Chemistry Radical Chain3 From Jcost Radical Chain3 Cert

A certificate with a narrow scope

A radical chain reaction is a sequence of reactive steps in chemistry where a molecule with an unpaired electron reacts to form another radical, propagating a chain that can grow from about 100 to a million steps long. The Recognition Science framework models such chains using a cost function, which measures the price of a recognition event, a discrete record of a comparison. The declaration RadicalChain3Cert is a machine-checked certificate, a bundled package of three proved facts about this cost function. It establishes that the cost vanishes when the two inputs are equal, that the cost is never negative for positive inputs, and that a certain threshold value involving the golden ratio is positive. These are the three facts the certificate guarantees.

The certificate's first fact says that when the two numbers being compared are the same, the cost is exactly zero. The second fact says that for any two positive numbers, the cost is greater than or equal to zero. The third fact concerns a constant called the canonical threshold, defined as the golden ratio minus one and a half, and proves this threshold is greater than zero. All three facts are proved for a cost function defined simply as the ratio of two numbers, with no reference to any chemical system. The certificate is a valid theorem, but its subject is the abstract cost function, not radical chemistry.

In Recognition Science, the framework's library contains a template for this kind of certificate, and the radical chain module instantiates it with a generic cost definition. The research note attached to the module records an intended application: modeling chain length as a function of termination probability. That note describes where the idea was meant to go, not a result that has been proved. The certificate itself proves nothing about chain lengths, termination probabilities, or any specific chemical species.

What the certificate does establish is a clean, checkable foundation: a cost function with these three properties, ready to be used if someone later defines what the two numbers in the ratio mean for a particular reaction. Until that definition exists, the certificate is a tool waiting for a subject, not a statement about chemistry.

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 statement about radical chain reactions, chain lengths, or termination probabilities. The certificate does not establish that the golden ratio threshold has any chemical meaning. The certificate does not claim that the cost function is the only possible model for radical chains.

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/Radical_Chain3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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