Chemistry Superconducting Tc Iron Between

The ordering theorem

Superconductors are materials that carry electric current with zero resistance below a critical temperature, Tc. The families differ sharply in how high that temperature can go: conventional metals like aluminum and lead sit below about 30 kelvin, iron-based compounds reach roughly 50 to 60 kelvin, and cuprates such as YBCO reach 90 to 130 kelvin. The Recognition Science framework models these families as occupying different rungs of a ladder built from the golden ratio phi, about 1.618. Its machine-checked library of formal theorems contains a declaration, iron_between, that proves a purely ordering statement: in the framework's model, the predicted Tc of iron-based superconductors lies strictly between that of cuprates and that of conventional superconductors.

The ladder itself is simple. The framework defines a phonon-route Tc proxy as (1/phi)^n, where n is a ladder step. The step mapping assigns conventional superconductors n = 6, MgB2 n = 5, iron-based n = 4, cuprates n = 3, and a hypothetical room-temperature family n = 1. Since 1/phi is about 0.618, a smaller step number gives a larger proxy value. The theorem tc_scaling proves that if n1 < n2 then (1/phi)^n1 > (1/phi)^n2. The iron_between theorem then applies this scaling twice: once comparing step 3 with step 4, and once comparing step 4 with step 6. The conclusion is exactly that cuprates exceed iron-based, and iron-based exceed conventional, in the framework's Tc proxy.

The framework calibrates its reference scale so that step 3 lands near 90 to 100 kelvin, matching the cuprate range. With a reference of 300 kelvin at step 1, the predicted Kelvin values are about 174 for cuprates, 108 for iron-based, and 40 for conventional. Those numbers are not measurements. The framework's own documentation calls the family classification a model, and the Kelvin values are predictions from that model, not experimental receipts. The iron_between theorem itself proves only the ordering, not any specific temperature.

What the declaration does not claim matters as much as what it proves. It does not prove that iron-based superconductors actually reach 108 kelvin in any lab. It does not derive the measured Tc values from first principles. It establishes a relative ordering inside a model whose step assignments are chosen, not forced. The theorem is real and machine-checked, but its content is the inequality, nothing more. A reader should take it as a structural claim about the framework's ladder, not as a new experimental result.

/-- Iron-based superconductors have intermediate Tc. -/
theorem iron_between :
    tcFamily .cuprate > tcFamily .ironBased ∧
    tcFamily .ironBased > tcFamily .conventional := by
  constructor
  · dsimp [tcFamily, familyLadderStep]
    exact tc_scaling 3 4 (by norm_num)
  · dsimp [tcFamily, familyLadderStep]
    exact tc_scaling 4 6 (by norm_num)

/-- 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

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

What this page does not claim

Iron-based superconductors are measured to reach 108 kelvin in any laboratory. The framework derives the measured Tc values of any superconductor family from first principles. The step assignments for each family are forced by the framework's theorems rather than chosen by its model.

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)

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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