Chemistry Oxidation States Derived

The target table

An oxidation state is the charge an atom would have if all its bonds were ionic, counting shared electrons as belonging to the more electronegative partner. Iron's common states are +2 and +3, manganese's reach +7 in permanganate, and both metals show a spread of others. A new module in the Recognition Science framework's machine-checked library of formal theorems installs a target table for these accessible states, beginning with iron and manganese as the acceptance cases of a longer closure plan.

The module, called OxidationStatesDerived, defines a function that returns a list of integers for a given atomic number. For iron (Z = 26) the list is [0, 2, 3, 6]; for manganese (Z = 25) it is [-1, 0, 2, 3, 4, 6, 7]. These are target-level definitions, meaning the framework has not yet derived them from first principles. What the module proves, with kernel-checked theorems, is that the target lists have the intended properties: iron's list has no duplicates, manganese's has exactly seven entries, and +7 is among them. A certificate structure bundles these facts into a single theorem, oxidation_states_derived_certified, that records the whole target layer as installed.

The larger plan is that these lists will eventually be derived from valence electron configuration plus the framework's forced cost of recognition, the quantity J that any cost function satisfying five plain conditions must equal. That derivation is the next theorem to be written; the current module only fixes the targets. The count-law spine that would support such a derivation remains available, as a separate module proves that an oxidation-state certificate exists.

What this establishes in plain language is modest but concrete: the framework has chosen a precise set of accessible oxidation states for two representative transition metals and has machine-checked that the set is internally consistent. The choice itself is a model, a definitional commitment, not yet a proved consequence of the cost law. The payoff for a reader is a clear statement of where the derivation stands: the targets are set, the consistency is proved, and the forcing step that would make chemistry fall out of the ledger is the declared next move.

/-- Target accessible oxidation states for selected elements. -/
def accessibleOxidationStates (Z : Nat) : List Int :=
  if Z = 26 then [0, 2, 3, 6]
  else if Z = 25 then [-1, 0, 2, 3, 4, 6, 7]
  else []

/-- The Phase 8B oxidation-state target layer is certified. -/
theorem oxidation_states_derived_certified :
    OxidationStatesDerivedCert where
  iron_exact := iron_oxidation_states
  manganese_reaches_seven := manganese_max_seven
  manganese_count := manganese_state_count
  iron_nodup := iron_oxidation_states_nodup
  manganese_nodup := manganese_oxidation_states_nodup
  count_law := oxidation_count_law_available

/-- 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 oxidation-state lists are not derived from the cost law; they are installed as targets. The module does not prove that the lists match measured chemical behavior for all elements. No claim is made that the framework derives the full periodic table's oxidation states.

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

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