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 glance

• Series: Foundations (Six Birds)
• Theme: Foundations & meta-theory
• Format: Tool spotlight
• Complexity: Deep cut
• Paper: SB

Source anchors

• SB §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 numerically
• QT §11 Mechanized results in Lean (label: app:lean)
• BC §3.5 What we certify versus what we simulate