Chemistry Reaction Mechanism From Jcost Reaction Mech Cert

What the certificate proves

In chemistry, a reaction mechanism describes the sequence of elementary steps that turns reactants into products. A concerted mechanism (one step, no intermediate) and a stepwise mechanism (two or more steps, with an intermediate) are the two classic alternatives. The Recognition Science framework, which derives structure from a forced cost function, has a declaration called ReactionMechCert that packages three proven facts about that cost. Those facts are general, not chemical: the cost is zero when the two inputs are equal, the cost is never negative for positive inputs, and a certain threshold constant is positive.

The certificate is a structure, a formal bundle, that holds three theorems. The first, cost_at_eq, says that for any nonzero real number r, the cost of r relative to itself is zero. The second, cost_nonneg, says that for any positive m and e, the cost is greater than or equal to zero. The third, threshold_pos, says that the constant phi minus 3/2 is positive, where phi is the golden ratio (about 1.618). These three facts are proved in the machine-checked library, and the certificate simply collects them into one object, with a proof that such a certificate exists.

What the certificate does not do is make any claim about actual chemical reactions. The cost function is defined as Jcost (m / e), where m and e are real numbers, but the declaration never says what m and e mean. The docstring records a research plan: for a concerted reaction, the cost of an activation ratio would be below the phi threshold, and for a stepwise reaction it would be above. But that plan is not implemented. There is no definition of m as, say, a bond order or an energy barrier, and no definition of e as a reference value. Without those definitions, the theorems apply to any positive real numbers, and the certificate proves nothing specific to chemistry.

The framework's own documentation is explicit about this. It says the module proves nothing specific to the subject, because the cost is defined without reference to one. The same body of text is shared verbatim with 2383 sibling modules, meaning the certificate is a template, not a tailored result. The threshold constant, phi minus 3/2, is about 0.118, a positive number, but its chemical significance, if any, remains a research note, not a theorem.

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]

structure ReactionMechCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The certificate does not prove that any real reaction is concerted or stepwise. The certificate does not define what m or e mean in chemical terms. The threshold constant's chemical significance is a research note, not a theorem.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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