Chemistry Reaction Network2 From Jcost Crnsteady State Cert

The certificate's scope

In the Recognition Science framework, a ledger (a discrete record of events) assigns a cost (a forced penalty) to each possible outcome. The declaration CRNSteadyStateCert is a machine-checked certificate: a formal object that packages three proved facts about the cost function J(x) = (x + 1/x)/2 - 1. It proves that J is zero when its input equals 1, that J is never negative for positive inputs, and that the golden-ratio constant phi minus 3/2 is positive. These are general properties of the cost function itself, stated for any real numbers.

The certificate's name suggests a chemical reaction network steady state, but the formal content does not mention chemistry. The definition domainCost(m, e) simply sets the cost as J(m/e), where m and e are arbitrary real numbers. The three proved facts apply to that ratio, whatever m and e represent. To make this a theorem about chemical steady states, one would need to define m and e in chemical terms, for example as a measured concentration and an equilibrium concentration. The certificate does not do that; it only proves facts about the abstract ratio.

This is an honest limitation, not a failure. The certificate is a small but sound building block: it shows that the cost function has the basic properties any equilibrium condition would need. It establishes that if m equals e, the cost is zero; if both are positive, the cost is never negative; and the threshold phi - 3/2 is positive. What it does not do is connect those facts to any specific chemical species, reaction, or network. The connection remains a research note, not a result.

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

The certificate does not define m and e in chemical terms. The certificate does not prove any statement about a specific reaction network. The certificate does not establish that J-cost zero is the equilibrium condition for any real chemical system.

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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