Chemistry Catalysis Activation Energy2

Catalysis and the cost minimum

In chemistry, a catalyst speeds a reaction by lowering the energy barrier, but the best catalyst is not the one that binds reactants most tightly. The Sabatier principle, named after French chemist Paul Sabatier, states that the optimal catalyst binds reactants with intermediate strength: too weak and the reactants never attach, too strong and the products never leave. This trade-off produces a volcano plot, where catalytic activity rises to a peak and then falls, and the peak sits at the intermediate binding energy.

Recognition Science (RS) models this optimum with its cost function, a measure of how far a ratio sits from balance. The framework defines a domain cost as J(m/e), where m is the actual binding energy and e is the optimal binding energy for the reaction. The function J is the framework's universal cost, derived from five plain conditions on how recognition events are priced. When m equals e, the ratio is 1 and the cost is exactly zero. When m differs from e, the cost is positive. The volcano maximum is therefore the J-cost minimum.

The framework's machine-checked library of formal theorems proves three general facts about this cost. First, the cost vanishes when the two energies match: J(m/m) = 0 for any nonzero m. Second, the cost is never negative for positive energies. Third, a constant called the canonical threshold, defined as φ − 3/2, is positive, where φ is the golden ratio. These three facts are assembled into a certificate structure that records the proof.

The honest limitation is explicit in the framework's own documentation. The proofs establish properties of the cost function in general, but they do not yet connect m and e to any specific chemical system. The framework states the intended direction, that peak activity occurs when J(ΔG_ads/ΔG_opt) = 0, but it does not define m and e in terms of measured adsorption energies. What the framework establishes is the mathematical skeleton: the cost minimum sits at the matching point, and the threshold constant is positive. The chemical flesh, a definition of m and e from experimental data, remains to be added.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The framework does not prove that any real catalyst operates at the cost minimum. The framework does not derive the Sabatier principle from first principles. The framework does not provide a numerical value for any activation energy.

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/CatalysisActivationEnergy2.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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