Chemistry Ferromagnetism Cobalt Ferromagnetic

Cobalt in the ledger

Cobalt is a hard, silvery metal that sits between iron and nickel in the periodic table. Its defining physical trait is ferromagnetism: below about 1394 kelvin, its atomic magnetic moments line up spontaneously, which is why cobalt makes permanent magnets and sticks to a fridge. The classical account of this behavior runs through quantum mechanics. The exchange interaction, a consequence of the Pauli exclusion principle, favors parallel spins in the partly filled d-orbitals, and the Stoner criterion U × D(E_F) > 1 decides whether that preference wins. Here U is the strength of the exchange interaction and D(E_F) is the density of electron states at the Fermi level. Cobalt satisfies the criterion comfortably, and its Curie temperature of 1394 K is the highest of the three common ferromagnetic metals.

In Recognition Science, the framework models ferromagnetism as a consequence of its ledger, a discrete record of recognition events from which physical structure is derived. Within that framework, the declaration cobalt_ferromagnetic is a theorem in the machine-checked library of formal theorems. It states, in the framework's own language, that the property isFerromagnetic holds for the element with atomic number 27. That property is defined to be true for the elements in a fixed list, which includes iron (26), cobalt (27), and nickel (28), as well as the rare earths gadolinium (64), terbium (65), and dysprosium (66). The theorem is proved by computation: the definition is evaluated and the result is checked by the machine.

The declaration does not itself prove that cobalt is ferromagnetic in the physical world. It proves that a particular formal definition, chosen by the framework's authors, labels cobalt as ferromagnetic. The physical content lives in the choice of that definition and in the surrounding theory that connects the ledger to quantum mechanics. The library does contain related theorems that give the framework's account more substance: cobalt has the highest Curie temperature among iron, cobalt, and nickel, and its domain wall width is smaller and its domain wall energy higher than iron's, which the library records as co_high_anisotropy. These are formal statements about numerical values assigned to each element, not measurements of real cobalt samples.

What a reader can safely take from cobalt_ferromagnetic is narrow but exact: inside the Recognition Science framework, the statement "cobalt is ferromagnetic" is a proved theorem, not an assumption or a hypothesis. The proof is a computation that the machine checked. The declaration does not claim to explain why cobalt is ferromagnetic, does not derive the Curie temperature from first principles, and does not compare its prediction against experimental data. Those tasks belong to other parts of the framework, and some of them remain open targets.

/-- Cobalt is ferromagnetic. -/
theorem cobalt_ferromagnetic : isFerromagnetic 27 = true := by native_decide

/-- Cobalt is ferromagnetic. -/
theorem cobalt_ferromagnetic : isFerromagnetic 27 = true := by native_decide

/-- Check if element is ferromagnetic. -/
def isFerromagnetic (Z : ℕ) : Bool :=
  Z ∈ ferromagneticElements ∨ Z ∈ rareEarthFerromagnets

/-- Cobalt is ferromagnetic. -/
theorem cobalt_ferromagnetic : isFerromagnetic 27 = true := by native_decide

/-- Co has highest Curie temperature among elemental ferromagnets. -/
theorem co_highest_curie :
    curieTemperature 27 > curieTemperature 26 ∧
    curieTemperature 27 > curieTemperature 28 := by
  simp only [curieTemperature]
  norm_num

/-- Co has highest anisotropy (narrowest walls, highest wall energy). -/
theorem co_high_anisotropy :
    domainWallWidth 27 < domainWallWidth 26 ∧
    domainWallEnergy 27 > domainWallEnergy 26 := by
  simp only [domainWallWidth, domainWallEnergy]
  norm_num

What this page does not claim

The declaration does not prove that real cobalt is ferromagnetic in the physical world. The declaration does not derive the Curie temperature of cobalt from the framework's axioms. The declaration does not compare its formal assignment against experimental measurements.

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/Ferromagnetism.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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