Chemistry Isotope Effect2 From Jcost Kie2 Cert

The certificate's scope

A kinetic isotope effect is the change in a reaction's speed when one atom is swapped for a heavier or lighter version of itself, such as hydrogen for deuterium. For a bond to hydrogen versus deuterium, measured rate ratios typically fall between about 2 and 7. The Recognition Science framework explores whether that range can be derived from its central cost function, cost, a measure of recognition effort that vanishes when two compared quantities are equal and grows as they diverge.

The declaration KIE2Cert appears in a machine-checked library of formal theorems. What it establishes is narrow and precise. It defines a domain cost as the framework's cost function applied to the ratio of two real numbers, m and e. It then proves three general facts about that construction: the cost is zero when the two numbers are equal, the cost is never negative for positive inputs, and a threshold value involving the golden ratio is positive. These three facts are packaged into a structure called KIE2Cert, and the library shows that this structure is inhabited, meaning the facts are consistent and provable.

In Recognition Science, the framework models the kinetic isotope effect by proposing that the rate ratio equals the golden ratio raised to a power, where the power counts bond recognition steps. At one step the ratio is about 1.618, at two steps about 2.618, and at three steps about 4.236, a range that overlaps the measured 2 to 7 window. This proposal is a research note, not a proved result. The certificate establishes nothing about chemistry because its definition of domainCost never connects m and e to actual atomic masses or bond energies.

What KIE2Cert does establish is that the cost function behaves sensibly on positive inputs: it bottoms out at equality and never goes negative. That is a necessary foundation for any later chemical claim, but it is not sufficient. The certificate's own documentation states plainly that it establishes nothing specific to the subject. To turn this into a theorem about isotope effects, someone would need to define m and e in terms of real masses and reaction steps, a step the current file does not take.

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)

What this page does not claim

KIE2Cert does not prove any specific kinetic isotope effect value. The golden-ratio power law is a research note, not a derived theorem. The certificate does not identify m and e with any physical quantities.

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/Isotope_Effect2_FromJCost.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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