Chemistry Reaction Selectivity2

Regioselectivity and its formal scaffold

Regioselectivity is the preference of a chemical reaction to form one structural isomer over another when several are possible. A classic example is addition to an unsymmetrical alkene: the major product might outnumber the minor by a fixed ratio. Chemists express this as a selectivity ratio, such as 62:38, and the ratio often follows regular patterns across related reactions.

The Recognition Science module builds a formal scaffold around one such pattern. It defines a cost function cost, a measure of the recognition effort a reaction pathway incurs, as J(m/e), where m and e are real numbers representing the major and minor product amounts. The module proves three facts about this function: it equals zero when m equals e, it is never negative for positive inputs, and a threshold constant phi minus 3/2 is positive. These are general properties of the cost function, not facts about any particular reaction.

The module also bundles these three facts into a structure called a regioselectivity certificate, and proves that this structure is inhabited, meaning the facts are consistent. This is a formal guarantee that the three properties can coexist, but it says nothing about whether real reactions satisfy them.

In Recognition Science, the framework models the selectivity ratio as phi to the n, where phi is the golden ratio and n is the number of recognition-rung advantages. At n equals 1, this predicts a ratio of phi to 1, about 1.618 to 1, which corresponds to roughly 62 percent selectivity. This is the research note's intended application: a moderate regioselectivity ratio that matches common experimental observations.

The machine-checked library of formal theorems proves only the three general cost properties. The step from those properties to a statement about real chemical reactions would require defining m and e in terms of reaction mechanisms, which the module does not do. The phi-based selectivity prediction is a research note, not a proved result.

What this means in practice: the formal scaffold is sound, but it is a template. The chemistry-specific content is an aspiration recorded in a comment, not a theorem. Any reader who wants a proved statement about regioselectivity must look for a module that defines m and e in chemical terms, which this one does not.

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 RegioselectCert := ⟨cert⟩

What this page does not claim

No theorem in this module proves any specific regioselectivity ratio for a real reaction. The phi-based selectivity prediction is not a proved result; it is a research note. The module does not define m and e in terms of reaction mechanisms.

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/Reaction_Selectivity2.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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