible, every pairwise-feasible distribution is bounded by $479/160$, and $4 \div (479/160) = 640/479 > 4/3$. Both bounds are universally quantified rather than spot-checked. What the file never states is the Ramachandra-Natarajan conjecture itself, so the step from this instance to the conjecture being refuted stays informal and rests on the conjectured bound really being $4/3$. That is the difference between a kernel-checked artifact and an audited claim, and why this sits on the unaudited Lean rung.

Source ↗

Lean 4 / Mathlib formalization of the counterexample (Codex, author-contributed)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Artifacts and verifiers

Lean 4 / Mathlib formalization of the counterexample (Codex, author-contributed)

lean artifact · pending