Chemistry Catalyst Selectivity From Jcost Selectivity Regime Count

Five selectivity regimes

In heterogeneous catalysis, a reaction can take several paths. The outcomes range from perfect selectivity, where only the desired product forms, to non-selective behavior, where many products appear. Between those extremes lie three intermediate cases: primary-product dominant, branching selectivity, and mixed. A machine-checked library of formal theorems proves that this list contains exactly five entries, no more and no fewer.

The theorem, named selectivityRegime_count, states that the cardinality of the SelectivityRegime type equals 5. The proof is a direct computation, verified by the Lean kernel without any unproved assumptions. The declaration also packages this fact into a certificate structure, catalystSelectivityCert, which records the five-regime count as a formal object.

In Recognition Science, the five regimes correspond to a five-dimensional configuration space. The framework models the branching point between these regimes using a canonical cost function J(φ). This cost function, which measures the forced expense of recognition events, gates when a catalyst switches from one selectivity regime to another. The framework's library proves the cost function's uniqueness from five plain conditions, and the selectivity theorem applies that structure to industrial chemistry.

The theorem establishes a count, not a mechanism. It proves that the classification has five members, but it does not predict which catalyst produces which outcome, nor does it derive the cost function's numerical values for specific reactions. The declaration is a structural result about the taxonomy itself.

theorem selectivityRegime_count :
    Fintype.card SelectivityRegime = 5 := by decide

theorem selectivityRegime_count :
    Fintype.card SelectivityRegime = 5 := by decide

theorem selectivityRegime_count :
    Fintype.card SelectivityRegime = 5 := by decide

What this page does not claim

The theorem does not predict which catalyst yields which selectivity outcome. The theorem does not derive numerical values for the cost function in specific reactions. The theorem does not prove that real catalysts must exhibit all five regimes.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/CatalystSelectivityFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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