Physics Flux Tube From Jcost

The vortex core size

A type II superconductor admits magnetic flux in discrete tubes, each with a normal core of radius ξ (the coherence length) surrounded by supercurrents that screen the field over the London penetration depth λ. The ratio κ = λ/ξ decides the regime: when κ exceeds 1/√2 ≈ 0.707, the material is type II and vortices can form. This is the classical Abrikosov picture, named for Alexei Abrikosov's 1957 work.

In Recognition Science (RS), the framework models the cost of a recognition event as a function J(x) = (x + 1/x)/2 - 1, which is zero when x = 1 and positive otherwise. The FluxTubeFromJCost development defines a domain cost as J(m/e), where m and e are real numbers. It establishes three general facts: the cost vanishes when m = e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive, where φ is the golden ratio.

The structure AbrikosovVortexCert packages these three facts as a certificate. Its plain-language meaning is narrow: it shows the cost function has the right shape to be a candidate measure of deviation from a matching condition, and that the golden ratio marks a positive threshold. The development proves nothing specific to superconductors, because the variables m and e are not defined in terms of vortex physics. The research note in the source records the intended direction: to identify m and e with physical quantities like the coherence length and penetration depth, and to connect the threshold φ - 3/2 ≈ 0.118 to the type II boundary κ > 0.707.

What the development establishes, in short, is a formal template. It proves the cost function's basic properties and the positivity of a specific constant, but the bridge to the Abrikosov vortex is not yet built. That bridge would require a definition of m and e in the superconductor's own terms, which the development does not provide.

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]

What this page does not claim

This development proves any statement specific to superconductors. The variables m and e are not defined in terms of coherence length or penetration depth. The connection between the threshold and the type II regime is a research note, not a theorem.

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/Physics/FluxTubeFromJCost.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.

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