Chemistry Reaction Selectivity2 Regioselect Cert

The certificate's scope

In organic chemistry, regioselectivity is the preference of a chemical reaction to form one structural isomer over another, such as one product out of two possible ones. The cost function, a mathematical measure of the price of a transformation, appears in the Recognition Science framework as a tool for comparing states. The declaration RegioselectCert is a machine-checked certificate, a formal object in the framework's library of verified theorems, that bundles three proven facts about a specific cost expression.

The certificate proves that the cost, defined as Jcost applied to the ratio of two positive real numbers, has three properties. First, the cost is zero when the two inputs are equal, meaning no cost is incurred for a transformation that changes nothing. Second, the cost is never negative for positive inputs, so it cannot reward a change. Third, a particular constant, the golden ratio minus 1.5, is positive. These are general mathematical facts about the cost function, not about any chemical reaction.

In Recognition Science, the framework models chemical selectivity as a ratio of products following a power of the golden ratio, phi^n, where n is a recognition-rung advantage. At n=1, the predicted ratio is phi to 1, roughly 1.618 to 1, corresponding to about 62% yield of the major product. This is a research note, a statement of where the idea was meant to go, not a proven result about chemistry.

The certificate itself proves nothing specific to chemistry, because its cost function is defined without reference to any chemical system. The declaration is shared verbatim with 2383 sibling modules, all using the same template. What would make it a theorem about regioselectivity is a definition of the two inputs, m and e, in a chemical reaction's own terms, which the library does not provide.

The practical consequence is clear: the certificate is a valid piece of formal mathematics, but it is not evidence for any chemical claim. A reader should treat the phi-based selectivity ratio as an unverified hypothesis, not a derived result. The framework's own documentation states this plainly, and the certificate's content confirms it.

structure RegioselectCert 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

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

structure RegioselectCert 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 any chemical regioselectivity claim. The phi to 1 ratio is not a derived result about any specific reaction. The certificate does not establish that the cost function applies to chemistry at all.

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)

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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