Condensed Matter Josephson Frequency Rs Josephson Freq Cert

What the certificate proves

The Josephson effect describes a current that flows between two superconductors separated by a thin insulating barrier. When a voltage V is applied, the current oscillates at a frequency f_J = 2eV/h, where e is the electron charge and h is Planck's constant. This gives the Josephson constant K_J = 2e/h ≈ 483.6 GHz/mV, a value measured so precisely that it now defines the volt in the International System of Units. The effect was predicted by Brian Josephson in 1962 and confirmed experimentally within a year.

In the Recognition Science framework, the declaration JosephsonFreqCert does not prove this formula. It proves three general facts about a cost function Jcost applied to a ratio m/e. First, the cost vanishes when m equals e. Second, the cost is nonnegative for positive inputs. Third, the constant phi − 3/2 is positive, where phi is the golden ratio. These are statements about the framework's cost function, not about superconductors.

The module defines domainCost (m e) as Jcost (m / e). The certificate bundles the three proved facts into a structure. The framework's machine-checked library of formal theorems shows the certificate is inhabited, meaning the structure has a valid instance. The docstring records a research note: the Josephson constant is determined by the fine-structure constant. That note is not a proved result.

What would make this a theorem about the Josephson effect is a definition of m and e in terms of physical quantities. The pack states this plainly: the module proves nothing specific to the subject because domainCost is defined without reference to one. The three facts hold for any positive real numbers m and e.

The practical consequence is a boundary. The framework's library contains a certificate named for the Josephson frequency, but the certificate's content is generic. A reader who wants the physical formula should rely on the measured Josephson constant, not on this declaration.

structure JosephsonFreqCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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)

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

What this page does not claim

The Josephson frequency formula f_J = 2eV/h is not proved by this declaration. The research note about the fine-structure constant is not a formal result. The certificate says nothing about superconductors or the physical Josephson effect.

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/Josephson_Frequency_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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