Chemistry Oxidation States Derived Oxidation States Derived Certified

The Phase 8B certificate

In chemistry, an oxidation state is a bookkeeping number that tracks how many electrons an atom appears to have gained or lost when it forms compounds. Iron commonly shows +2 and +3, as in ferrous and ferric salts; manganese is famous for reaching +7 in the purple permanganate ion. The Recognition Science framework's ledger, a discrete record of such electron-accounting events, aims to derive these accessible states from a deeper counting law rather than take them from experiment.

The declaration oxidation_states_derived_certified is a machine-checked certificate, a bundled set of formal statements verified by a computer, that installs the target table for two elements. For iron, atomic number 26, the certificate fixes the list [0, 2, 3, 6]. For manganese, atomic number 25, it records the list [-1, 0, 2, 3, 4, 6, 7], confirms that +7 appears, and checks that neither list contains duplicates. It also carries a reference to an earlier module that proves a count-law spine, a separate result about how many oxidation states a configuration should permit.

The certificate is honest about its own status. Its own docstring says the definitions are still target-level for the selected elements: the lists are installed as goals, not derived. The next theorem, the text states, must derive these lists from valence occupation plus J-cost removal, where J-cost is the framework's forced recognition cost. So the certificate proves only that the target table is internally consistent and that the count-law certificate remains available; it does not prove that the table follows from the framework's first principles.

What the certificate does establish is a clean, checkable starting point. A reader can verify in seconds that iron's list has no duplicates, that manganese's list has seven entries, and that +7 is among them. The certificate bundles these checks into one named object, making the Phase 8B target layer a stable platform for the derivation that must come next. The payoff is not a new chemical fact; it is a formal promise about what the framework intends to explain.

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

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

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

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

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

/-- 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

What this page does not claim

The certificate does not derive the oxidation-state lists from valence occupation or J-cost; that derivation remains a stated next step. The certificate does not claim that the lists match measured chemistry; it installs them as targets. The certificate does not cover elements beyond iron and manganese.

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: