Chemistry Equilibrium Constant From Jcost Keq Cert

What KeqCert proves

In chemistry, the equilibrium constant K measures how far a reaction proceeds before it stops changing. The van't Hoff equation, d(lnK)/d(T⁻¹) = −ΔH/R, connects that constant to temperature and the heat of reaction. Recognition Science (RS) has a machine-checked library of formal theorems, and one declaration in it, KeqCert, is a certificate: a packaged collection of proved statements about a cost function, a mathematical measure of the price of recognizing one quantity as another. The certificate proves three general facts about the function J(x) = (x + 1/x)/2 − 1.

First, the cost vanishes when the two quantities are equal. If m and e are positive real numbers, and m/e equals 1, then the cost J(m/e) is exactly 0. Second, the cost is never negative for positive inputs: for any positive m and e, J(m/e) is greater than or equal to 0. Third, a specific threshold value, the golden ratio φ minus 3/2, is positive. These three facts are proved in the library's file EquilibriumConstantFromJCost.lean, and the certificate simply bundles them together with a proof that such a certificate exists.

The name suggests a connection to equilibrium constants, but the certificate itself proves nothing specific to chemistry. The cost function is defined as J(m/e) with no reference to any chemical system. The docstring for the module says this plainly: the code proves general facts about Jcost, and it proves nothing about equilibrium constants because the quantities m and e are not defined in chemical terms. The van't Hoff equation and the RS reference temperature appear in a research note, not in the proved theorems. To make KeqCert a theorem about chemistry, someone would need to define m and e using chemical quantities like enthalpy or entropy.

What KeqCert does establish is a small, solid foundation. It shows that the cost function behaves sensibly: zero cost at equality, nonnegative values everywhere, and a positive threshold. These are the kinds of properties any reasonable cost function should have. The certificate is inhabited, meaning the three properties are consistent and provable together. That is a real result, but it is a result about the cost function, not about chemistry. The gap between the name and the content is a reminder that in a formal library, the name of a declaration does not determine what it proves.

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]

theorem cert_inhabited : Nonempty KeqCert := ⟨cert⟩

What this page does not claim

KeqCert does not prove any statement about equilibrium constants or chemical reactions. The certificate does not derive the van't Hoff equation or any temperature dependence of K. The name KeqCert does not imply that the cost function has been connected to 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/EquilibriumConstantFromJCost.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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