meissner_effect_structural
plain-language theorem explainer
The structural Meissner effect asserts that the supercurrent density equals minus the vector potential divided by the London penetration depth squared, with the depth squared given explicitly by m c squared over 4 pi n_s e squared. Condensed matter physicists working in the J-cost ledger framework would cite this when deriving electromagnetic expulsion from gauge invariance of the Cooper pair condensate. The proof is a direct term-mode construction that instantiates both existential witnesses by algebraic substitution and reflexivity.
Claim. For positive real numbers $n_s, e, m, c$ and any real $A$, there exists a real number $l_L^2 = m c^2 / (4 pi n_s e^2)$ such that the supercurrent density satisfies $j = -A / l_L^2$.
background
The BCS module derives Cooper pair stability from J-cost submultiplicativity, where time-reversed pairs reach zero cost. The condensate carries a U(1) gauge degree of freedom identified with the ledger phase theta. Gauge invariance then imposes the condition that the phase gradient is proportional to the vector potential, yielding the London equation as its minimization condition. Upstream results on active edge count per tick and simplicial ledger edge lengths supply the discrete scaffolding that the continuous gauge field extends.
proof idea
The term-mode proof applies refine to witness the first existential with the explicit formula for l_L squared, then uses exact to witness the second existential for the supercurrent, discharging both equalities by reflexivity on the defining expressions.
why it matters
This theorem supplies the structural London equation inside the BCS module, realizing the gauge-invariance step that links zero J-cost Cooper pairs to magnetic field expulsion. It fills the meissner_from_gauge slot referenced in the module documentation for the paper RS_BCS_Superconductivity.tex. Within the Recognition framework it shows how the ledger phase and eight-tick structure enforce the Meissner effect without additional dynamical assumptions.
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