Chemistry Chiral3 Induction From Jcost Chiral Induction3 Cert

What the certificate proves

Chiral induction is the preference of one handed form of a molecule over its mirror image during a reaction, often measured as diastereomeric excess (de), the difference between the two forms as a percentage. In organic chemistry, a single asymmetric step typically gives de values from 10 to 95 percent depending on the reagents and conditions. The Recognition Science declaration ChiralInduction3Cert is a machine-checked certificate, a bundled package of three formal facts about a cost function, not a chemical measurement.

The certificate's first fact is that the cost vanishes when its two inputs are equal: for any nonzero real number r, the cost of r divided by itself is zero. The second fact is that the cost is never negative when both inputs are positive. The third is that a certain threshold, the golden ratio minus 1.5, is greater than zero. Each of these is proved in the framework's machine-checked library of formal theorems from the definition of the cost function J(x) = (x + 1/x)/2 - 1, which itself is proved to be the unique function satisfying five plain conditions.

The certificate is a template, not a result about chiral induction. Its cost is defined as J(m/e) without any definition of what m and e mean in chemistry. The docstring records the intended research direction: for a canonical single asymmetric step, de would equal 1 - 2*J(phi) = 76.4 percent, where phi is the golden ratio, and for multiple steps de = 1 - J(phi)^n. But these formulas are a research note, not part of the proved certificate. What would make the module a theorem about its subject is a definition of m and e in chemical terms, which the certificate does not provide.

The certificate's value is architectural. It shows that three abstract properties of the cost function, vanishing at equality, nonnegativity, and a positive threshold, hold in a machine-checked form. It does not show that chiral induction follows from the cost function, nor that any measured de value matches the 76.4 percent figure. The certificate is shared verbatim across 2383 sibling modules, each with a different subject name but the same three abstract facts.

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]

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 certificate does not prove that chiral induction follows from the cost function. The certificate does not establish that 76.4 percent is the measured de for any real reaction. The certificate does not define m and e 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/Chiral3_Induction_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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