The London equation jₛ = − (nₛ e²/mₑ c) A, which underlies the Meissner effect in superconductors, is here derived as an unconditional theorem of the four PDL axioms C1–C4, without invoking any condensed-matter physics or phenomenological postulate. The derivation proceeds in three steps. First, Lemma 1 establishes that the fraction of violated triangles ν in any mixed transition of the PDL programme satisfies ν = (1/4) ‖ψ₁ − Tᵏ ψ₁‖²Hcyc, where T = −iτ₂ is the half-cycle operator of D33 and k ∈ 0, 1, 2 labels the violation type. This identity is verified exhaustively over all 768 configurations of D29 with zero exception, in exact integer arithmetic. Second, Theorem 1 proves that axiom C4 (minimisation of ν) is equivalent to minimising ‖ψ₁ − ψ₂‖²Hcyc between neighbouring K₄ units in the coherence volume VC, and that the unique minimum is attained at ψ₁ = ψ₂, i. e. , at uniform phase φ = const over VC — the London gauge. Third, Corollary 1 combines this result with the PDL probability current of D32, the U (1) phase freedom of D46, and the orbital coherence tensor of D48 to obtain the London equation with coherence density ncoh = ρcoh (N) /mₚ in place of the electron superfluid density. The coherence nucleon density ncoh (N) = σ (N) Rₛurf/ (VC Rₜot) ∈ ℚ (√5) is an unconditional theorem of C1–C4 (D48). The resulting PDL London penetration depth is λL, PDL (N=40) ≈ 7. 25 × 10⁻¹⁵ m ≈ 34 λC, where λC = ℏ/ (mₚ c) is the proton Compton wavelength. The Meissner analogy — exponential screening of vector potentials over λL, PDL — follows as a corollary without any condensed-matter input. No free parameter enters at any step. This document resolves OP-London of DM v18.
Cédric Laubscher (Mon,) studied this question.