Physics Josephson Inductance3 From Jcost

A formal placeholder

A Josephson junction is a pair of superconductors separated by a thin barrier; it carries a supercurrent that depends on the phase difference between the two sides. Its kinetic inductance, the inertia of the current-carrying Cooper pairs, is classically written L_J = hbar / (2e * I_c), where I_c is the critical current and e is the electron charge. The framework's module proposes to evaluate that inductance at a special operating current, I = J(phi) * I_c, where J is the cost function and phi is the golden ratio. At that current the inductance becomes L_J0 / J(phi), roughly 8.47 times the base value, an amplification factor of J(phi)^(-1).

The module itself does not prove that the Josephson formula follows from the cost function. Its definition, domainCost, a discrete record of recognition events, is simply Jcost (m / e), a ratio of two real numbers with no physical meaning attached. The Lean file proves three general facts about this cost: it vanishes when m equals e, it is nonnegative for positive inputs, and phi minus 3/2 is positive. These are properties of the cost function alone, true for any positive ratio, not facts about superconductivity. The docstring says plainly that the paragraph about Josephson inductance is a research note recording where the idea was meant to go, not a result.

What the module does establish is a template. It packages the three cost facts into a structure called JosephsonL3Cert and shows that structure is inhabited. That is a formal way of saying the cost function satisfies its own basic sanity checks: zero at unity, no negative costs, and a positive threshold. The same body of theorems appears in 2383 sibling modules, each with a different subject name but the same universally quantified content. The single source of truth is the template module DomainCostTemplate; this file adds the Josephson name without adding Josephson physics.

For the module to become a theorem about a Josephson junction, someone would need to define m and e in the junction's own terms, as physical quantities with units and meaning. That definition does not exist here. The honest summary: the framework proves the cost function behaves, and the research note sketches a hope that the cost function explains kinetic inductance amplification. The proof and the hope are separate objects, and the module keeps them visibly separate.

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]

theorem cert_inhabited : Nonempty JosephsonL3Cert := ⟨cert⟩

What this page does not claim

The Josephson inductance formula L_J = hbar / (2e * I_c) is not derived from the cost function in this module. The numerical factor 8.47 is a research note, not a proved consequence of the Lean theorems. The module does not establish that the golden ratio governs any measurable property of a Josephson junction.

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/Josephson_Inductance3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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