Materials Room Tsuperconductor Candidate

The phi-ladder prediction

Superconductors carry electric current with zero resistance, but only below a material-specific critical temperature T_c. The highest confirmed examples, hydrogen-rich compounds under extreme pressure, reach around 250 to 260 K. A machine-checked library of formal theorems, built inside the Recognition Science framework, now derives a simple ladder for T_c: each integer step up multiplies the critical temperature by the golden ratio φ ≈ 1.618.

The ladder starts from a reference point: magnesium diboride (MgB₂) at 39 K, labeled rung 0. The framework's derivation, formalized in the module RoomTSuperconductorCandidate, defines T_c at rung k as referenceTc times φ^k. The next rungs follow: rung 1 gives 63 K, rung 2 gives 102 K, rung 3 gives 165 K, rung 4 gives 267 K, and rung 5 gives 432 K. The observed values for cuprates (YBCO at 92 K, Bi-2223 at 110 K) and for LaH₁₀ at 170 GPa (250-260 K) sit near the predicted rungs, though the framework treats these as empirical checks, not as part of the proof.

Rung 5 sits above 300 K, meaning room-temperature operation is structurally permitted for any hydride that achieves rung-5 pairing strength. The framework proves four properties of this ladder: every rung has positive T_c, each step multiplies by exactly φ, the sequence strictly increases, and the adjacent ratio equals φ. These are collected in a certificate structure, RoomTSuperconductorCert, which bundles the four theorems into a single object. The certificate is a formal guarantee that the ladder behaves as claimed, with zero unproved assumptions in the framework's logic.

In plain language, the framework establishes a precise, testable consequence of its phi-ladder structure: if a material reaches the fifth rung of pairing strength, its critical temperature should exceed 300 K. The prediction is sharp, and the formal certificate makes the mathematical structure auditable. Whether any real hydride achieves that pairing strength at ambient pressure remains an experimental question, not a theorem.

/-- T_c at φ-ladder rung `k`. -/
def tcAtRung (k : ℕ) : ℝ := referenceTc * phi ^ k

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

theorem tcAtRung_strictly_increasing (k : ℕ) :
    tcAtRung k < tcAtRung (k + 1) := by
  rw [tcAtRung_succ_ratio]
  have hk : 0 < tcAtRung k := tcAtRung_pos k
  have hphi_gt_one : (1 : ℝ) < phi := by
    have := Constants.phi_gt_onePointFive; linarith
  have : tcAtRung k * 1 < tcAtRung k * phi :=
    mul_lt_mul_of_pos_left hphi_gt_one hk
  simpa using this

theorem tc_adjacent_ratio (k : ℕ) :
    tcAtRung (k + 1) / tcAtRung k = phi := by
  rw [tcAtRung_succ_ratio]
  have hpos : 0 < tcAtRung k := tcAtRung_pos k
  field_simp [hpos.ne']

What this page does not claim

No real material has been confirmed to achieve rung-5 pairing strength at ambient pressure. The observed values for cuprates and LaH₁₀ are empirical checks, not derived predictions within the framework. The framework does not derive the existence of any specific room-temperature superconductor material.

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/RoomTSuperconductorCandidate.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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