Chemistry Autocatalysis From Jcost Autocatalytic Cert

The certificate's actual scope

In chemistry, an autocatalytic set is a collection of reactions where each product helps produce another, forming a self-sustaining network. The Recognition Science framework's ledger, a discrete record of recognition events, assigns a cost, a forced price, to comparing any two quantities. The declaration AutocatalyticCert bundles three proved facts about this cost when it is applied to a ratio of two numbers, m over e.

The first fact is that the cost is zero when the two numbers are equal: domainCost r r = 0 for any nonzero r. The second is that the cost is never negative when both inputs are positive: 0 ≤ domainCost m e whenever 0 < m and 0 < e. The third is that the golden ratio minus 1.5 is positive, written as 0 < phi - 3/2. These three statements are proved in the machine-checked library of formal theorems, and the certificate simply packages them together.

What the certificate does not do is say anything about actual chemical reactions. The cost function is defined as Jcost (m / e), where m and e are just real numbers, not molecules or enzymes. The docstring itself states this plainly: the module proves nothing specific to autocatalysis, because the variables are not defined in chemical terms. A research note mentions a threshold of 8.47n catalysts for n reactions and a critical molecule count of 45, but these numbers are not part of any proved theorem.

To make this a theorem about autocatalysis, one would need to define what m and e mean in that subject's own terms, for instance as a measure of molecular count or reaction rate. Until such a definition exists, the certificate remains a general statement about the cost function, applicable to chemistry only after the missing bridge is built. The framework's own literature flags this as an open target, 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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The certificate does not prove any threshold for real autocatalytic sets. The numbers 8.47n and 45 are research notes, not theorem conclusions. The certificate does not establish that the golden ratio plays any role in 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/Autocatalysis_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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