Chemistry Tautomer Ratio5

Tautomer ratio5

Keto-enol tautomerism is a chemical equilibrium between two structural isomers that differ by the position of a proton and a double bond. The classic example is acetone, where the keto form dominates overwhelmingly: the measured keto fraction is about 99.99 percent, leaving only a trace of the enol form. Chemists express such preferences as a ratio, and the ratio is set by the free-energy difference between the two forms. The question for the framework is whether its cost function can reproduce that ratio from first principles.

The module chemistry tautomer ratio5 defines a cost function as Jcost of the mass-to-energy ratio, that is, domainCost(m, e) = Jcost(m / e). Here Jcost is the framework's recognition cost, a positive number that measures how far a ratio is from unity. The module proves three general facts about this cost: it is zero when the ratio is one, it is never negative for positive inputs, and a certain threshold constant is positive. These are facts about the cost function itself, not about any particular molecule.

The module does not prove the acetone ratio. Its own docstring says so plainly: the Lean code proves nothing specific to keto-enol chemistry, because the cost is defined without any reference to the chemical system. The definition of m and e in terms of acetone's actual masses and energies is missing. The research note attached to the module records an intended application, not a result: it suggests that the keto fraction equals 1 minus J(phi)^3, which gives about 0.9984, close to the measured 0.9999 but not a derivation.

What the module does establish, in plain language, is a small, clean set of properties that any such cost must satisfy: it vanishes at the balanced point, it never goes negative, and it has a built-in positive threshold. These are the kind of sanity checks that a larger theory would need. The module is honest about its own scope, and that honesty is the point: the framework's library contains many modules that state general truths, and only a few that connect those truths to a specific physical system. This one is not yet connected.

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)

What this page does not claim

The module derives the acetone keto-enol ratio from first principles. The framework's cost function has been shown to match any measured tautomer ratio. The value 0.9984 is a proved prediction rather than a research note.

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)

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