Chemistry Reaction Network2 From Jcost

Reaction network steady states

A chemical reaction network is a set of species that interconvert through reactions, like A turning into B and B back into A. Its steady state, also called equilibrium, is the condition where the concentrations of all species stop changing over time. In classical chemistry, finding that state usually means solving a system of equations that balance the rates of every forward and reverse reaction. The Recognition Science framework models the same idea with a different tool: a cost function, a measure of how far a system sits from balance, where zero means equilibrium.

The module in question, chemistry reaction network2 from jcost, defines the cost for a single species as J(m/e), where m is the current concentration and e is the equilibrium concentration. The framework's cost function J(x) equals (x + 1/x)/2 minus 1, and it has three proved properties that matter here. First, when m equals e, the cost is exactly zero. Second, for any positive concentrations, the cost is never negative. Third, a threshold constant involving the golden ratio phi minus 1.5 is always positive. These three facts assemble into a certificate, a formal object that packages the conditions any steady state must satisfy.

In Recognition Science, the machine-checked library of formal theorems proves these three general facts about J(m/e): it vanishes when m = e, it is nonnegative on positive inputs, and phi - 3/2 > 0. What the module does not do is prove anything specific to chemistry, because the cost is defined as J(m/e) without reference to a particular reaction network. The docstring says plainly: the paragraph above is a research note recording where the idea was meant to go, not a result. To make this a theorem about a real network, one would need a definition of m and e in that network's own terms, such as concentrations tied to specific reaction rates.

The plain-language takeaway is this: the framework supplies a universal equilibrium condition, cost equals zero, and proves its basic mathematical hygiene. The bridge from that general condition to a specific chemical system remains open. The module is a template, not a finished theory of any particular reaction network.

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]

What this page does not claim

This module proves anything specific to any particular chemical reaction network. The cost function J(m/e) is derived from chemical principles rather than assumed as a definitional choice. The golden ratio threshold phi - 3/2 has a demonstrated chemical meaning in this module.

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/Reaction_Network2_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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