Chemistry Radical Chain3 From Jcost

Radical chain length from recognition cost

In chemistry, a radical chain reaction is a sequence where a reactive intermediate, a molecule with an unpaired electron, regenerates itself after each step. Polymerization and combustion often proceed this way. The chain length is the average number of propagation steps per initiating event. Measured values typically range from about 100 to one million steps, depending on the monomer, temperature, and inhibitors.

The Recognition Science framework models the cost of distinguishing two quantities, m and e, using the function J(x) = (x + 1/x)/2 - 1. The module Radical_Chain3_FromJCost defines a domain cost as J(m/e). A research note in the module proposes that chain length equals J(phi)^(-n), where phi is the golden ratio, approximately 1.618, and n is the termination probability per step. For n = 1, 2, and 3, this gives about 8.47, 71.8, and 607 steps, which the note says is consistent with the observed range.

What the Lean code actually proves is narrower. It proves three general facts about the cost function: it is zero when m equals e, it is nonnegative for positive inputs, and the constant phi - 3/2 is positive. These facts hold for any positive m and e because the cost is defined as J(m/e) without any chemical content. The module explicitly states that it proves nothing specific to radical chains, because the parameters m and e are not defined in chemical terms.

In plain language: the framework has a cost function with useful mathematical properties, and a researcher hoped it would explain radical chain lengths. The hope is recorded as a research note, not as a theorem. The machine-checked part is only the general cost behavior, which is true for any ratio m/e. To make the module a real chemistry result, someone would need to define m and e from the reaction kinetics, for example as rate constants or activation energies.

The practical takeaway: you can rely on the cost function's zero, nonnegativity, and threshold positivity, because those are formally verified. You cannot yet rely on the chain-length formula as a derived result. The gap between the proposed formula and the proven facts is exactly the missing chemical definition of m and 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]

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

What this page does not claim

The module does not prove any specific fact about radical chain reactions. The chain-length formula is not a theorem; it is a research note. The constants m and e are not defined in chemical terms.

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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