Chemistry Autocatalysis From Jcost

Autocatalysis and the cost threshold

Autocatalysis is a chemical system in which the product of a reaction helps make more of itself. A set of molecules is autocatalytic when each member is produced by reactions that require at least one other member as a catalyst, forming a closed loop of mutual production. The classic question is how many distinct molecules are needed before such a self-sustaining network can exist. In the theory of reflexively autocatalytic food-generated sets, the answer depends on the ratio of catalysts to reactions: a network with n reactions needs more than 8.47n catalysts to achieve catalytic closure.

Recognition Science approaches the same threshold from a different starting point. The framework's central object is the cost of recognition, a number that measures the price of matching one thing against another, and its core theorem proves that any cost function meeting five plain conditions must take the form J(x) = (x + 1/x)/2 - 1. The framework models an autocatalytic network by setting the cost of a reaction to J(m/e), where m is the number of catalysts and e is the number of reactions. When the two are equal, the cost is exactly zero: the network is perfectly balanced. When they differ, the cost is positive, and the framework proves this cost is never negative for positive inputs.

The framework then defines a canonical threshold as phi - 3/2, where phi is the golden ratio, approximately 1.618. This number is positive, which the framework proves, and it marks the point where the cost function's curvature changes sign. The framework's library, a machine-checked collection of formal theorems, establishes three general facts about this setup: the cost vanishes at equality, it is nonnegative, and the threshold is positive. These are general properties of the cost function itself, not specific results about any particular chemical network.

In Recognition Science, the module is a template rather than a completed chemical theory. The cost function is defined without reference to any specific chemical species, so the three proved facts apply to any system that adopts this cost. What would turn the template into a theorem about actual autocatalysis is a definition of m and e in chemical terms, such as counting specific catalysts and reactions in a real network. The framework's research note records the intended direction: catalytic closure requires more than 8.47 catalysts per reaction, and the minimal autocatalytic set needs 45 molecules, a number the note calls gap-45. Those numbers are research targets, not proved results.

What the module does establish, in plain language, is that the cost of imbalance is never negative and that a specific golden-ratio threshold is positive. These facts guarantee that the cost function behaves sensibly as a measure of network imbalance. The step from this template to a concrete chemical prediction remains open, waiting for a chemical definition of the variables.

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

The module does not prove that any real chemical network requires 45 molecules. The 8.47n threshold is a research note, not a proved theorem. The cost function's properties are general and do not yet apply to any specific chemical species.

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)

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Derived articles

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