Chemistry Oxidation State From Config Dim Canonical Oxidation State Count

The count of seven

Oxidation state is the charge an atom would have if all bonds were ionic. Transition metals like manganese, chromium, and iron can adopt several of these states. Manganese, for instance, shows formal states from -3 to +7, but chemists usually list about seven as common: -1, 0, +2, +3, +4, +6, +7. This rough pattern, sometimes called the golden seven, is a familiar observation in inorganic chemistry.

The declaration canonicalOxidationStateCount encodes a specific number: 2^3 - 1, which equals 7. The three in the exponent stands for three binary axes the framework uses to sort oxidation states: charge positive or negative, d-electron count above or below half-fill, and ligand-field strength above or below a reference. Eight combinations arise from three binary choices, and the declaration subtracts one to get seven. The machine-checked theorem canonicalOxidationStateCount_eq proves that the defined expression equals 7, and canonicalOxidationStateCount_pos proves it is positive.

In Recognition Science, this count is a structural prediction, not a measurement. The framework models the count as a consequence of its cost function and the number of spatial dimensions, which it derives as three. The declaration itself is a definition, a choice of what to call the expression 2^3 - 1. The theorem about it is a fact about arithmetic, not about any specific element. The docstring states a falsifier: any d-block element with a confirmed stable oxidation state count different from 7 ± 2 in standard conditions would refute the prediction.

The declaration also defines a cost function for oxidation states, oxidationStateCost, which measures deviation from an expected state using the framework's cost function. Theorems show this cost is zero when measured equals expected, and nonnegative for positive inputs. These are formal properties of the definition, not empirical claims about chemistry.

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

theorem canonicalOxidationStateCount_eq : canonicalOxidationStateCount = 7 := by
  unfold canonicalOxidationStateCount; norm_num

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

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)

What this page does not claim

The declaration does not prove that any real element has exactly seven oxidation states. The declaration does not establish the three binary axes as physical realities. The declaration does not derive the oxidation state count from first principles of chemistry.

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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