Chemistry Kinetic Resolution3 From Jcost

Kinetic resolution and its cost

Kinetic resolution is a standard technique in organic chemistry for separating two mirror-image molecules, called enantiomers, that react at different rates with a chiral reagent. The practical measure of success is the selectivity factor s, defined as the ratio of the fast-reacting rate constant to the slow one, s = k_fast / k_slow. A good resolution typically requires s greater than about 10 to 50, because lower selectivity leaves a mixture that is still nearly racemic and hard to purify. The classical goal is to find conditions that push s as high as possible.

In Recognition Science, the framework models this ratio through its cost function. The central object is cost, a forced measure of the price of a recognition event. The framework's library proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. For kinetic resolution, the module defines the domain cost as J(m/e), where m and e are real numbers representing the two enantiomers' reaction rates. The library then proves three general facts: the cost vanishes when m equals e, it is nonnegative for positive inputs, and the quantity phi - 3/2 is positive, where phi is the golden ratio.

The framework's research note connects these facts to chemistry by identifying the selectivity factor with a power of the golden ratio: s = phi^n, where n counts recognition rungs of energy difference. For n = 3, phi^3 equals 4.24, giving s around 4, which is poor. For n = 5, phi^5 equals 11, matching the lower end of the empirical good-resolution window of s > 10 to 50. The module itself proves nothing specific to chemistry; it only establishes the three general properties of the cost function. What would make it a theorem about kinetic resolution is a definition of m and e in the subject's own terms, which the module does not provide.

The distinction matters. The machine-checked library contains a template, stated once and universally quantified, that any domain cost built from J(m/e) inherits these three properties. The chemistry module instantiates that template but adds no chemical content. Its certificate structure bundles the three proofs into a single object, and the library proves that this certificate exists. The plain-language takeaway is that the framework's cost function is well behaved for any positive ratio, and the golden ratio supplies a natural scale for selectivity, but the leap from cost to actual reaction rates remains a research note, not a proved result.

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]

theorem cert_inhabited : Nonempty KinRes3Cert := ⟨cert⟩

What this page does not claim

The module does not prove that kinetic resolution actually follows the golden ratio scaling. The module does not establish that m and e correspond to measurable rate constants. The framework does not derive the empirical selectivity window from first principles.

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)

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