Chemistry Oxidation States Derived Manganese Max Seven

The manganese target table

Oxidation state is the charge an atom would have if all its bonds were ionic, a bookkeeping device chemists use to track electron transfer. Manganese, element 25, is the classic example of a transition metal with a wide range: it appears in compounds from the −3 state in some carbonyls to the +7 state in the purple permanganate ion, MnO4−. The +7 state is the headline because it strips the atom of all seven of its valence electrons, the maximum for any element in the first transition series.

In the Recognition Science framework, a ledger, a discrete record of events, underlies the framework's account of physical structure. The framework's machine-checked library of formal theorems, a collection of proofs verified by computer, contains a file that installs a target table of accessible oxidation states for selected elements. The table is a definition, a list of integers chosen for iron and manganese, not a theorem about nature. For manganese, the list is [−1, 0, 2, 3, 4, 6, 7]. The declaration named manganese_max_seven proves one small fact about that list: the integer 7 belongs to it. A companion theorem proves the list has length 7 and no duplicates.

What the declaration does not do is derive the list from any deeper principle. The file's own documentation says the definitions are still target-level and that a future theorem must derive these lists from valence occupation plus J-cost removal. J-cost is the framework's forced cost of recognition, the quantity its central theorem pins down, but that derivation has not been written. The declaration is a certified target, a placeholder that records what the framework eventually hopes to explain, not an explanation itself.

The practical consequence is modest but concrete: the framework now has a machine-checked statement that manganese's target list includes +7, that the list has seven entries, and that none repeat. That is enough to build on, and it is honest about what remains open. The next step, deriving the list from valence occupation and the cost function, is the actual scientific claim the framework wants to make.

/-- Manganese reaches +7. -/
theorem manganese_max_seven :
    (7 : Int) ∈ accessibleOxidationStates 25 := 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 []

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

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

What this page does not claim

Manganese's +7 state is derived from first principles; it is only listed as a target. The target table applies to elements beyond iron and manganese. The framework explains why manganese, rather than another element, reaches +7.

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)

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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: