selectivityRegime_count
The theorem establishes that the inductive enumeration of selectivity regimes for heterogeneous catalysts contains exactly five elements. Chemists classifying branching behaviors under the J-cost functional would cite this cardinality when enumerating possible reaction outcomes. The proof is a single decision tactic that exhausts the finite inductive type.
claimThe finite type of canonical selectivity regimes has cardinality five: $|S| = 5$, where $S$ consists of perfect selectivity, primary-product dominant, branching selectivity, mixed, and non-selective.
background
The module introduces five canonical selectivity regimes for heterogeneous catalysts, identified with configDim D = 5. These regimes are perfect selectivity, primary-product dominant, branching selectivity, mixed, and non-selective. The Recognition canonical J(φ) band gates the branching point between them. The result rests on the inductive definition of the selectivity regime type, which derives DecidableEq, Repr, BEq, and Fintype instances.
proof idea
The proof applies the decide tactic, which computes the cardinality of the finite inductive type by exhaustive enumeration of its five constructors.
why it matters in Recognition Science
This cardinality is assigned directly to the five_regimes field of the catalyst selectivity certificate. It supplies the enumeration step for the B10 industrial chemistry depth, connecting the discrete regime count to the J-cost functional and the five-regime classification in the Recognition framework.
scope and limits
- Does not prove physical realizability of any regime.
- Does not derive J-cost thresholds separating the regimes.
- Does not extend to continuous parameter spaces or dynamical evolution.
formal statement (Lean)
27theorem selectivityRegime_count :
28 Fintype.card SelectivityRegime = 5 := by decide
proof body
29