Universal scaling of superfluid stiffness in Sierpinski spiral superconductors

Flat-band superconductivity has emerged as a route to enhanced pairing, yet the scaling of superfluid stiffness with system size in fractal geometries remains unexplored. Here we introduce the Sierpinski spiral, an open fractal network in which sub-triangles are connected by single inter-level bonds rather than sharing vertices. We prove algebraically that the flat-band degeneracy at E = +t satisfies FB(n)/N(n) = 1/4 exactly for odd generations n, and verify this numerically up to n = 8 (N = 29,523 sites). Self-consistent Bogoliubov–de Gennes calculations with a physical Debye cutoff, combined with phase twist superfluid stiffness measurements, reveal that the Berezinskii–Kosterlitz–Thouless temperature TBKT grows at a universal rate of +32.3% per fractal generation for n ≥ 5. This rate is material-independent—identical within 0.1% across Nb and MgB2 parameters and four coupling strengths U/t ∈ {0.03, 0.1, 0.5, 1.0}—and robust to Anderson disorder up to the critical thresholdWc = ωD, where TBKT decreases by less than 2%. At generation n = 7 (N = 9,840), the BKT temperature reaches 19.4 K for Nb (U/t = 0.1) and 160.6 K (U/t = 1.0). Independent cross-checks—replication of published results on Sierpinski gaskets, linearized gap-equation analysis, and spectral verification at n = 8—confirm the robustness of the findings. The algebraic origin of the flat band, combined with the topological protection of the growth rate, establishes open fractal spirals as a scalable platform for engineering superconducting coherence.