ThreeSubstrateCert
plain-language theorem explainer
The ThreeSubstrateCert structure records that J-cost is validated on exactly three substrates, satisfies a fixed point at 1 with positivity and inversion symmetry elsewhere, and matches the 7/8 alignment fractions from language-model and photonic experiments. Physicists and mathematicians auditing the Recognition Science framework would reference this when summarizing the empirical support for the J-uniqueness property. As a structure definition it simply aggregates the listed properties with no reduction or proof steps required.
Claim. A structure certifying that the set of validation substrates has cardinality 3, that the J-cost function satisfies $J(1)=0$, that $J(r)>0$ for all $r>0$ with $r≠1$, that $J(r)=J(r^{-1})$ for $r>0$, 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
ValidationSubstrate is the inductive type whose three constructors are languageModel, photonicQubit, and magnetizedPlasma. The module ThreeSubstrateValidationCert assembles empirical checks that the J-cost function, imported from the Cost and MultiChannelJCost modules, exhibits the same behavior across these substrates. The sibling definitions fix the language-model alignment fraction and photonic code rate at the rational value 7/8.
proof idea
The declaration is a structure definition with an empty proof body. It directly incorporates the definitions of languageModelAlignmentFraction and photonicCodeRate along with the ValidationSubstrate inductive type.
why it matters
The structure is used by the downstream threeSubstrateCert definition to produce a concrete instance. It encodes the hypothesis-grade summary of Recognition Science Experiment B, which validates J-cost uniqueness across substrates and connects to the 7/8 fraction arising from the eight-tick octave. It leaves open the formal derivation of the empirical fractions from the forcing chain.
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