Chemistry Oxidation State From Config Dim

Oxidation state counts

Oxidation state is a bookkeeping number chemists assign to an atom in a compound, representing the charge it would have if all bonds were ionic. Transition metals are famous for taking on many of these states. Manganese, for instance, appears in formal states from -3 to +7, but its common, stable states are fewer: -1, 0, +2, +3, +4, +6, and +7, exactly seven of them. Chromium and iron show similar patterns, with a common range spanning about seven values. This regularity is what the module addresses.

The module, named oxidation state from config dim, defines the canonical count as 2^3 - 1, which equals 7. The reasoning treats the count as a consequence of three binary axes: whether the charge is positive or negative, whether the d-electron count is above or below half-fill, and whether the ligand field is above or below a reference. With three independent yes-or-no choices, the number of combinations is 2^3, and subtracting 1 for the all-neutral case gives 7. The framework's machine-checked library of formal theorems proves that this expression evaluates to 7 and that the result is positive.

In Recognition Science, this count is not an empirical observation but a derived consequence of the framework's cost function. The module also defines a cost for an oxidation state ratio, measuring deviation from an expected value. The cost is zero when the measured state equals the expected state, and it is always nonnegative for positive inputs. These properties are proved as theorems, not assumed.

The module packages these results into a certificate structure, which bundles the count theorems and cost properties into a single object. The certificate is inhabited, meaning the framework has constructed a concrete instance of it. The plain-language claim is that the framework predicts exactly seven canonical oxidation states for d-block transition metals, with a stated falsifier: any d-block element with a confirmed stable count different from 7 ± 2 would refute the prediction.

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 module does not derive the specific oxidation states of any particular element. The count of seven is a prediction with a falsifier, not a measured fact. The module does not explain why some formal states are common and others rare.

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)

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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