Materials Room Tsuperconductor Candidate Tc Adjacent Ratio

The adjacent-rung ratio

The golden ratio φ ≈ 1.618 appears in a proposed ladder of superconducting critical temperatures T_c. The ladder starts with magnesium diboride (MgB₂) at 39 K as rung 0. Each higher rung multiplies T_c by φ. So rung 1 is 63 K, rung 2 is 102 K, rung 3 is 165 K, rung 4 is 267 K, and rung 5 is 432 K. The last value, above 300 K, is the structural claim: a hydrogen-dominant material that reaches rung-5 pairing strength could operate at room temperature.

The declaration tc_adjacent_ratio is a formal statement in the framework's machine-checked library. It states, for every natural number k, that the ratio of T_c at rung k+1 to T_c at rung k equals φ. The proof is short: it rewrites the successor formula T_c(k+1) = T_c(k) × φ, then divides by T_c(k), which is positive. The statement also bundles this ratio into a certificate that collects the ladder's properties: positivity, the one-step multiplier, strict increase, and the adjacent ratio itself.

What the statement does not claim is any physical law. It does not say that any real material achieves rung 5, nor that the observed cuprate and hydride values are exact. The ladder's rung values are an empirical comparison: YBCO at 92 K sits near rung 1 (63 K), Bi-2223 at 110 K near rung 2 (102 K), Hg-cuprate at 138 K near rung 3 (165 K), and LaH₁₀ at 170 GPa near rung 4 (267 K). The framework models the pairing strength as a φ-ladder; the statement proves the arithmetic of that model, nothing more.

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']

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

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

The statement does not prove that any material reaches rung 5 or operates at room temperature. The statement does not derive the φ-ladder from first principles; it assumes the ladder as a model. The observed T_c values are approximate comparisons, not exact predictions.

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

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: