Chemistry Superconducting Tc Mgb2 Between

The MgB2 ordering theorem

Magnesium diboride (MgB2) is a superconductor, a material that carries electric current without resistance below a critical temperature (Tc). Its Tc of about 39 K made it famous in 2001 as an unusually warm metallic superconductor, warmer than conventional ones like niobium-tin but cooler than the iron-based and cuprate families discovered later. The theorem mgb2_between, proved in the framework's machine-checked library of formal theorems, establishes exactly this ordering: the framework's model assigns MgB2 a Tc that sits strictly between iron-based superconductors and conventional BCS superconductors.

The framework, Recognition Science, builds physics from a discrete record of events called a ledger. Its central result forces the golden ratio φ (about 1.618) as a universal scaling constant. The superconducting model applies this scaling to the energy gap that binds Cooper pairs, the paired electrons that carry supercurrent. The model places each superconductor family on a ladder of steps: conventional at step 6, MgB2 at step 5, iron-based at step 4, cuprates at step 3. The Tc proxy is (1/φ)^n, so a lower step number means a higher Tc. The theorem mgb2_between proves, from this definition alone, that the iron-based proxy exceeds the MgB2 proxy and the MgB2 proxy exceeds the conventional proxy.

The proof is elementary. Since 1/φ is a positive number less than 1, its powers decrease as the exponent grows: (1/φ)^4 > (1/φ)^5 > (1/φ)^6. The theorem combines two applications of this monotonicity fact, one for steps 4 and 5, one for steps 5 and 6. The machine-checked library records this as a single theorem with a complete formal proof, so the ordering claim is not an empirical guess but a derived consequence of the model's definitions.

What the theorem does not claim is just as important. It does not prove that MgB2's actual measured Tc of about 39 K is exactly predicted by the model. The model's Kelvin scale is calibrated by choosing a reference value of 300 K at step 1, and the mapping of families to ladder steps is a definitional choice, not a derivation. The theorem only establishes a relative ordering among the model's own Tc proxies. It says nothing about the physical mechanism of superconductivity in MgB2, which relies on strong electron-phonon coupling, nor does it claim the framework has derived the BCS theory itself. The ordering is a structural statement within the model, not a new experimental measurement.

/-- MgB2 is between iron-based and conventional. -/
theorem mgb2_between :
    tcFamily .ironBased > tcFamily .mgb2 ∧
    tcFamily .mgb2 > tcFamily .conventional := by
  constructor
  · dsimp [tcFamily, familyLadderStep]
    exact tc_scaling 4 5 (by norm_num)
  · dsimp [tcFamily, familyLadderStep]
    exact tc_scaling 5 6 (by norm_num)

/-- Phonon-route Tc proxy at ladder step `n`. -/
noncomputable def tc_phonon (n : Nat) : ℝ := (1 / Constants.phi) ^ n

/-- Map superconductor family to φ-ladder step. -/
def familyLadderStep : SuperconductorFamily → ℕ
| .conventional => 6
| .mgb2 => 5
| .ironBased => 4
| .cuprate => 3
| .theoretical => 1

/-- Tc decreases with ladder step: if `n₁ < n₂` then `tc_phonon n₁ > tc_phonon n₂`.
    Since 0 < 1/φ < 1, we have (1/φ)^n₁ > (1/φ)^n₂ when n₁ < n₂.
    The proof is elementary: for 0 < a < 1, a^n is strictly decreasing in n. -/
theorem tc_scaling (n₁ n₂ : Nat) (h : n₁ < n₂) : tc_phonon n₁ > tc_phonon n₂ := by
  dsimp [tc_phonon]
  have hφpos : 0 < Constants.phi := Constants.phi_pos
  have hφ_gt_1 : 1 < Constants.phi := Constants.one_lt_phi
  have ha_pos : 0 < (1 / Constants.phi) := by positivity
  have ha_lt_one : (1 / Constants.phi) < 1 := by
    rw [div_lt_one hφpos]
    exact hφ_gt_1
  -- For 0 < a < 1 and n₁ < n₂, a^n₂ < a^n₁
  exact pow_lt_pow_right_of_lt_one₀ ha_pos ha_lt_one h

What this page does not claim

The theorem does not predict the exact measured Tc of MgB2 in Kelvin. The theorem does not describe the physical pairing mechanism in MgB2. The theorem does not derive the BCS theory of superconductivity.

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

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