Materials Bcspairing From Phi Ladder Pairing Strength Adjacent Ratio

The adjacent ratio theorem

In the study of superconductors, the ratio of the energy gap to the critical temperature, Δ₀/(k_B T_c), is a dimensionless number that measures how strongly electrons bind into Cooper pairs. In conventional BCS theory this ratio is famously universal at about 1.76. The Recognition Science framework models this quantity as a sequence indexed by integer rungs on a ladder, where each rung corresponds to a material class. The framework defines the pairing strength at rung k as φ^k, where φ is the golden ratio, approximately 1.618.

The theorem pairingStrength_adjacent_ratio proves that the ratio of the pairing strength at rung k+1 to the pairing strength at rung k is exactly φ, for every integer k. This is a formal statement in the machine-checked library of formal theorems, verified with zero axioms beyond the standard logical ones. The proof is short: it unfolds the definition, applies the power rule, and simplifies. The theorem also implies that the sequence is strictly increasing, since φ is greater than one.

In Recognition Science, this is part of a broader prediction. The framework proposes that real material classes sit on this ladder: conventional BCS superconductors like tin and lead at rung 0 with Δ/T_c near 1.76, intermediate-coupling materials like NbN at rung 1 near 2.85, and strong-coupling materials like MgB₂ at rung 2 near 4.61. The framework further predicts that high-temperature cuprates and pnictides occupy rungs 3 and 4, with values between 4.6 and 12.1, a range that covers the empirically observed 5 to 12 across YBCO, BSCCO, FeSe, and LiFeAs.

What the theorem does not claim is that real materials actually sit on this ladder. The theorem is a statement about a defined sequence, not about measured data. The empirical comparison is a separate check, and the numbers for real materials are approximate. The theorem does not derive the golden ratio from physics; it takes φ as given by the framework's constants. It also does not prove that the ratio is universal in nature; it only proves that the defined sequence has this property.

theorem pairingStrength_adjacent_ratio (k : ℕ) :
    pairingStrength (k + 1) / pairingStrength k = phi := by
  rw [pairingStrength_succ_ratio]
  have hpos : 0 < pairingStrength k := pairingStrength_pos k
  field_simp [hpos.ne']

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

What this page does not claim

No measured material is proven to sit on the phi-ladder; the empirical comparison is a separate check. The golden ratio is not derived from physics in this theorem; it is taken as given by the framework's constants. The theorem does not prove that the ratio is universal in nature, only that the defined sequence has this property.

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: