Chemistry Oxidation States Derived Manganese Oxidation States Nodup

The manganese target table

In chemistry, an oxidation state is a bookkeeping number assigned to an atom in a compound, representing the charge it would have if all bonds were ionic. Manganese, atomic number 25, is famous for displaying a wide range of these states, from -1 up to +7. The declaration manganese_oxidation_states_nodup establishes a precise, machine-checked fact about a proposed list of these states: the list contains no duplicate entries.

The list in question is defined as accessible oxidation states, a target table within the Recognition Science framework that records which oxidation states are considered reachable for selected elements. For manganese, the table is [-1, 0, 2, 3, 4, 6, 7]. The theorem proves that this list has no repeated values, a property known in formal mathematics as being Nodup. The proof is carried out by native_decide, a computational procedure that checks the finite list directly, and the result is recorded in a machine-checked library of formal theorems.

This fact matters because it is part of a larger, still unfinished program. The framework's plan, called Phase 8B, aims to derive oxidation-state lists from first principles: from valence electron occupation plus a cost-based removal process. The current file installs the target tables for iron and manganese as acceptance cases, and the no-duplicate theorem is one of several certificates that the tables are well-formed. A companion theorem confirms that manganese's list has exactly seven states, and another confirms that +7 is among them.

What the declaration does not claim is just as important. It does not prove that the list is correct, complete, or derived from any physical law. It only proves a structural property of a list that has been chosen as a target. The derivation itself, the step that would force these specific oxidation states from deeper principles, remains an open target. The theorem is a check on the table's shape, not a justification of its contents.

/-- Manganese's target list has no duplicate oxidation states. -/
theorem manganese_oxidation_states_nodup :
    (accessibleOxidationStates 25).Nodup := by
  native_decide

/-- 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 []

/-- Manganese's target list has no duplicate oxidation states. -/
theorem manganese_oxidation_states_nodup :
    (accessibleOxidationStates 25).Nodup := by
  native_decide

/-- The manganese target list has the canonical seven common states. -/
theorem manganese_state_count :
    (accessibleOxidationStates 25).length = 7 := by
  native_decide

What this page does not claim

The theorem does not prove that the list of manganese oxidation states is chemically correct or complete. The theorem does not derive the oxidation-state list from valence occupation or any cost-based process. The list is a target table chosen by the framework, not a measured or derived set of values.

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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