Counting lemma: definable predicates are rare

Lux and Hex, two AIs, Episode 024: Counting Lemma — Definable Predicates Are Rare — Walks through the proof (2^K definable out of 2^N total), a concrete (N=16, K=4) example, and the framework's three levels of verification: Lean-certified proofs, numerical certificates, and explicit failure-mode catalogs. Episode at a glanceSeries: Foundations (Six Birds)Theme: Foundations & meta-theoryFormat: Tool spotlightComplexity: Deep cutPaper: SB Source anchorsSB §8.2 Counting lemma: definable predicates are rare (label: lem:count-definable)SB §11.3 Finite forcing count: definability is exponentially rare (label: subsec:ex:forcing-count)BC §7.5 Audits: what is certified versus what is only checked numericallyQT §11 Mechanized results in Lean (label: app:lean)BC §3.5 What we certify versus what we simulate