Chemistry Oxidation State From Config Dim Canonical Oxidation State Count Pos

The canonical count

An oxidation state is the hypothetical charge an atom would carry if all its bonds were ionic, a bookkeeping device chemists use to track electron transfer in redox reactions. Transition metals are famous for displaying many of them: manganese alone shows formal states from -3 to +7. The declaration canonicalOxidationStateCount_pos is a small, precise statement inside a machine-checked library of formal theorems. It proves that the number 7, which the framework calls the canonical oxidation state count, is greater than zero. That is all the declaration itself establishes: a positivity fact, not a chemical law.

The number 7 is not arbitrary in this account. The framework's Count Law at dimension 3 predicts that the canonical count of oxidation states for d-block transition metals is 2^3 - 1 = 7. The three binary axes it imagines are whether the charge is positive or negative, whether the d-electron count sits above or below half-fill, and whether the ligand field is above or below some reference. The definition canonicalOxidationStateCount sets this count as 2^3 - 1, and two theorems in the same file verify the arithmetic: one proves the expression equals 7, the other proves 0 < 7. The positivity theorem is the latter, a formal consequence of the definition and of ordinary arithmetic.

What the declaration does not claim is that real chemistry always produces exactly seven oxidation states. The framework's own docstring names a falsifier: any d-block element with a confirmed, stable oxidation state count different from 7 ± 2 in standard inorganic conditions would refute the prediction. Manganese, for instance, lists ten formal states, and the framework counts only the seven it considers common. The positivity theorem is a statement about the defined number, not about the periodic table. It guarantees the count is a positive integer, which is a necessary property for any sensible count, but it says nothing about whether that count matches measurement.

The theorem also feeds a larger structure. The file packages the positivity result together with other facts into an OxidationStateCert, a certificate that bundles the count equality, the positivity, and two properties of a cost function that measures deviation from an expected state. The certificate is inhabited, meaning the framework can exhibit one. This is the framework's way of saying the arithmetic and the cost properties are consistent, not that the chemical prediction is confirmed. The empirical check against measured oxidation state ranges remains a separate, open comparison.

theorem canonicalOxidationStateCount_pos : 0 < canonicalOxidationStateCount := by
  rw [canonicalOxidationStateCount_eq]; norm_num

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

theorem canonicalOxidationStateCount_pos : 0 < canonicalOxidationStateCount := by
  rw [canonicalOxidationStateCount_eq]; norm_num

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

What this page does not claim

This answer does not claim that the positivity theorem confirms the chemical prediction of seven oxidation states. This answer does not claim that the framework derives the existence of any specific transition metal oxidation state. This answer does not claim that the Count Law at dimension 3 is a proved theorem about chemistry; it is a prediction with a named falsifier.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: