Gaussian Products and an Order Dichotomy for Principal Ideals

Abstract

We identify the algebraic principal ideal generated by a Gaussian product of two Hilbert--Schmidt operators. Its deterministic generator is obtained by multiplying the square tails of their singular-value sequences; in a fixed diagonal basis, one probability-one event gives the identity for all decreasing square-summable weights. For the induced operation on principal ideals, we prove an order dichotomy for arbitrary families of strictly positive probes. A family jointly determines every inclusion if and only if its tail stable-rank profiles have no common eventual upper bound. Equivalently, each prescribed countable input system admits one probe from that family reflecting all inclusions. If the condition fails, one common fibre contains a copy of $\mathcal P(\mathbb N)/\mathrm{Fin}$ preserving finite sums and intersections. Its generators are continuously parametrized and arbitrarily close in Hilbert--Schmidt norm, while their deterministic output ratios tend to one at any prescribed rate. The order dichotomy follows from a bound on all probes that lose one noninclusion and a separated block construction under a scalar tail budget. Further consequences include generic separation under every positive summable envelope and exact cardinal thresholds for joint detection and single-probe separation.