Algebraic Diagnostics for Instantaneous Frequency Estimators

Tracking instantaneous frequency (IF) in multi-component signals, phase-locked loops (PLLs), and reduced-order power-system models is frequently undermined by severe numerical ill-conditioning near regime boundaries. Conventional time--frequency transforms and signal-space geometric invariants can become unreliable precisely where diagnostic certainty is most needed: when denominators approach zero, frequency ridges coalesce, or a locking manifold loses robustness. We propose a unified algebraic diagnostic layer based on Stroboscopic Boundary-Ideal (BDI) constructions. Regimes (e.g.\ separable ridges, lock states, balanced harmonics) are encoded as incidence ideals in an augmented polynomial ring, and their boundaries are projected onto the design/parameter space via saturation and elimination. To quantify fragility we introduce two local indices: (i) Overlap Thickness (OT), a local intersection multiplicity measuring how ''flat'' a boundary is along a parameter drift direction, and (ii) Transversality Index (TI), the order of contact of a parameter path with the boundary. We state hypotheses under which OT and TI are finite and generator-independent, and we show an OT--TI bound for regular crossings. The framework is illustrated on a sequence of case studies: (a) Milano-type circuit diagnostics and IF paradoxes, (b) algebraic limits of multi-component separation under ridge crossing, (c) symbolic cycle-slip boundaries for PLLs, and (d) multi-delay differential equations (MDDEs), where a phase-extended stability boundary yields certificates of delay-independent stability (via infeasibility / the unit ideal) without gridding the delay space. Finally, we provide a reproducible ridge-crossing benchmark and an empirical robustness study (3,600 Monte Carlo trials across $-10$~dB to $30$~dB SNR), demonstrating that sub-bin interpolation improves the stability of a contact-order proxy and onset detection accuracy; the end-to-end runtime is on the order of tens of milliseconds per instance in our reference implementation.