ValidationSubstrate
plain-language theorem explainer
ValidationSubstrate enumerates the three physical realizations on which J-cost is tested in the Recognition Science framework. Researchers citing ALEXIS Exp B results reference it when stating that language models, photonic qubits, and magnetized plasma all obey the same fixed-point and descent properties. The declaration is a plain inductive type with three constructors that automatically derives decidability and finiteness instances.
Claim. Let $S$ be the finite set whose elements are the language-model substrate, the photonic-qubit substrate, and the magnetized-plasma substrate.
background
The module records an empirical certificate that the J-cost function, defined by the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y), yields identical predictions on three independent substrates. J-cost vanishes at the fixed point x = 1 and is positive for all other positive r, with symmetry J(r) = J(r^{-1}). The local setting is the hypothesis-grade validation of T5 J-uniqueness across language-model layers, photonic code rates, and plasma convergence data.
proof idea
The declaration is an inductive definition with three constructors and zero proof lines. It derives DecidableEq, Repr, BEq, and Fintype by the standard type-class mechanism for finite inductive types.
why it matters
ValidationSubstrate supplies the carrier set for the downstream ThreeSubstrateCert structure, which asserts that all three substrates share the fixed point J(1) = 0, the strict descent property, and the inversion symmetry of J-cost. It thereby anchors the empirical side of the T5 J-uniqueness step in the forcing chain. The certificate remains hypothesis-grade; further formalization of the reported alignment fractions would discharge the remaining empirical gap.
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