Explicit Full-Spark and Quantitatively Phase-Retrievable Gabor Windows via Algebraic Perturbations

Abstract

We construct explicit finite Gabor windows whose orbits are simultaneously full spark and phase retrievable. In every cyclic dimension, algebraic specialization preserves all Gabor minors while removing every zero of the self-ambiguity function, thereby resolving the explicit simultaneous- construction problem arising in finite nilpotent-group phase retrieval. We isolate the underlying regularization principle: a sufficiently small algebraic perturbation transfers full spark from one seed while retaining an ambiguity gap from another. In every prime dimension \(p\geq5\), this gives an explicit full-spark window with normalized ambiguity margin at least \(c_0p^{-1/2}\), for an absolute constant \(c_0>0\); the order is sharp by the finite Moyal identity. Chinese-remainder tensorization followed by the same regularization extends the \(N^{-1/2}\) scale to squarefree cyclic dimensions having a bounded number of prime factors. The resulting full-data lifted intensity map has a dimension-independent lower stability bound on each such family. A second explicit seed gives a polynomial ambiguity margin, of order \(N^{-2}\), in every cyclic dimension, and we prove that this order is sharp within the geometric-seed construction. We also identify why the sharper cubic-chirp argument itself cannot cross that boundary: the unmodified cubic chirp has systematic ambiguity zeros at every modulus \(p^a\), \(a\geq2\). For a smaller fixed character perturbation and \(p\geq2^{14}\), we additionally prove uniform stability after a fixed fraction of arbitrary erasures on each fixed two-atom Gabor subspace. This last statement is deliberately local to one two-atom subspace and does not compare signals supported on different pairs.