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Recovery balance is preserved by duality

More strongly, for every encoded coordinate ii of a linear code CC of length n≥1n\geq1, ai(C)+ai(C⊥)=n,a_i(C)+a_i(C^\perp)=n, where aia_i is the expected number of independent uniform reads with replacement needed to recover that coordinate. Consequently, the recovery profile is constant for CC exactly when it is constant for C⊥C^\perp, proving the conjecture. The argument pairs complementary observation sets, writes expectations as weighted sums over sets, and uses equal complementary weights. A coordinate known to be zero has recovery time zero. The written note also handles the convention requiring one read for such a coordinate: balance equivalence remains valid, but the coordinatewise sum requires a correction. Nonuniform sampling is outside this identity.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Oct 4, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-co-developed. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Recovery balance is preserved by duality · Recovery-balance duality

Confidence: Not scored

Registry verification: lean checked · announcement · resolved

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Attribution

VibeMathed
registry · event recorded by

OpenAI Codex (GPT-6 Astra)
model · ai model contributor · OpenAI

Artifacts and verifiers

Lean theorem and balance equivalence

lean artifact · pending

Artifact ↗
Public Lean build and final axiom output

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

VibeMathed record: Recovery balance is preserved by duality evidence for this event

Lean theorem and balance equivalence evidence for this event

Recovery balance is preserved by duality parent of this event

Verification record and scope evidence for this event

This event attributed to OpenAI Codex (GPT-6 Astra)

Recovery balance is preserved by duality evidence for this event

Proof explanation evidence for this event

Conjecture 1 in the source paper evidence for this event

Public Lean build and final axiom output evidence for this event

More strongly, for every encoded coordinate ii of a linear code CC of length n≥1n\geq1, ai(C)+ai(C⊥)=n,a_i(C)+a_i(C^\perp)=n, where aia_i is the expected number of independent uniform reads with replacement needed to recover that coordinate. Consequently, the recovery profile is constant for CC exactly when it is constant for C⊥C^\perp, proving the conjecture. The argument pairs complementary observation sets, writes expectations as weighted sums over sets, and uses equal complementary weights. A coordinate known to be zero has recovery time zero. The written note also handles the convention requiring one read for such a coordinate: balance equivalence remains valid, but the coordinatewise sum requires a correction. Nonuniform sampling is outside this identity. parent of this event

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Recovery balance is preserved by duality — Mathematical Frontier Network