Chemistry Catalysis Chiral From Jcost

Chiral catalysis and the cost function

Chiral catalysis is the art of making one mirror-image version of a molecule while suppressing the other. In asymmetric catalysis, the standard measure of success is enantiomeric excess (ee), the difference between the two forms divided by their sum. A good catalyst reaches above 90 percent ee; excellent ones exceed 98 percent. The question is whether a mathematical cost function from Recognition Science can predict these numbers.

Recognition Science starts from a simple picture: reality keeps a ledger, a discrete record of recognition events, and the cost of each recognition is forced by five plain conditions. The framework proves those conditions yield exactly one cost function, J(x) = (x + 1/x)/2 - 1. A machine-checked library of formal theorems, written in the Lean proof assistant, verifies this and its consequences. The library is a collection of statements each checked by a computer kernel, with no hidden assumptions beyond the standard logical axioms.

The module CatalysisChiralFromJCost defines a domain cost as J applied to the ratio of two masses, m and e. The Lean code proves three general facts about this cost: it vanishes when m equals e, it is never negative for positive inputs, and a certain threshold involving the golden ratio is positive. These are true statements about the cost function itself, and they hold for any positive numbers m and e.

What the module does not do is connect those numbers to chemistry. The definition of domainCost uses the ratio m/e without saying what m and e mean for a catalyst. The docstring records a research note: setting ee = 1 - 2J(phi)^2 gives about 97.2 percent, and ee = 1 - J(phi)^2 gives about 98.6 percent, matching excellent catalysts. But this is an idea about where the work might go, not a proved result. The Lean code itself proves nothing specific to chiral catalysis.

In plain language: the framework has a sharp tool, a cost function with proved properties, and a plausible guess about how to apply it to catalysis. The guess is promising because the numbers land in the right range, but the bridge from cost to chemistry is missing. Until someone defines m and e in terms of actual catalyst properties, the 97 and 98 percent figures remain a research aspiration, not a derived prediction.

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

No claim that the framework derives enantiomeric excess for any real catalyst. No claim that the 97.2 and 98.6 percent figures are measured or predicted values. No claim that the golden ratio threshold has a chemical interpretation.

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

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