Ill-conditioning of the discrete inverse conductance problem as exponential dependence on the earlier span: a column-residual certificate, partly machine-checked, and its measurement

The companion preprints measured the exponential growth of the condition number of the Jacobian of the Dirichlet-to-Neumann map of a resistor network with respect to its edge conductances, and found that persistent homology of the nearest-neighbour structure of the Jacobian columns sees the depth only through a slow power law. This preprint measures the quantity that carries the rate: the distance of a column to the span of the columns that precede it, with columns ordered by depth. Such residuals bound the smallest singular value from above; one of the bounds is machine-checked in Lean 4 (any lower bound on the action of the matrix is at most every column residual), the other is the standard leave-one-out identity. On a square disk the minimum depth-ordered residual of each layer falls 1.243 decades per layer against 1.272 for the smallest singular value (preregistered band 0.85 to 1.15 of the ratio; measured 0.977), and the median column, not only the worst, falls exponentially (0.656 decades per layer, correlation -0.999), seven times faster than the nearest-neighbour topology. Same-layer dependence costs a further 0.6 to 1.8 decades per layer; a preregistered bound of 1.5 was refuted. On the {7,3} hyperbolic tiling the worst column falls 0.40 decades per layer over four layers, a third of the square's rate, and most columns stay well separated. Two design errors of the author (an explicit Jacobian too large for memory, bands set from a smaller pilot) are recorded with their deviations. Elementary Lean lemmas on the symmetry of the DtN matrix are included. Double precision, no noise; nothing here concerns holography. Companions: doi:10.5281/zenodo.23228685, doi:10.5281/zenodo.23241463, doi:10.5281/zenodo.23244556.