Physics Superconducting Flux Quantum

The flux quantum

The superconducting flux quantum, written Φ₀, is the smallest discrete unit of magnetic flux that can pass through a superconducting loop. Its measured value is about 2.0678 × 10⁻¹⁵ webers. The standard formula is Φ₀ = h/(2e), where h is the Planck constant and e is the elementary charge. The factor 2 in the denominator appears because the current in a superconductor is carried by Cooper pairs, two electrons bound together, so the effective charge is 2e. This quantization is a direct consequence of the fact that the superconducting wavefunction must be single-valued around the loop, a condition first recognized by Fritz London in 1950 and confirmed experimentally by Bascom Deaver and William Fairbank in 1961.

The flux quantum is one of the cleanest demonstrations that macroscopic quantum effects are real. A superconducting ring threaded by a magnetic field will trap flux only in integer multiples of Φ₀, no matter how the field is applied. This is not a small correction; it is a hard constraint on the magnetic field itself. The phenomenon underlies the operation of SQUIDs, superconducting quantum interference devices, which are the most sensitive magnetometers known, capable of detecting fields a billion times weaker than the Earth's magnetic field.

In Recognition Science, the framework models the factor 2 in Φ₀ = h/(2e) as a recognition doubling, a single step in a discrete counting process. The framework defines a cost function J(x) = (x + 1/x)/2 − 1 that measures the price of a recognition event, and it proves three general facts about this cost. First, the cost vanishes when its two inputs are equal, meaning a perfect match costs nothing. Second, the cost is never negative for positive inputs. Third, a particular threshold value, the golden ratio minus 1.5, is always positive. These are proved in a machine-checked library of formal theorems, but they are general properties of the cost function, not specific to superconductivity.

The framework's module on the flux quantum does not yet connect these general facts to the physics of Cooper pairs. The module defines a cost on the ratio of two real numbers m/e, proves the three general facts above, and packages them into a certificate object. What is missing is a definition of m and e in terms of the superconducting phase itself. The docstring records the intended direction: one J-cost cycle on the phase space should correspond to one flux quantum. That identification is a research note, not a proved result.

What the module does establish, in plain language, is that the cost function has the right shape to serve as a measure of mismatch. It is zero at perfect agreement, always nonnegative, and has a built-in positive threshold. Whether that threshold is the flux quantum remains open. The general facts are proved; the bridge to the measured value of Φ₀ is not yet built.

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

The framework proves that the flux quantum equals h/(2e); that identification is a research note, not a result. The module establishes any property specific to superconductivity beyond the general cost facts. The framework derives the measured value of the flux quantum.

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/SuperconductingFluxQuantum.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

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