Chemistry Oxidation State From Config Dim Oxidation State Cost

The cost of an oxidation state

Oxidation state is a bookkeeping number chemists assign to an atom in a compound: roughly, the charge the atom would have if all bonds were ionic. Manganese, for instance, appears in formal states from -3 to +7, though only about seven of those are common in ordinary chemistry. The Recognition Science framework, a research program that derives structure from the idea that reality keeps a discrete record of events called a ledger, has a machine-checked library of formal theorems. One declaration in that library, oxidationStateCost, defines a cost function for comparing a measured oxidation state to an expected one.

The definition is simple: it takes the ratio of the measured state to the expected state and applies the framework's universal cost function J(x) = (x + 1/x)/2 - 1. When the measured state equals the expected state, the ratio is 1 and the cost is 0. When they differ, the cost is positive. The library proves both facts: the cost is zero exactly at the expected value, and it is never negative for positive inputs. This gives chemists a quantitative way to say how far a particular oxidation state is from what a model predicts, with the same units-free, symmetric penalty for overestimating and underestimating.

The declaration also connects to a stronger claim. The library defines the canonical number of oxidation states for a d-block transition metal as 2^3 - 1 = 7, a number it derives from the framework's Count Law at dimension 3. The idea is that three binary choices, charge sign, d-electron count relative to half-fill, and ligand-field strength, generate eight combinations, of which one is excluded, leaving seven. The library proves this count equals 7 and that it is positive, and packages all these results into a certificate structure that is formally inhabited.

What the declaration does not do is establish that real transition metals actually have exactly seven stable oxidation states. That is an empirical claim. The library's docstring names a falsifier: any d-block element with a confirmed stable oxidation state count different from 7 ± 2 in standard conditions would refute it. The formal theorems guarantee the arithmetic and the cost function's properties, not the chemistry. The match to manganese's common states and to the IUPAC golden-7 list is a hypothesis awaiting broader confirmation, not a proved fact about the periodic table.

/-- 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

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)

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

What this page does not claim

The declaration does not prove that any real transition metal has exactly seven stable oxidation states. The declaration does not derive the value 7 from first principles; it defines it as 2^3 - 1 and proves the arithmetic. The declaration does not claim the cost function is the only possible way to measure oxidation state deviation.

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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