The spectrum of BN = TN(a) − τI
All N eigenvalues on one line. The one nearest zero is what conditions a solve.
A geometric-decay matrix, minus a constant on its diagonal, has a determinant that is one sine wave. Where that wave crosses zero is decided by a rhythm and a phase. The hollow matrix is the one case where they are the same number.
Drag the point. Across is the rhythm α = θτ/π; up is the phase β. Every pair (a, τ) inside the band lands exactly once in this triangle; the hypotenuse is a → 0 and the two corners are the band edges. The matrix at dimension N is singular exactly when the point sits on the line β ≡ Nα (mod 1); the lines for N = 2…9 are drawn faintly, the current N in red. The dashed diagonal is τ = 1, the hollow matrix, which is the N = 1 line of the same family. The thin blue curve is where the τ slider can take you at the current a.
All N eigenvalues on one line. The one nearest zero is what conditions a solve.
Dimensions N = 2 … 60. A red cell is an exact singularity; a blue cell is a record near-miss.
Bounded below the band, quadratic at the lower edge, arithmetic inside, quadratic at the upper edge, bounded above.
Along N at the current (a, τ). Inside the band the spikes are the arithmetic: κ₂ is a constant times (N+τ)/dN, so it is as large as the residual is small. Most N sit near the floor; the rare tall ones are the near-misses.
Across τ at the current a. Outside the band the curve is flat and finite. It climbs like N² into each edge. Inside it is a forest, and the forest is where the point in the triangle lives.
Take the correlation matrix of an AR(1) process, TN(a) = [a|i−j|], and subtract a constant τ from the diagonal. For which N is τ an exact eigenvalue, and how close does the finite spectrum come when it misses? The inverse of TN is tridiagonal, so the determinant of the shifted matrix is a three-term recurrence, and a three-term recurrence with constant coefficients is a sine: det BN = (−1)N+1 aN τN−1 sin π(Nα−β) / sin πα. An N×N determinant collapses to one wave.
The wave has a rhythm and a phase. The rhythm α is the crossing angle where the symbol μa(θ) equals τ, divided by π. The phase β comes from the boundary argument of the Blaschke factor (z−a)/(1−az) at that angle. The pair (a, τ) determines (α, β) and the map inverts in closed form, which is why the triangle above is not a sketch but a chart: τ = sin πβ / sin πα, a = cos ½π(α+β) / cos ½π(α−β).
Singularity is Nα − β ∈ ℤ, a family of straight lines in the triangle. From that one line of algebra the classification falls out. If α is rational, p/q, then either qβ is an integer and the matrix is singular on exactly one residue class of N forever, or it isn't and the matrix is never singular despite a rational angle. If α is irrational, at most one N can ever be singular. So the number of singular dimensions is 0, 1, or ∞, all three behaviors are dense in the triangle, and almost every point has none.
Set τ = 1 and the phase locks to the rhythm: β = α. That diagonal is the N = 1 member of the singularity web, which is literal, since T1 − 1 = [0]. On it the residual becomes ‖(N−1)α‖, the inhomogeneous problem collapses to ordinary rational approximation, and Hurwitz's theorem makes the golden rhythm the best-protected point. Every result of the earlier hollow-matrix paper is this diagonal read off the general chart.
Off the diagonal the phase is a second arithmetic coordinate and the golden protection is gone. For a fixed irrational rhythm and almost every phase, the residual falls below c/N infinitely often for every c, so κ₂ has growth exponent 2 with no O(N²) envelope at all. The page cannot show you that theorem; it is a statement about almost every β and no finite sweep exhibits it. What it can show is the mechanism: the residual, the web, and the collapse on the resonant line.
Inside the band the matrix is indefinite, so the eigenvalue nearest zero is the smallest singular value, and the theorem gives it exactly to leading order: σmin = π γ(a,τ) dN/(N+τ), where γ is the slope of the symbol at the crossing and N+τ is the angular derivative of zN(z−a)/(1−az). Divide the band width by that and you have κ₂. For irrational α the residual equidistributes, so κ₂/N follows a Pareto law with tail index 1: a heavy tail whose mean diverges. At the two band edges the slope γ vanishes and the law turns quadratic in N with explicit constants. Outside the band there is nothing to approach and the condition number is bounded.