lm_above_threshold
The declaration establishes that the language model alignment fraction exceeds one half. Researchers auditing Recognition Science validation certificates cite it to confirm the J-cost model meets the majority threshold on the language-model substrate. The proof is a one-line term-mode reduction that unfolds the rational definition and normalizes the comparison.
claimThe language model alignment fraction satisfies $7/8 > 1/2$.
background
The module ThreeSubstrateValidationCert supplies a Lean certificate for J-cost validations across language models, photonic qubits, and magnetized plasma. languageModelAlignmentFraction is defined as the rational 7/8, standing for the fraction of MLP layers in which J-cost beats cross-entropy. The module states that all three substrates share the fixed point at x=1 and the descent direction from the J-cost gradient.
proof idea
The term proof unfolds languageModelAlignmentFraction to the literal 7/8 and invokes norm_num to discharge the inequality 7/8 > 1/2.
why it matters in Recognition Science
The result anchors the language-model component of the ThreeSubstrateValidationCert, which the module presents as hypothesis-grade empirical support for J-cost uniqueness. It aligns with the Recognition Science landmarks of J-uniqueness (T5), the self-similar fixed point (T6), and the multi-channel J-cost extension. No downstream theorems yet consume it.
scope and limits
- Does not derive the 7/8 value from the Recognition Composition Law or the phi-ladder.
- Does not address the photonic or plasma substrates.
- Does not prove the experimental 96.4% figure reported in the module doc.
- Does not claim the fraction is exact rather than a convenient rational proxy.
formal statement (Lean)
47theorem lm_above_threshold : languageModelAlignmentFraction > 1/2 := by
proof body
Term-mode proof.
48 unfold languageModelAlignmentFraction; norm_num
49
50/-- Photonic code rate: 7/8. -/