Chemistry Oxidation States Derived Iron Oxidation States

Iron's target table

Iron, atomic number 26, most often appears in compounds with oxidation states +2 and +3, as in ferrous and ferric salts. It also forms compounds in the 0 state, such as iron carbonyl, and in the +6 state, as in the ferrate ion. Chemists list these four accessible states, 0, +2, +3, and +6, when describing the element's redox chemistry.

The Recognition Science framework, which derives physical structure from a forced cost of recognition events, maintains a machine-checked library of formal theorems. In that library, a definition called accessibleOxidationStates assigns to each atomic number a list of target oxidation states. For iron, the definition returns [0, 2, 3, 6]. A theorem named iron_oxidation_states proves, by direct computation, that the list for atomic number 26 is exactly that four-element list. A companion theorem confirms the list has no duplicate entries.

This is a target table, not a derivation. The docstring for the module states plainly that the definitions are still target-level for the selected elements, and that the next theorem must derive these lists from valence occupation plus J-cost removal. The theorem establishes what the target list is, and that it is internally consistent, but it does not explain why iron, rather than some other element, has these particular states. The count-law spine, a separate module, proves a broader statement about oxidation-state counts, and the library records that this certificate remains available.

What the declaration does not claim is as important as what it proves. It does not claim that the list [0, 2, 3, 6] is derived from first principles within the framework. It does not claim that these are the only oxidation states iron can exhibit in nature, nor that the list matches every measured oxidation state of iron. It does not claim that the framework's cost function, J(x) = (x + 1/x)/2 - 1, produces this list. The theorem is a formal record of a chosen target, installed as part of a periodic-table closure plan, awaiting a future derivation.

/-- Iron's accessible oxidation states in the Phase 8 target table. -/
theorem iron_oxidation_states :
    accessibleOxidationStates 26 = [0, 2, 3, 6] := by
  native_decide

/-- Iron's target list has no duplicate oxidation states. -/
theorem iron_oxidation_states_nodup :
    (accessibleOxidationStates 26).Nodup := by
  native_decide

/-- Phase 8B certificate: oxidation-state targets are installed. -/
structure OxidationStatesDerivedCert : Prop where
  iron_exact : accessibleOxidationStates 26 = [0, 2, 3, 6]
  manganese_reaches_seven : (7 : Int) ∈ accessibleOxidationStates 25
  manganese_count : (accessibleOxidationStates 25).length = 7
  iron_nodup : (accessibleOxidationStates 26).Nodup
  manganese_nodup : (accessibleOxidationStates 25).Nodup
  count_law : Nonempty OxidationStateFromConfigDim.OxidationStateCert

/-- The count-law oxidation certificate remains available. -/
theorem oxidation_count_law_available :
    Nonempty OxidationStateFromConfigDim.OxidationStateCert :=
  OxidationStateFromConfigDim.cert_inhabited

What this page does not claim

The theorem does not derive iron's oxidation states from the framework's cost function. The list [0, 2, 3, 6] is not claimed to be a complete enumeration of all oxidation states iron can exhibit in real compounds. The theorem does not assert that the framework's cost function J(x) produces this particular list.

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/OxidationStatesDerived.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: