Chemistry Rate Constant From Phi Ladder Eyring Rate Cert

The certificate's scope

The Eyring rate constant is a classical result in chemical kinetics. It gives the rate of a reaction as k = kT/h × K‡, where kT/h is a frequency from thermal motion and K‡ is the equilibrium constant for reaching the transition state. In the Recognition Science framework, the plan was to express K‡ as exp(−ΔG‡/RT) with the activation free energy ΔG‡ set by the framework's cost function J at the golden ratio φ, giving a canonical rate of about kT/h × 0.889. That plan appears in the research note, but it is not what the formal declaration proves.

The declaration named EyringRateCert is a structure in the framework's machine-checked library of formal theorems. It bundles three general facts about the function domainCost(m, e) = J(m/e). First, the cost is zero when the measured and expected values are equal. Second, the cost is nonnegative for positive inputs. Third, the threshold φ − 3/2 is positive. These are properties of the cost function itself, universally quantified over real numbers, and they hold for any positive measured and expected values. The library proves each one, and the certificate simply collects them into a single object that is inhabited, meaning such a certificate exists.

What the declaration does not do is connect those facts to chemistry. The cost function is defined as J(m/e) with no reference to a reaction, a rate, or a transition state. Nothing in the module defines what m and e mean for an Eyring rate. The research note says where the idea was meant to go, but the formal content stops at the three general properties. The module is shared verbatim with 2554 sibling modules; the content is stated once, universally quantified, in a template module, and this chemistry module adds no subject-specific definition.

The distinction matters for anyone reading the framework's library. A certificate that proves cost is zero at equality and nonnegative elsewhere is a sound piece of mathematics, but it is not a derivation of the Eyring equation. To make it a theorem about reaction rates, the module would need to define m and e in chemical terms, such as a measured rate and an expected rate from the φ-ladder. Until that definition exists, the EyringRateCert establishes only the general cost facts, and the chemistry remains a research aspiration, not a proved result.

structure EyringRateCert 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

theorem cert_inhabited : Nonempty EyringRateCert := ⟨cert⟩

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem cert_inhabited : Nonempty EyringRateCert := ⟨cert⟩

What this page does not claim

The certificate does not derive the Eyring equation or any reaction rate. No chemical meaning is assigned to the measured and expected values in the cost function. The canonical rate kT/h × 0.889 is a research note, not a proved result.

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/RateConstantFromPhiLadder.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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