validationSubstrateCount
The theorem establishes that the inductive type ValidationSubstrate enumerating the three empirical testbeds for J-cost has cardinality exactly 3. Researchers constructing the three-substrate certificate cite it to fix the substrate count before bundling fixed-point and descent properties. The proof is a one-line decision that evaluates the derived Fintype instance on the three constructors.
claimLet $V$ be the inductive type with constructors languageModel, photonicQubit, and magnetizedPlasma. Then $|V| = 3$.
background
The Three-Substrate Validation Certificate module records empirical checks that J-cost outperforms cross-entropy in language-model layers, achieves 7/8 code rate in photonic qubits, and converges near the fixed point in magnetized plasma. ValidationSubstrate is the inductive type carrying exactly these three cases and deriving DecidableEq together with Fintype. Upstream, cost is defined in ObserverForcing as the J-cost of a recognition event and in MultiplicativeRecognizerL4 as the derived cost of a comparator on positive ratios.
proof idea
The proof is a one-line wrapper that applies the decide tactic to the equality Fintype.card ValidationSubstrate = 3. The tactic succeeds because the inductive definition derives Fintype, allowing direct computation of the cardinality from the three listed constructors.
why it matters in Recognition Science
The result supplies the substrate count field to the threeSubstrateCert definition, which also records the shared fixed point at x = 1, descent direction, symmetry, and language-model alignment fraction. It thereby anchors the structural claim that J-cost uniqueness (T5) holds uniformly across the three independent substrates. The surrounding module remains at hypothesis grade because the empirical performance numbers are not Lean-proved.
scope and limits
- Does not prove numerical agreement of J-cost values across the three substrates.
- Does not establish the descent direction or symmetry properties.
- Does not certify the reported experimental figures such as 96.4 percent alignment or 7/8 code rate.
- Does not extend the certificate to any substrate beyond the three enumerated constructors.
Lean usage
def threeSubstrateCert : ThreeSubstrateCert where three_substrates := validationSubstrateCount fixed_point := shared_fixed_point descent := shared_descent symmetry := shared_symmetry lm_alignment := lm_fraction_eqformal statement (Lean)
31theorem validationSubstrateCount : Fintype.card ValidationSubstrate = 3 := by decide
proof body
Term-mode proof.
32
33/-- All three substrates have the same J-cost fixed point at x = 1. -/