Timelike Supersymmetric Geometries in Seven-Dimensional Supergravity

Seven-dimensional supergravity poses a uniquely intricate supersymmetry problem, combining topologically massive higher-form gauge dynamics, distinct Clifford algebra properties, and rich geometric structures. We first establish the complete formulation of the maximal Cartan-gauged theory in terms of the gauge-invariant higher-form flux, including both the bosonic action and supersymmetry variations. This formulation, which also makes the smooth ungauged limit manifest, is achieved by requiring a consistent truncation to the minimal Cartan-gauged theory in the equal-charge limit. We construct the complete bilinear system for a timelike Killing vector, where additional internal orientation of the symplectic-Majorana spinor arises already in the scalar sector, uniquely among pure minimal ungauged theories. In the equal-charge sector, we decouple the algebraic constraints into SU$(3)$-isotypic components, and analyse the differential bilinear relations, determining the permitted dilaton gradient, fluxes, fibration twist and intrinsic torsion of the base. We prove that with gauging and active Cartan flux, the complex scalar bilinear vanishes identically. We also construct a Killing gauge for the higher-form potential compatible with the self-duality condition. We show that with gauging but no active Cartan flux, the Killing vector also acts as a generator of the gauged R-symmetry, and the base structure is enlarged from SU$(3)$ to U$(3)$. We find that BPS-flow solutions realize this setting. With supersymmetry, we isolate a single independent component of the 3-form equation across an exhaustive four-class partition for both theories within this setup. Applied to known and proposed solutions in both theories, this classification exposes unexplored sectors and provides an accessible toolbox for constructing and testing solutions and developing new geometric and holographic applications.