Condensed Matter Bcs Coherence Length Rs

The coherence length

The BCS coherence length, usually written ξ, is the distance over which two electrons stay paired in a superconductor. In ordinary superconductors like aluminum or lead, it ranges from about 100 to 1000 nanometers. In high-temperature cuprate superconductors, it is much shorter, often around 1 to 5 nanometers. That difference matters: a short coherence length means the paired electrons are tightly bound, which changes how the material responds to magnetic fields and defects. The standard theory, named after Bardeen, Cooper, and Schrieffer from 1957, predicts this length from the material's electron density and gap energy.

Recognition Science (RS) offers a structural way to organize these values. The framework models the coherence length as a power of the golden ratio φ ≈ 1.618, written as ξ = φ^k nanometers. Three examples appear in the research note: φ^15 ≈ 1364 nm for a conventional long-range superconductor, φ^7 ≈ 29 nm for a cuprate with short coherence, and φ^8 ≈ 47 nm for an intermediate case. This is a definitional choice, a way to place known values on a ladder, not a derivation from first principles. The note itself says this paragraph is a research note recording where the idea was meant to go, not a result.

What the module actually proves is much narrower. In the machine-checked library of formal theorems, the file defines a cost function J(m/e) and proves three general facts about it: the cost vanishes when the two inputs are equal, it is never negative for positive inputs, and the quantity φ − 3/2 is positive. These are properties of the cost function from the framework's core, applied to a ratio of two numbers labeled m and e. The module proves nothing specific to superconductors, because it never defines what m and e mean in that subject. The structure assembles these three facts into a certificate, and a theorem confirms that such a certificate exists.

In plain language: the page records an ambition and a proof. The ambition is to place BCS coherence lengths on a φ-power ladder. The proof is a small, correct statement about a general cost function. The gap between them is the missing definition of m and e in superconductor terms. Until that definition exists, the module is a template applied to a name, not a theorem about physics. What a reader can take away is the distinction: a research note can point where a theory might go, while a formal proof only certifies what it actually shows.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module derives the BCS coherence length from first principles. The φ^k values are measured quantities; they are a proposed structural placement. The proof establishes any property of real superconductors.

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/BCS_Coherence_Length_RS.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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