Chemistry Tautomer Ratio5 Tautomer5 Cert

The certificate's scope

Keto-enol tautomerism is the reversible shift of a hydrogen atom and a double bond within a molecule. Acetone, the simplest ketone, exists almost entirely in the keto form, with the enol form present only in traces. The Recognition Science framework's declaration Tautomer5Cert is a certificate, a machine-checked collection of formal theorems, that was intended to address this ratio.

The certificate establishes three general facts about the framework's cost function, the forced penalty for recognition events. First, the cost is zero when the two quantities being compared are equal. Second, the cost is never negative for positive inputs. Third, a particular constant derived from the golden ratio is positive. These statements are universally quantified over real numbers; they hold for any positive inputs, not for any specific chemical system.

The module defines a domain cost as J(m/e), where J is the framework's cost function. But the declaration never defines what m and e mean for a ketone or an enol. The docstring records the intended idea: a keto fraction of 1 minus J(phi)^3 gives about 0.9984, close to the measured 0.9999 for acetone. That paragraph is a research note, not a result that has been formally checked. The machine-checked part establishes nothing specific to tautomerism.

What the certificate does establish is that the cost function behaves sensibly at equality and stays nonnegative, and that a golden-ratio threshold is positive. These are structural facts about the framework's own mathematics. The chemical application remains a stated intention, not a derivation. The certificate's name promises more than its content delivers.

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 Tautomer5Cert 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 establish that acetone's keto fraction is 0.9984 or any other specific value. The certificate does not establish that the framework's cost function applies to chemical systems at all. The certificate does not derive the measured keto-enol ratio for any molecule.

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/TautomerRatio5.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.

Derived articles

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