Chemistry Superconducting Tc Superconductor Family

Superconductor families

Superconductors are materials that carry electric current without resistance below a critical temperature, Tc. They are commonly grouped into families by the mechanism that pairs electrons: conventional superconductors such as aluminum and niobium rely on lattice vibrations, while cuprates and iron-based compounds pair through other channels. The Recognition Science framework defines a ledger, a discrete record of recognition events, and models each family as occupying a rung on a ladder of energy scales. The framework's machine-checked library of formal theorems, a collection of proofs verified by computer, contains a declaration, SuperconductorFamily, that names five families: conventional, magnesium diboride (MgB2), iron-based, cuprate, and a hypothetical room-temperature case.

The declaration assigns each family a ladder step, a natural number that sets its energy scale. Conventional superconductors sit at step 6, MgB2 at step 5, iron-based at step 4, cuprates at step 3, and the theoretical room-temperature case at step 1. The critical temperature follows a simple rule: Tc is proportional to (1/φ)^n, where φ is the golden ratio, approximately 1.618, and n is the ladder step. Since 1/φ is less than one, a smaller step gives a larger Tc. The library proves this ordering: cuprates exceed iron-based materials, iron-based exceed MgB2, and MgB2 exceeds conventional superconductors. It also proves the ratio between cuprate and conventional Tc equals φ³, about 4.236.

The framework calibrates the scale so that step 3, the cuprate rung, lands near 90 to 100 kelvin, matching the measured range for yttrium barium copper oxide and bismuth strontium calcium copper oxide. Step 1, the theoretical maximum, corresponds to roughly 300 kelvin, room temperature. The definition also relates the BCS ratio, the dimensionless gap-to-temperature constant of about 1.76 in the standard theory, to a φ-derived expression, 2 log φ + 1, which evaluates to about 1.96. The library proves this expression lies between 1.7 and 2.1, a loose bracket around the measured value.

What the declaration does not claim is as important as what it proves. The ladder steps themselves, 6 for conventional and 3 for cuprates, are definitional choices, not derived results. The framework chooses these assignments to fit observed families; it does not prove they must take these values. The 300 kelvin reference is likewise a calibration, not a prediction. The proved theorems concern only the ordering that follows from the ladder once the steps are fixed. The framework does not claim to explain why cuprates pair electrons through d-wave symmetry or why iron-based materials use spin fluctuations; those mechanisms are named in the docstring as context, not as formal results. The φ-derived BCS ratio is an approximation in the right ballpark, not an exact match to the standard 1.764.

/-- Superconductor family classification. -/
inductive SuperconductorFamily
| conventional   -- BCS phonon-mediated (Al, Pb, Nb, etc.)
| mgb2           -- MgB2 enhanced phonon
| ironBased      -- Fe-based pnictides/chalcogenides
| cuprate        -- High-Tc cuprates (YBCO, BSCCO)
| theoretical    -- Hypothetical room temperature

/-- 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

/-- Cuprates have higher Tc than conventional superconductors. -/
theorem cuprate_gt_conventional :
    tcFamily .cuprate > tcFamily .conventional := by
  dsimp [tcFamily, familyLadderStep]
  exact tc_scaling 3 6 (by norm_num)

/-- 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)

/-- 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)

/-- Ratio between cuprate and conventional Tc follows φ^3.
    (1/φ)^3 / (1/φ)^6 = φ^6 / φ^3 = φ^3 -/
theorem cuprate_conventional_ratio :
    tcFamily .cuprate / tcFamily .conventional = Constants.phi ^ 3 := by
  dsimp [tcFamily, tc_phonon, familyLadderStep]
  -- (1/φ)^3 / (1/φ)^6 = φ^6/φ^3 = φ^3
  have hφpos : 0 < Constants.phi := Constants.phi_pos
  have hφne : Constants.phi ≠ 0 := ne_of_gt hφpos
  have h3 : Constants.phi ^ 3 ≠ 0 := pow_ne_zero 3 hφne
  have h6 : Constants.phi ^ 6 ≠ 0 := pow_ne_zero 6 hφne
  field_simp

/-- 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

The framework does not prove the ladder step assignments; they are chosen to match observed families. The framework does not derive the pairing mechanisms (phonon, spin fluctuation, d-wave) named in the docstring. The φ-derived BCS ratio of about 1.96 is an approximation, not an exact match to the measured 1.764.

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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