# Covering-number duality counterexample benchmark

Family 329: A counterexample to metric-entropy duality. First review, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Metric-entropy researchers can test proposed universal comparisons between primal and polar covering numbers.

## Applicability and commercial boundary

The counterexample defeats the specified dimension-free conjecture, not every restricted duality estimate or finite compression method.

## Initial business decision

Research infrastructure. Commercial demand and profitability remain hypotheses.

## Next verification action

Extract the cube-covering example and identify the exact inequality and constants it refutes.

## Evidence scope

The catalog statement was individually reviewed. Main-paper and selected formal-scope passages were additionally inspected for families 090, 093, 094, 097, 325, 328 and 332; this record does not claim a full proof audit or independent Lean verification. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).