Chemistry Kinetic Resolution3 From Jcost Kin Res3 Cert

A certificate and its limits

Kinetic resolution is a chemist's way to separate two mirror-image molecules that react at different speeds. The selectivity factor s measures how well the separation works: it is the ratio of the fast reaction rate to the slow one, so s = k_fast / k_slow. A good practical resolution needs s above roughly 10 to 50, depending on the reaction. The Recognition Science framework, which builds structure from a forced recognition cost, offers a template for thinking about such ratios.

In that framework, the recognition cost J(x) is a function that measures the price of recognizing one quantity as another; it is zero when the two are equal and grows as they diverge. The declaration KinRes3Cert packages three general facts about this cost when it is applied to a ratio m/e. First, the cost is exactly zero when m equals e. Second, for positive m and e, the cost is never negative. Third, a threshold value built from the golden ratio phi is positive, specifically phi minus 1.5 is greater than zero. These three facts are proved in the machine-checked library of formal theorems, and the certificate simply bundles them into one object.

The intended chemical story is that selectivity s equals phi raised to a power n, where n counts recognition rungs of energy difference. That would give s about 4 at phi cubed and about 11 at phi to the fifth, matching the lower end of practical resolution. But the formal declaration does not establish any of that. The definition of domainCost uses the ratio m/e without saying what m and e mean for a real reaction. The certificate proves facts about the abstract cost function, not about any actual chemical system. To make it a theorem about kinetic resolution, someone would need to define m and e in the reaction's own terms, for example as rate constants or activation energies.

The practical lesson is a caution about provenance. A formal proof is only as strong as its definitions. KinRes3Cert is a valid certificate for three general inequalities, and it is shared verbatim with 2383 sibling modules that each point at a different intended subject. The chemical content lives in the research note, not in the proof. A reader who wants a theorem about kinetic resolution should look for a module where the ratio is defined from reaction rates; this one does not provide that bridge.

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

KinRes3Cert does not prove that any real chemical reaction achieves a selectivity factor of phi^n. The declaration does not define m and e in terms of rate constants, activation energies, or any other chemical quantity. The certificate does not establish that the golden ratio governs kinetic resolution in practice.

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/Kinetic_Resolution3_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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