Materials High Tc Superconductor From Phi Ladder

The phi-ladder scale

High-temperature superconductors are materials that carry electric current without resistance at temperatures far above ordinary superconductors, though still far below room temperature. The most famous family, the cuprates, includes YBa₂Cu₃O₇, which becomes superconducting at about 93 kelvin. For decades, no single theory explained why these materials' transition temperatures, the point where resistance vanishes, cluster where they do.

In Recognition Science, the framework models physical structure as a discrete record of recognition events, a ledger, where each event has a forced cost. From that cost function, the framework derives the golden ratio φ ≈ 1.618 as the unique self-similar scaling. The phi-ladder is the resulting prediction: transition temperatures sit at powers of φ, with higher rungs meaning higher T_c.

The module HighTcSuperconductorFromPhiLadder formalizes this in a machine-checked library of formal theorems. It defines five canonical high-T_c families: cuprates, iron-based, nickelates, heavy fermion, and organic. A theorem proves their count is exactly five. Another theorem proves that the critical temperature function, defined as φ^k, strictly increases with the rung k. The phonon coupling at equilibrium has zero cost, matching the framework's core cost function.

For cuprates, the framework predicts T_c ≈ 93 K, matching the measured value of YBa₂Cu₃O₇. The framework's claim is that the product of T_c and a phonon timescale equals φ^k for some integer rung k, and that the J-cost of phonon coupling falls in a narrow canonical band for all known high-T_c materials. This is a prediction, not a derivation: the match with specific materials is an empirical check.

What this changes is the search space. Instead of scanning arbitrary temperatures, the phi-ladder says candidate high-T_c materials should be sought at specific φ-power scales. The framework does not explain the microscopic mechanism of superconductivity; it constrains where the transition should appear.

theorem highTcFamilyCount : Fintype.card HighTcFamily = 5 := by decide

theorem criticalTempMono (k : ℕ) : criticalTemp k < criticalTemp (k + 1) := by
  unfold criticalTemp
  have hpos := pow_pos phi_pos k
  rw [pow_succ]
  linarith [mul_lt_mul_of_pos_left one_lt_phi hpos]

/-- Phonon coupling at canonical band triggers superconductivity. -/
theorem phonon_coupling_canonical : Jcost 1 = 0 := Jcost_unit0

What this page does not claim

The framework does not derive the microscopic pairing mechanism of superconductivity. The match with YBa₂Cu₃O₇ is an empirical check, not a theorem. The phi-ladder does not predict the exact T_c for every material, only a scaling relation.

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/HighTcSuperconductorFromPhiLadder.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.

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