Materials Room Tsuperconductor Candidate Tc At Rung Pos

A positive temperature at every rung

Superconductivity is the flow of electrical current with zero resistance, and its key material property is the critical temperature T_c, the point below which that flow begins. A proposed theory of high-temperature superconductors arranges candidate materials on a ladder: each rung up multiplies the predicted T_c by the golden ratio φ ≈ 1.618. The declaration tcAtRung_pos establishes the most basic sanity condition of that ladder: every rung, from the reference point at magnesium diboride (MgB₂, 39 K) upward, carries a positive critical temperature. No rung dips to zero or below, which would make the ladder physically meaningless.

The proof is short and purely formal. The definition sets T_c(k) = φ^k in dimensionless units calibrated so that rung 0 equals 1. Since φ is a positive constant greater than 1, any power of it is positive, and multiplying by the positive reference value keeps the result positive. The machine-checked library of formal theorems verifies this for every natural number k in one step, using the positivity of φ and the fact that a positive number raised to any power stays positive. The declaration is one of four properties bundled into a certificate for the ladder; the others state that each step multiplies by φ, that the sequence strictly increases, and that adjacent ratios equal φ exactly.

What tcAtRung_pos does not claim is any empirical content. It does not assert that any real material reaches rung 5 or any other rung; it only says that if the ladder's definition is accepted, the numbers it produces are positive. The ladder's comparison with measured materials, such as YBCO at 92 K against rung 1's 63 K or LaH₁₀ at 250–260 K against rung 4's 267 K, is a separate empirical check, not part of this theorem. The declaration also says nothing about whether room-temperature superconductivity is physically achievable; it merely confirms that the framework's own model does not produce a nonsensical negative or zero temperature at any step.

In Recognition Science, this positivity result is a hygiene check on the φ-ladder construction. It guarantees that the model's internal arithmetic is coherent before any comparison with experiment begins. For a reader, the practical takeaway is simple: the framework's predicted T_c values form a strictly increasing, positive sequence, and the formal proof of that fact is machine-verified, not a hand-wave.

theorem tcAtRung_pos (k : ℕ) : 0 < tcAtRung k := by
  unfold tcAtRung referenceTc
  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  linarith [this]

theorem tcAtRung_pos (k : ℕ) : 0 < tcAtRung k := by
  unfold tcAtRung referenceTc
  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  linarith [this]

structure RoomTSuperconductorCert where
  tc_pos : ∀ k, 0 < tcAtRung k
  one_step_ratio : ∀ k, tcAtRung (k + 1) = tcAtRung k * phi
  strictly_increasing : ∀ k, tcAtRung k < tcAtRung (k + 1)
  adjacent_ratio_eq_phi : ∀ k, tcAtRung (k + 1) / tcAtRung k = phi

What this page does not claim

No empirical claim that any real material reaches rung 5 or any other rung. No assertion that room-temperature superconductivity is physically achievable. No statement about the accuracy of the ladder's comparison with measured materials.

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)

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