Materials Bcspairing From Phi Ladder Pairing Strength Strictly Increasing

The phi-ladder pairing strength

In the mathematics of superconductivity, the BCS theory of Bardeen, Cooper, and Schrieffer (1957) predicts a universal ratio between the energy gap at zero temperature and the critical temperature: 2Δ₀/(k_B T_c) ≈ 3.53 for weakly coupled conventional superconductors. The Recognition Science framework takes a different starting point. It defines a dimensionless pairing strength Δ₀/(k_B T_c) that sits on a ladder of values governed by the golden ratio φ ≈ 1.618. The base rung is set to 1, and each step up multiplies the previous value by φ. The theorem pairingStrength_strictly_increasing proves, in the framework's machine-checked library of formal theorems, that each rung is strictly larger than the one before it: pairingStrength(k) < pairingStrength(k+1) for every natural number k.

The proof is short and transparent. The definition gives pairingStrength(k) = φ^k. The theorem pairingStrength_succ_ratio shows pairingStrength(k+1) = pairingStrength(k) · φ. Since every pairing strength is positive (pairingStrength_pos) and φ > 1, multiplying a positive number by φ makes it strictly larger. The chain of inequalities closes with the golden ratio's defining property: φ² = φ + 1, so φ > 1. This is a purely formal result about the sequence 1, φ, φ², φ³, ...; it does not by itself say which material sits on which rung.

The framework's empirical bench, documented alongside the theorem, assigns conventional BCS superconductors like Sn, Pb, and Nb to rung 0 with Δ/T_c ≈ 1.76; intermediate-coupling materials like NbN and V₃Si to rung 1 with ≈ 2.85; and strong-coupling materials like Pb-Bi alloys and the σ-band of MgB₂ to rung 2 with ≈ 4.61. The ratios between adjacent rungs are φ and φ², matching the ladder's construction. The same ladder predicts high-T_c cuprates and pnictides at rungs 3 and 4, with Δ/T_c between 4.6 and 12.1, covering the empirically observed 5 to 12 range across YBCO, BSCCO, FeSe, and LiFeAs. These assignments are empirical comparisons, not consequences of the theorem.

What the theorem itself establishes is narrow and precise: the sequence of powers of the golden ratio is strictly increasing. It does not prove that any real superconductor follows this ladder. The mapping from rung numbers to material classes is a modeling choice, and the numerical agreement with measured ratios is an empirical check, not a derived law. The theorem's value is structural: it certifies that the ladder has no plateaus or descents, so if the identification of rungs with materials holds, the ordering of pairing strengths across material classes is forced in one direction only.

theorem pairingStrength_strictly_increasing (k : ℕ) :
    pairingStrength k < pairingStrength (k + 1) := by
  rw [pairingStrength_succ_ratio]
  have hk : 0 < pairingStrength k := pairingStrength_pos k
  have hphi_gt_one : (1 : ℝ) < phi := by
    have := Constants.phi_gt_onePointFive; linarith
  have : pairingStrength k * 1 < pairingStrength k * phi :=
    mul_lt_mul_of_pos_left hphi_gt_one hk
  simpa using this

/-- Pairing strength at φ-ladder rung `k`. -/
def pairingStrength (k : ℕ) : ℝ := referenceStrength * phi ^ k

theorem pairingStrength_succ_ratio (k : ℕ) :
    pairingStrength (k + 1) = pairingStrength k * phi := by
  unfold pairingStrength
  rw [pow_succ]; ring

theorem pairingStrength_pos (k : ℕ) : 0 < pairingStrength k := by
  unfold pairingStrength referenceStrength
  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  linarith [this]

What this page does not claim

The theorem does not prove that any real superconductor's measured Δ/T_c follows the phi-ladder. The mapping of material classes to rung numbers is a modeling choice, not a derived consequence. The framework does not derive the fine-structure constant or any coupling constant from first principles.

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/Materials/BCSPairingFromPhiLadder.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: