Chemistry Isotope Effect2 From Jcost

Isotope effects and the cost function

A kinetic isotope effect is the change in a chemical reaction's speed when one atom is swapped for a heavier isotope of the same element. For hydrogen versus deuterium, the rate ratio kH/kD typically falls between 2 and 7. The module chemistry isotope effect2 from jcost applies a specific cost function, Jcost, to the ratio of masses m/e, and proves three general facts about that function.

The first fact is that the cost is zero when the two masses are equal. The second is that the cost is never negative for positive masses. The third is that a particular threshold value, phi minus 3/2, is positive, where phi is the golden ratio. These are all proved in the machine-checked library of formal theorems, using only the definition of Jcost and basic properties of real numbers.

In Recognition Science, the framework models the cost of a recognition event, where a ledger, a discrete record of events, forces a specific cost function J(x) = (x + 1/x)/2 - 1. The module defines the domain cost as J(m/e), but it does not define what m and e mean in chemical terms. The docstring notes the intended connection: the rate ratio might be phi to the power of the number of bond recognition steps, giving values like 1.618, 2.618, and 4.236, which fits the observed 2-7 range. But this is a research note, not a proved theorem.

The three proved facts are universal statements about the cost function itself. They hold for any positive real numbers m and e. What would make this a theorem about chemistry is a definition of m and e in terms of actual isotope masses and reaction steps. The module does not provide that. It proves only that the cost function has the basic properties of being zero at equality, nonnegative, and having a positive threshold.

In plain language: the module checks that the cost function behaves sensibly when applied to a mass ratio. It does not yet show that this cost function explains any measured isotope effect. The chemistry claim, that KIE equals phi to the power of the number of steps, is a prediction awaiting a definition that connects the abstract masses to real chemical bonds.

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]

structure KIE2Cert 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 module does not prove that any measured isotope effect equals phi to a power. The module does not define what m and e represent in a chemical reaction. The three proved facts are about the cost function, not about chemistry.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: