Materials Bcspairing From Phi Ladder Pairing Strength Pos

The ladder of pairing strengths

In conventional superconductors, the ratio of the energy gap to the critical temperature, 2Δ₀/(k_B T_c), is famously close to 3.53 for many materials. Recognition Science starts from a different, dimensionless ratio: Δ₀/(k_B T_c), without the factor of 2. The framework models this ratio as pairing strength, a measure of how strongly electrons bind into Cooper pairs at the transition temperature. Its library defines a sequence of these strengths, one for each integer rung of a ladder built on the golden ratio φ, and the declaration pairingStrength_pos proves that every value in this sequence is greater than zero.

The proof itself is short. The framework defines the pairing strength at rung k as φ raised to the power k, starting from a reference strength of 1. Since φ is positive, any power of φ is positive, and the theorem follows. A companion theorem shows that each step up the ladder multiplies the strength by φ, so the sequence grows without bound. The declaration pairingStrength_pos is the foundation for these later statements: without positivity, division by the strength would be undefined, and the ratio of adjacent rungs could not be computed.

In Recognition Science, this ladder is not a free choice. The framework derives φ from a forcing chain that starts with a cost function for recognition events, and it proposes that real materials occupy specific rungs. The empirical bench is rough: conventional superconductors such as Sn and Pb sit near rung 0 with Δ/T_c ≈ 1.76, intermediate-coupling materials like NbN near rung 1 with ≈ 2.85, and strong-coupling ones like MgB₂ near rung 2 with ≈ 4.61. The framework predicts that high-temperature cuprates and pnictides occupy rungs 3 and 4, with strengths between 4.6 and 12.1, covering the measured 5 to 12 range for YBCO, BSCCO, FeSe, and LiFeAs.

What the declaration does not claim is just as important. The theorem only establishes positivity of a defined sequence; it says nothing about whether real materials actually follow this ladder. The comparison to measured values is an empirical check, not a proof. The framework's own library makes this distinction explicit: the ladder structure is a prediction with a named falsifier, namely a material whose measured Δ/T_c falls off the predicted rungs by more than the experimental resolution. The machine-checked theorems guarantee internal consistency, not physical truth.

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

What this page does not claim

The theorem does not prove that any real material follows the φ-ladder. The empirical agreement with measured values is not a theorem but a comparison. The reference strength of 1 is a definitional choice, not a derived result.

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)

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