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In the tetrad formulation of gravity, the so-called simplicity constraintsplay a central role. They appear in the Hamiltonian analysis of the theory, andin the Lagrangian path integral when constructing the gravity partitionfunction from topological BF theory. We develop here a systematic analysis ofthe corner symplectic structure encoding the symmetry algebra of gravity, andperform a thorough analysis of the simplicity constraints. Starting from aprecursor phase space with Poincar\'e and Heisenberg symmetry, we obtain thecorner phase space of BF theory by imposing kinematical constraints. Thisamounts to fixing the Heisenberg frame with a choice of position and spinoperators. The simplicity constraints then further reduce the Poincar\'esymmetry of the BF phase space to a Lorentz subalgebra. This picture provides aparticle-like description of (quantum) geometry: The internal normal plays therole of the four-momentum, the Barbero-Immirzi parameter that of the mass, theflux that of a relativistic position, and the frame that of a spin harmonicoscillator. Moreover, we show that the corner area element corresponds to thePoincar\'e spin Casimir. We achieve this central result by properly splitting,in the continuum, the corner simplicity constraints into first and second classparts. We construct the complete set of Dirac observables, which includes thegenerators of the local $\mathfrak{sl}(2,\mathbb{C})$ subalgebra of Poincar\'e,and the components of the tangential corner metric satisfying an$\mathfrak{sl}(2,\mathbb{R})$ algebra. We then present a preliminary analysisof the covariant and continuous irreducible representations of theinfinite-dimensional corner algebra. Moreover, as an alternative path toquantization, we also introduce a regularization of the corner algebra andinterpret this discrete setting in terms of an extended notion of twistedgeometries.

Edge modes of gravity. Part III. Corner simplicity constraints