Complex spectrum of the partial theta function

We study the complex spectrum of the partial theta function $$\mathrm{\Theta }(q,x)=\sum _{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{x}^j,\mathrm{\mid }q\mathrm{\mid }<1,$$ where a spectral value is a parameter for which $\mathrm{\Theta }(q,\cdot )$ has a multiple zero. Since the function is defined here only for $\mathrm{\mid }q\mathrm{\mid }<1$, all spectral values are strictly inside the unit disk; boundary points on $\mathrm{\mid }q\mathrm{\mid }=1$ occur only as accumulation points of the spectrum. The paper combines two complementary points of view. Near the unit circle we prove that every point of $\mathrm{\mid }q\mathrm{\mid }=1$ is an accumulation point of the spectrum; the proof uses explicit spectral factors of truncations, the Jacobi triple product, and a boundary-window lifting argument near roots of unity. Inside a fixed subdisk, illustrated for $\mathrm{\mid }q\mathrm{\mid }\le 0.8$, the true spectrum is locally finite and must be separated carefully from the much larger branch loci of truncations and Jensen polynomials. We give a truncation-seeded Newton procedure which produces a discrete list of candidate spectral values, explain the caustic/escaping-root mechanism in finite approximants, and record numerical monodromy experiments using a radial convention: for a spectral point $q_{\ast }$, roots are labelled at the point $0.1q_{\ast }\mathrm{/}\mathrm{\mid }{q}_{\ast }\mathrm{\mid }$ on the small circle and then continued along the straight radial segment to $q_{\ast }$. This convention gives a coherent set of collision labels in the disk, treats negative real spectral values from the base point $-0.1$, and leads to a preliminary rational-direction heuristic for radial monodromy.