Chemistry Oxidation State From Config Dim Oxidation State Cost Nonneg

The cost of a wrong oxidation state

Oxidation states are the bookkeeping numbers chemists assign to atoms in a compound: +2 for iron in FeO, +3 for iron in Fe2O3, and so on. The Recognition Science framework treats this bookkeeping as a case of its general cost function, a measure of how far a measured value sits from an expected one. The declaration oxidationStateCost_nonneg establishes a simple fact about that measure: when both the measured and expected oxidation states are positive real numbers, the cost is never negative. It proves that the framework's penalty for a wrong oxidation state cannot be a reward, a credit, or a negative number.

The cost function itself is J(x) = (x + 1/x)/2 - 1, evaluated at the ratio of measured to expected state. At the expected value the ratio is 1 and the cost is 0. Move away in either direction and the cost rises. The theorem oxidationStateCost_nonneg is a formal consequence of the fact that J is never negative for positive arguments, a property the framework's library proves for the cost function generally. The declaration assembles this into a certificate, a packaged proof that the oxidation-state cost is nonnegative, along with related facts about the canonical count of seven oxidation states for d-block metals.

What the declaration does not claim matters as much as what it proves. It does not assert that any particular element actually has seven stable oxidation states, nor that the framework's count of seven is a measurement. The count of seven, expressed as 2^3 - 1, is a prediction with a named falsifier: any d-block element with a confirmed stable oxidation state count different from 7 ± 2 under standard conditions. The nonnegativity theorem is a structural fact about the cost function, not an empirical claim about chemistry. It says that if you use this cost, you get a nonnegative number; it does not say the cost is the right one for any real compound.

The theorem's value is architectural. It guarantees that the framework's oxidation-state bookkeeping cannot produce a negative penalty, which keeps the cost function usable as a measure of deviation. A reader can rely on that property without knowing which elements the framework's count of seven fits. The nonnegativity is a floor beneath the framework's chemistry claims, not a substitute for testing them against the periodic table.

theorem oxidationStateCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) :
    0 ≤ oxidationStateCost m e := by
  unfold oxidationStateCost; exact Cost.Jcost_nonneg (div_pos hm he)

/-- J-cost on oxidation state ratio: deviation from the expected state. -/
def oxidationStateCost (measured expected : ℝ) : ℝ :=
  Cost.Jcost (measured / expected)

theorem oxidationStateCost_at_expected (s : ℝ) (h : s ≠ 0) :
    oxidationStateCost s s = 0 := by
  unfold oxidationStateCost; rw [div_self h]; exact Cost.Jcost_unit0

/-- Count Law at D = 3: 2^3 - 1 = 7 canonical oxidation states. -/
def canonicalOxidationStateCount : ℕ := 2 ^ 3 - 1

What this page does not claim

The theorem does not prove that any real element has seven stable oxidation states. The theorem does not say the cost function is the correct one for any particular compound. The count of seven is a prediction, not a measurement.

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

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