Chemistry Reaction Coordinate From Jcost Reaction Coord Cert

Reaction coordinate certificate

A reaction coordinate traces how a chemical reaction progresses from reactants to products. The certificate named ReactionCoordCert is a machine-checked collection of formal theorems about a cost function, not a chemical result. It establishes three general facts about the function J(x) = (x + 1/x)/2 - 1, applied to a ratio of two numbers called measured and expected. The certificate proves the cost is zero when the two numbers are equal, that the cost is never negative for positive inputs, and that a certain threshold value phi - 3/2 is greater than zero.

The three facts are deliberately modest. The first says that when the measured value equals the expected value, the cost is exactly zero, which matches the intuitive idea that a perfect match costs nothing. The second says that for any two positive numbers, the cost is always zero or positive, never negative. The third fixes a specific constant, the golden ratio phi minus 1.5, and proves it is positive. Each of these is a theorem in the framework's machine-checked library of formal theorems, meaning the proofs have been verified mechanically.

What the certificate does not do is connect these facts to chemistry. The cost function is defined as J(measured / expected), but the variables measured and expected are never defined in chemical terms. The certificate does not say what a reaction coordinate is, what the transition state is, or how the ratio should be computed from molecular properties. A research note attached to the module mentions Hammond's postulate, which says transition state structure resembles reactants for exothermic reactions, and suggests a position along the reaction coordinate of about 0.882. That note is a research idea, not a proved result.

The same three theorems appear in 2554 sibling modules, each with a different subject heading but the same formal content. The certificate would become a theorem about chemistry only if someone defined measured and expected in chemical terms. As it stands, ReactionCoordCert is a template with a chemical name, not a chemical statement.

theorem domainCost_at_equilibrium (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 (measured expected : ℝ) : ℝ := Jcost (measured / expected)

What this page does not claim

The certificate does not establish any chemical fact about reaction coordinates or transition states. The value 0.882 for exothermic reactions is a research note, not a proved result. The certificate does not prove Hammond's postulate or any version of it.

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/ReactionCoordinateFromJCost.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.

Derived articles

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