Materials Room Tsuperconductor Candidate Tc At Rung Succ Ratio

The rung-to-rung step

The declaration tcAtRung_succ_ratio is a formal theorem about a ladder of predicted critical temperatures, the temperatures below which a material superconducts. The ladder assigns each integer rung k a temperature T_c(k). The theorem states that moving up one rung multiplies the temperature by exactly φ, the golden ratio, about 1.618. In symbols: T_c(k + 1) = T_c(k) × φ. This is not a numerical coincidence; it is a proved identity in the framework's machine-checked library of formal theorems, derived from the definition of the ladder itself.

The ladder is anchored to a reference rung. Rung 0 is set to magnesium diboride (MgB₂) at 39 kelvin, a real superconductor. From that anchor, the theorem forces the rest of the ladder: rung 1 at about 63 K, rung 2 at about 102 K, rung 3 at about 165 K, rung 4 at about 267 K, and rung 5 at about 432 K, which sits above room temperature. The framework's account predicts that a hydrogen-dominant structure achieving rung-5 pairing strength could operate at ambient pressure. The theorem itself, however, only establishes the ratio between adjacent rungs; it says nothing about whether any material actually reaches a given rung.

The same theorem yields two immediate consequences, both proved in the same file. First, the sequence is strictly increasing: each rung is warmer than the one below. Second, the ratio of consecutive temperatures is exactly φ. These follow from the multiplication rule and from the fact that φ is greater than 1. The certificate structure RoomTSuperconductorCert bundles these properties into a single object, a formal witness that the ladder behaves as claimed.

In Recognition Science, this ladder is one concrete application of the φ-power scaling that the framework derives from its cost function. The framework models physical constants and mass ratios on a φ-power ladder, and this superconducting temperature ladder is a materials-science instance of the same pattern. The theorem does not prove that any hydride reaches rung 5; that remains a structural prediction, not a formal result. The measured values listed alongside the ladder, such as YBCO at 92 K and LaH₁₀ at 250 to 260 K, are empirical checks, not consequences of the theorem.

What the theorem changes for a reader is the shape of the prediction. If the framework's pairing-strength ladder is right, then the gap between known superconducting families and a room-temperature candidate is not arbitrary; it is a fixed multiplicative step. The theorem makes that step precise and machine-checked, so the claim about the ratio is not a hope or a fit, it is a derived identity. Whether the step is physically realized in any specific compound is a separate, open question.

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

/-- Reference T_c (RS-native dimensionless 1, calibrated at MgB₂ = rung 0). -/
def referenceTc : ℝ := 1

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

What this page does not claim

The theorem does not prove that any material reaches rung 5 or operates at room temperature. The measured values listed alongside the ladder are empirical checks, not consequences of the theorem. The theorem does not derive the fine-structure constant or any other coupling constant.

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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