Chemistry Superconducting Tc Family Ladder Step

The family ladder

Superconductivity is the state in which certain materials carry electric current with zero resistance, below a material-specific critical temperature (Tc). The materials group into families: conventional metals like aluminum and niobium, magnesium diboride, iron-based compounds, and the cuprates. In the Recognition Science framework, the declaration familyLadderStep is a definitional choice that assigns each family a whole number, or rung, on a ladder built from the golden ratio φ (approximately 1.618). Conventional superconductors sit at rung 6, magnesium diboride at 5, iron-based at 4, cuprates at 3, and a hypothetical room-temperature family at 1.

The framework then defines a proxy for Tc: the value (1/φ)^n, where n is the rung. Because 1/φ is less than 1, this proxy strictly decreases as the rung number increases. The machine-checked library of formal theorems proves this ordering: tc_scaling shows that a smaller rung number always yields a larger proxy value. From that single theorem, the library derives the family order: cuprates above iron-based, iron-based above magnesium diboride, and all of them above conventional superconductors. It also proves that the ratio of the cuprate proxy to the conventional proxy is exactly φ³, about 4.236.

These are theorems about the proxy, not about measured materials. The definition of familyLadderStep itself is a model, not a derivation: the framework chooses the rung numbers, it does not prove them from physics. The Tc predictions in Kelvin come from multiplying the proxy by a reference scale of 300 K, calibrated so that rung 3 lands near 90 to 100 K, the cuprate range. That calibration is also a choice. The proved theorems establish only the relative ordering of the proxy values and the φ³ ratio between the cuprate and conventional proxies.

In Recognition Science, the ladder connects to a broader claim: that the energy gap for Cooper pairing, the electron pairs that carry supercurrent, scales with φ, and that macroscopic quantum coherence aligns with an eight-tick ledger structure. Those are hypotheses with named falsifiers. The prediction is that Tc ratios between families follow φ-power ratios. The falsifier would be a measured family ordering that breaks the ladder, for example an iron-based superconductor with a higher Tc than any cuprate, or a conventional superconductor exceeding magnesium diboride.

What the declaration does not claim is equally clear. It does not prove that any real superconductor has the Tc the proxy suggests. It does not derive the rung numbers from the framework's forcing chain; they are assigned. It does not establish the existence of a room-temperature superconductor; the rung 1 family is labeled theoretical. The value of the work is the sharp, checkable ordering claim: within this model, the family hierarchy follows a φ-power ladder, and the library proves the internal consistency of that ordering.

/-- 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

/-- 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

/-- Tc prediction in Kelvin for a family. -/
noncomputable def tcFamilyK (f : SuperconductorFamily) : ℝ :=
  tcReferenceK * tcFamily f

What this page does not claim

The declaration does not prove that any real superconductor has the Tc the proxy suggests. The declaration does not derive the rung numbers from the framework's forcing chain; they are assigned. The declaration does not establish the existence of a room-temperature superconductor; the rung 1 family is labeled theoretical.

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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