threeSubstrateCert
plain-language theorem explainer
The definition assembles a record certifying that language models, photonic qubits, and magnetized plasma all obey the same J-cost relations: fixed point at unity, positive cost off equilibrium, inversion symmetry, and alignment fractions of 7/8. Researchers summarizing Recognition Science Exp B would cite the certificate when listing the common mathematical predictions verified across the three substrates. Construction occurs by direct field assignment from prior rfl equalities and shared theorems on the fixed point, descent, and symmetry.
Claim. Let $J$ be the J-cost function. The certificate asserts that exactly three validation substrates exist, that $J(1)=0$, that $J(r)>0$ whenever $r>0$ and $r≠1$, that $J(r)=J(r^{-1})$ for all $r>0$, that the language-model alignment fraction equals $7/8$, that the photonic code rate equals $7/8$, and that the alignment fraction equals $(2^3-1)/2^3$.
background
The module supplies a Lean certificate for RS Experiment B, which checks J-cost behavior on language models (96.4% layer alignment), photonic qubits (code rate 7/8), and magnetized plasma (convergence to x=1.036). Jcost is the cost function whose unit value is zero and whose positivity and symmetry properties are inherited from the Recognition Composition Law. Upstream theorems establish the fixed point by Jcost_unit0, descent by Jcost_pos_of_ne_one, and symmetry by Jcost_symm; the alignment fractions are recorded by rfl equalities and the explicit reduction to (2^3-1)/2^3.
proof idea
The definition is a direct structure constructor. It sets the substrate count from validationSubstrateCount and populates the remaining fields by substituting the shared_fixed_point, shared_descent, and shared_symmetry theorems together with the rfl equalities lm_fraction_eq, photonic_code_rate_rfl, and seven_eighths_from_F2_cube.
why it matters
The definition supplies the Lean summary of the three-substrate validation in Recognition Science Exp B. It collects the J-cost uniqueness properties that underlie the fixed point, descent, and symmetry shared by all three substrates, consistent with the T5 J-uniqueness step of the forcing chain. The module documentation states the certificate is at hypothesis grade and that the Lean content confirms J-cost uniqueness across the substrates; no downstream theorems depend on it.
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