Chemistry Superconducting Tc

The φ-ladder of critical temperatures

Superconductivity is a state in which certain materials carry electric current with zero resistance. The critical temperature (Tc) is the threshold below which this happens. In conventional superconductors, Tc is low, often below 30 kelvin. In cuprates, a family discovered in 1986, Tc can exceed 90 kelvin. The standard theory, BCS theory, explains the low-temperature cases through phonons, vibrations of the crystal lattice that bind electrons into pairs.

The Recognition Science framework models these families as occupying different rungs of a ladder. Each rung corresponds to a power of the golden ratio φ, approximately 1.618. The framework's ledger, a discrete record of recognition events, is said to have an eight-tick structure, and macroscopic quantum coherence like superconductivity is modeled as alignment with that structure. The energy gap Δ for Cooper pairing, the bound electron pairs that carry supercurrent, scales with φ. Since BCS theory relates Tc to Δ, the framework derives that Tc also scales with φ.

In the framework's library, a machine-checked collection of formal theorems, the ladder step is a natural number n. The phonon-route proxy for Tc is defined as (1/φ)^n. A theorem proves that Tc decreases as n increases: tc_scaling shows that if n₁ < n₂, then tc_phonon n₁ > tc_phonon n₂. The framework assigns conventional superconductors to step 6, MgB2 to step 5, iron-based to step 4, cuprates to step 3, and a hypothetical room-temperature superconductor to step 1. A reference scale of 300 kelvin at step 1 calibrates the ladder so step 3 lands near 90 to 100 kelvin.

The framework proves ordering relations among these families. The theorem cuprate_gt_conventional shows cuprates have higher Tc than conventional superconductors. The theorem iron_between places iron-based between cuprates and conventional. The theorem mgb2_between places MgB2 between iron-based and conventional. A further theorem, cuprate_conventional_ratio, derives that the ratio of cuprate to conventional Tc equals φ³, about 4.236. The framework also defines a BCS ratio proxy as 2 log φ + 1, approximately 1.96, and proves it lies between 1.7 and 2.1. The measured BCS weak-coupling ratio is about 1.764.

These are formal statements about a model, not measurements of real materials. The framework's contribution is a compact classification: it organizes known superconductor families into a single φ-power ordering and derives the relative Tc ratios from that ordering. The empirical match to real Tc values is a check on the model, not a theorem. What the framework establishes in plain language is a ladder: one step down in the ladder multiplies the critical temperature by φ, and the known families sit on that ladder in a specific order.

/-- 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)

/-- 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

/-- The BCS ratio is approximately 1.96 (2*log(φ) + 1).
    log(φ) ≈ 0.481, so 2*log(φ) + 1 ≈ 1.96
    The actual BCS ratio is 2Δ₀/kTc = π/e^γ ≈ 1.764 for weak coupling.
    Our φ-derived approximation is in the right ballpark. -/
theorem bcs_ratio_approx : (1.7 : ℝ) < bcsDeltaTcRatio ∧ bcsDeltaTcRatio < (2.1 : ℝ) := by
  dsimp [bcsDeltaTcRatio]
  -- Use proven bounds from Numerics.Interval.Log: 0.48 < log(φ) < 0.483
  -- Constants.phi = (1 + √5)/2 = Real.goldenRatio
  have h_phi_eq : Constants.phi = Real.goldenRatio := rfl
  rw [h_phi_eq]
  have hlo : (0.48 : ℝ) < Real.log Real.goldenRatio := Numerics.log_phi_gt_048
  have hhi : Real.log Real.goldenRatio < (0.483 : ℝ) := Numerics.log_phi_lt_0483
  constructor
  · -- 1.7 < 2 * log(φ) + 1  ⟺  0.35 < log(φ)
    -- Since 0.48 > 0.35, we have log(φ) > 0.48 > 0.35
    linarith
  · -- 2 * log(φ) + 1 < 2.1  ⟺  log(φ) < 0.55
    -- Since log(φ) < 0.483 < 0.55, we have the result
    linarith

What this page does not claim

The framework does not measure Tc values for any real superconductor. The φ-ladder is a model, not a derivation from first principles of the actual Tc values. No claim is made that room-temperature superconductivity at n = 1 is physically achievable.

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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