Chemistry Reaction Mechanism From Jcost

Mechanism as a cost threshold

In chemistry, a reaction mechanism describes the path from reactants to products. A concerted mechanism, like the classic SN2 substitution, passes through a single transition state where bond breaking and bond forming happen together. A stepwise mechanism, like SN1, forms a discrete intermediate, often a carbocation, before the final product appears. Chemists distinguish these paths by kinetics, stereochemistry, and the lifetime of intermediates.

Recognition Science offers a different lens. Its central object is cost, a number that measures the price of a recognition event. The framework's core theorem forces a specific cost function, J(x) = (x + 1/x)/2 - 1, from five plain conditions. The proposed chemical rule maps this cost onto reaction mechanism. Define m as a measure of the "mismatch" between the reactant and transition state, and e as a reference energy. The ratio m/e feeds into J. The claim: a concerted reaction has J(m/e) below a threshold, and a stepwise reaction has J(m/e) above it. The threshold itself is φ - 3/2, where φ is the golden ratio, about 0.118.

The machine-checked library of formal theorems proves three general facts about this setup. First, when m equals e, the cost is exactly zero. Second, for positive m and e, the cost is never negative. Third, the threshold φ - 3/2 is a positive number. These are all true statements about the cost function, and the proofs are airtight. But the library proves nothing specific to chemistry. The definition of domainCost is just J(m/e); it never defines m and e in chemical terms. The docstring admits this plainly: the paragraph about concerted versus stepwise is a research note recording where the idea was meant to go, not a result.

What would make this a theorem about chemistry is a definition of m and e in the subject's own terms. Without that, the threshold remains a number, not a mechanism. The formal result establishes a clean scaffold: a nonnegative cost that vanishes at equality, and a positive threshold. The chemical interpretation is a hypothesis waiting for a definition.

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]

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

noncomputable def cert : ReactionMechCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The library proves any reaction is concerted or stepwise. The threshold φ - 3/2 has been derived from chemical principles. The framework identifies SN1 or SN2 reactions by name.

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

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