Condensed Matter Cuprate Tc From Phi Ladder

The cuprate doping target

In condensed matter physics, cuprate high-temperature superconductors show their highest transition temperature T_c at a specific hole doping level, the optimal doping p_opt. For many families of cuprates, that optimal value sits near 0.16 holes per copper atom, a number measured in experiments across different compounds. The question is why this particular fraction appears so often.

The Recognition Science framework offers one candidate answer in a research note: p_opt = 0.16 is approximately J(φ)/0.73, where J is the framework's cost function, φ is the golden ratio, and 0.73 is called the gap-to-pi charge ratio. The note states that T_c is maximal at this doping. This is a hypothesis about cuprates, not a theorem. The framework's own library does not yet contain a definition of the measured or expected quantities in cuprate terms, so the identification of 0.16 with J(φ)/0.73 is an unproven proposal.

What the machine-checked library does prove is narrower. In the module CuprateTcFromPhiLadder, the cost function J(x) = (x + 1/x)/2 - 1 is applied to the ratio of a measured value to an expected value. Three theorems hold for any positive inputs: the cost is zero when measured equals expected, the cost is never negative, and the canonical threshold φ - 3/2 is positive. These are general properties of the cost function, true for any subject matter, not specific to superconductors.

The module also defines a structure called CuprateTcCert, a certificate that packages these three facts together, and proves the certificate is inhabited. This is a formal way of saying the three general properties are consistent and provable. It does not connect the cost function to any physical measurement of cuprates. The bridge from the general cost function to the specific claim about p_opt = 0.16 is missing; that bridge is the open research problem.

In plain language: the framework has a tool, the cost function, that behaves sensibly (zero at equality, never negative). The cuprate claim would require defining what "measured" and "expected" mean for hole doping in a superconductor, then showing the ratio lands at the golden-ratio-derived value. That definition does not exist in the library yet. The module proves the tool works, not that the tool applies to cuprates.

theorem domainCost_at_equilibrium (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty CuprateTcCert := ⟨cert⟩

What this page does not claim

No theorem connects the cost function to cuprate superconductivity. The value p_opt = 0.16 is not derived from the framework; it is a research note. The module does not prove T_c is maximal at any doping.

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/CondensedMatter/CuprateTcFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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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: