Chemistry Chiral3 Induction From Jcost

Chiral induction and its cost

Chiral induction is the art of making one handed version of a molecule preferentially. When a reaction creates a new stereocenter, chemists measure the outcome as diastereomeric excess (de), the percentage point gap between the major and minor diastereomers. A typical single-step asymmetric synthesis gives de values from 10% to 95%, depending on the catalyst and substrate. The Recognition Science framework offers a striking formula: de = 1 − 2·J(φ) = 76.4%, where J is the framework's cost function and φ is the golden ratio. For multiple steps, the proposed formula generalizes to de = 1 − J(φ)ⁿ.

The cost function J(x) = (x + 1/x)/2 − 1 measures the price of recognition events in the framework's ledger, a discrete record of such events. The framework proves, in a machine-checked library of formal theorems, that any cost function satisfying five plain conditions must equal this J. The golden ratio φ ≈ 1.618 emerges as the unique self-similar scaling, and the framework's constants set hbar = φ⁻⁵ and G = φ⁵/π. The chiral formula plugs φ into J to get the 76.4% figure.

What the module actually proves is far narrower. The Lean code establishes three general facts about Jcost(m/e): it vanishes when m = e, it is nonnegative for positive inputs, and φ − 3/2 > 0. These are properties of the cost function itself, not of any chemical system. The module defines domainCost(m, e) = Jcost(m/e) without specifying what m and e mean for a chiral reaction. The docstring is explicit: the paragraph above is a research note recording where the idea was meant to go, not a result.

The gap matters. To turn the general cost theorem into a statement about chiral induction, one must define m and e in chemical terms, for example as rates or concentrations of the two diastereomeric pathways. Until such a definition exists, the 76.4% figure remains a hypothesis with a named falsifier: any measured de outside the predicted band, given the step count, would refute it. The machine-checked part is real, but it is mathematics about J, not chemistry about chirality.

In Recognition Science, the framework models the cost of recognition as forced, not chosen, and the chiral formula is one proposed application of that forced cost. The honest summary: the framework proves the cost function's universal form, proposes a chiral induction efficiency from it, and the Lean module certifies only the cost function's basic analytic properties. The chemical bridge, defining m and e, remains open.

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)

structure ChiralInduction3Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The module proves any statement about actual chiral molecules or reactions. The 76.4% de value is a measured or derived chemical result. The framework derives the fine-structure constant alpha. The Riemann Hypothesis is proved.

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/Chiral3_Induction_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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