ThreeSubstrateCert
ThreeSubstrateCert packages the J-cost fixed-point, descent, and symmetry properties observed in language-model, photonic, and plasma experiments into a single certificate. A researcher working on Recognition Science foundations would cite it when confirming that the same J-cost behavior appears across three independent substrates together with the shared 7/8 alignment rate. The definition declares the record type whose concrete instance is supplied downstream by threeSubstrateCert using shared lemmas.
claimA three-substrate validation certificate is a record asserting that the set of validation substrates has cardinality three, that the J-cost function satisfies $J(1)=0$, that $J(r)>0$ for every positive real $r$ different from 1, that $J(r)=J(r^{-1})$ for every positive real $r$, that the language-model alignment fraction equals $7/8$, that the photonic code rate equals $7/8$, and that the language-model alignment fraction equals $(2^3-1)/2^3$.
background
The module summarizes Recognition Science Experiment B, which tests J-cost on language models, photonic qubits, and magnetized plasma. ValidationSubstrate is the inductive type with three constructors corresponding to these substrates. The constants languageModelAlignmentFraction and photonicCodeRate are each set to the rational number 7/8, and the former is additionally identified with $(2^3-1)/2^3$ to link to the F_2-cube structure. Upstream results provide these constant definitions and the enumeration of the three substrates. The shared lemmas establish the fixed point at unity, the descent property, and the inversion symmetry for the J-cost function. This places the certificate in the local setting of hypothesis-grade empirical validation rather than axiomatic derivation.
proof idea
As a structure definition the declaration introduces the type ThreeSubstrateCert with no attached proof. The fields are populated in the downstream construction threeSubstrateCert by referencing validationSubstrateCount for the cardinality, shared_fixed_point for the fixed-point equation, shared_descent for the inequality, shared_symmetry for the equality under inversion, and lm_fraction_eq for the alignment value.
why it matters in Recognition Science
This structure earns its place by serving as the Lean carrier for the three-substrate validation in Recognition Science Experiment B. It is instantiated by threeSubstrateCert and thereby supplies the empirical support for the J-cost properties that follow from T5 J-uniqueness in the forcing chain. The 7/8 rate connects to the eight-tick octave (T7) and the F_2-cube structure. The certificate touches the open question of how far the empirical alignment extends beyond the three tested substrates.
scope and limits
- Does not derive the listed J-cost properties from the Recognition Composition Law.
- Does not contain the experimental datasets or error bars.
- Does not cover substrates outside the enumerated three.
- Does not prove uniqueness of the 7/8 rate from first principles.
formal statement (Lean)
59structure ThreeSubstrateCert where
60 three_substrates : Fintype.card ValidationSubstrate = 3
61 fixed_point : Jcost 1 = 0
62 descent : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
63 symmetry : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹
64 lm_alignment : languageModelAlignmentFraction = 7/8
65 photonic_rate : photonicCodeRate = 7 / 8
66 f2_cube_connection : languageModelAlignmentFraction = (2^3 - 1 : ℚ) / 2^3
67