Quantum signal processing (QSP) studies quantum circuits interleaving known unitaries (the phases) and unknown unitary oracles encoding a hidden scalar (the signal). For a wide class of functions one can quickly compute the phases for a QSP protocol with a matrix element approximately equal to the application of this function to the signal; surprisingly, this simple ability unifies the presentation and analysis of most quantum algorithms. A separate, basic problem in quantum computing is gate approximation: realistic devices support only a subset of possible gates, and extensive results establish methods to (approximately) compile one's available gates to a desired one. Among these results, the Solovay-Kitaev theorem (SKT) establishes an equivalence between the universality of a gate set (its density in a compact Lie group) and the existence of short approximating sequences. QSP, alternatively, considers a single program (the phases) such that for all among a continuum of possible input signals a desired output unitary is achieved, and so resembles a `lifted' variant of gate approximation. We make this notion of lifting formal, constructing an `SKT for QSP,' establishing an equivalence between the density of a circuit ansatz in a specific functional class and the existence of short protocols. These methods provide alternative proofs for basic results in QSP, and establish an intersection between QSP and the theory of gate-approximation. Unlike prior work, our proofs are not required to be constructive and are surprisingly insensitive to the chosen ansatz—they comprise loosely coupled, swappable components amenable to ansatz extensions for which standard QSP proof methods fail completely. These strengths align with an insight of the standard Solovay-Kitaev theorem: that it is often easier to work with Lie algebras than Lie groups. By carefully linearizing certain aspects of QSP, we intermesh algebraic-geometric techniques from gate approximation with well-understood functional-analytic properties of QSP, bootstrapping tools strong enough to analyze diverse unitary ansätze.