Chemistry Gas Phase3 Reaction From Jcost

A threshold for gas reactions

In gas-phase chemistry, a reactive collision cross section measures the effective area a molecule presents for a reaction to occur when it meets another molecule. A common starting estimate is the total collision cross section, the area for any contact, multiplied by a probability factor. For many reactions, this factor sits near 0.118, meaning roughly 11.8 percent of all collisions at the threshold temperature lead to reaction. With a typical collision cross section of 100 square angstroms, this gives a reactive cross section near 11.8 square angstroms, a value consistent with many measured gas-phase reactions.

The Recognition Science framework models this probability factor as a value of its cost function J. The framework's central result proves that any cost function meeting five plain conditions must equal J(x) = (x + 1/x)/2 - 1. For the golden ratio phi, about 1.618, J(phi) equals phi - 3/2, which is approximately 0.118. The framework's machine-checked library of formal theorems proves three general facts about this cost function: it vanishes when its two inputs are equal, it never goes negative for positive inputs, and the threshold value phi - 3/2 is positive. These facts hold for any positive real numbers m and e, not for any specific chemical system.

In Recognition Science, the module defines its domain cost as J(m / e), where m and e are positive real numbers. The library proves that this cost is zero when m equals e, nonnegative whenever both inputs are positive, and that the canonical threshold phi - 3/2 is positive. These three facts assemble into a certificate structure, a formal package that any future theorem about this cost can rely on. The certificate is inhabited, meaning the library demonstrates that such a package exists.

What the module does not do is connect these general facts to any specific chemistry. The definitions of m and e as molecular masses, energies, or any other chemical quantities are absent. The docstring records a research note that the reactive cross section should equal J(phi) times the collision cross section, but this is an intended direction, not a proved result. The library itself states this plainly: the module proves nothing specific to gas-phase reactions because its cost function is defined without reference to one.

The practical consequence is a clean separation. The framework's cost function supplies a candidate probability factor, and the numerical agreement with common gas-phase estimates is suggestive. But until m and e are defined in chemical terms and the reactive cross section formula is proved from those definitions, the chemistry remains a hypothesis, not a theorem. The machine-checked part is the general cost function, and the chemistry is the open question.

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

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]

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

What this page does not claim

The reactive cross section formula sigma_react = J(phi) * sigma_collision is not proved in the module. The 11.8 percent value is a research note, not a theorem. No specific chemical system is modeled by the module's cost function.

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/Gas_Phase3_Reaction_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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