Chemistry Radical Stability3 From Jcost Radical Stab3 Cert

What the certificate proves

In organic chemistry, radical stability follows a familiar order: methyl radicals are least stable, then primary, secondary, and tertiary radicals most stable. Each additional alkyl group attached to the radical center stabilizes it. The Recognition Science framework models this stabilization with a cost function, a mathematical device that assigns a nonnegative penalty to a ratio of two quantities. The declaration RadicalStab3Cert is a certificate, a packaged set of three proved facts about that cost function.

The three facts are general properties of the cost function J(x) = (x + 1/x)/2 - 1, not chemical observations. First, the cost is zero when the two inputs are equal. Second, the cost is never negative when both inputs are positive. Third, a certain constant, phi - 3/2, is positive; phi is the golden ratio, about 1.618, so this constant is about 0.118. These are proved in the machine-checked library of formal theorems, meaning the reasoning has been verified step by step by a computer.

The certificate itself contains no chemistry. The cost function is defined as Jcost(m/e) without any definition of what m and e mean in a chemical context. The radical stability order and the numerical claim that three substitutions give 0.354 stabilization appear only in a research note above the formal code, not in any proved theorem. The certificate proves the mathematics; the chemistry would require a separate definition linking m and e to alkyl groups and energy.

What the certificate does establish is a foundation: any future work that defines m and e chemically can rely on these three proved facts. The framework's own account of radical stability, as stated in the research note, is a hypothesis, not a theorem. The gap between the general cost function and the specific chemical claim is exactly what remains open.

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 the radical stability order methyl < primary < secondary < tertiary. The certificate does not establish that three alkyl groups stabilize a radical by 0.354 eV. The certificate does not define what m and e mean 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_Stability3_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.

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