Logarithmic boundary depth and the conditioning of the discrete inverse conductance problem on hyperbolic lattices

Abstract

Version 1.1 (revised after peer review; the review and the point-by-point response are included in code.zip under paper/reviews). Changes from v1.0: (a) the flat-lattice condition numbers that are singular in double precision are now computed in 512-bit ball arithmetic with certified radii, and the growth law is tested against preregistered rival forms; (b) conditioning under uniform and two-decade log-uniform disorder and under high-contrast defects is added (the advantage persists; the gap widens under strong disorder); (c) a dimension-matched probe control refuted our own preregistered prediction and corrects v1.0: at matched probe count and matched dimension the hyperbolic advantage is at most half a decade and reversed at N~112; (d) the degree-three premise of Proposition 1 is argued for general L. Every added computation was preregistered before it was run. Preprint, data and code. We study the discrete inverse conductance (Calderón) problem on layer-truncated hyperbolic {7,3} tilings and matched square and triangular lattice disks. The condition number of the Dirichlet-to-Neumann sensitivity Jacobian grows polynomially on the hyperbolic lattice (log10 kappa = 3.89 at N = 847, inside a preregistered band) and exceeds float64 resolution by N ~ 400 on flat lattices, because boundary depth scales as O(log N) versus O(sqrt N). Exact GF(p) ranks show all tested networks are identifiable: the difference is conditioning. The advantage persists under probe matching (post hoc identifiable-subspace metric) and for the Neumann-to-Dirichlet map. For the RC version, a preregistered finite-size prediction was refuted; a domain-monotonicity argument shows the Dirichlet spectral gap decreases to a positive limit, bounding the RC relaxation time uniformly in N. The RC dynamics are cross-validated with SciPy BDF and the rusty-SUNDIALS CVODE solver against the exact matrix-exponential solution and the static Dirichlet-to-Neumann map. Files: paper.pdf; dataset.zip (graphs, DtN matrices, conditioning, exact ranks, spectral gaps, integrator controls, RC step responses; CC BY 4.0); code.zip (simulator, analysis scripts, and the RC-network benchmark contributed to rusty-SUNDIALS; MIT).