The conformal algebra of the scale window: so(2,1), its three mirrors, and the hyperbolic geometry of the substrate

The scale window of the Spectral Quantum Gravity (SQG) framework carries the mode operator H = −∂²_ω + c₀/ω², which together with the dilatation D and the special conformal generator K closes the de Alfaro–Fubini–Furlan algebra sl(2,R) ≅ so(2,1). This companion collects, in one place and with uniform grading, what that algebra fixes and what it does not.Three generators, three times: the three conjugacy classes as three candidate generators of emergent time (parabolic window Hamiltonian, hyperbolic logarithmic scale flow, elliptic equidistant clock).Three mirrors: the three reflections that appear in the framework are, as checked on the lattice, three different operations: an antiunitary observer mirror that is CRT and exchanges the two valleys with opposite chirality; a half-rotation that reaches the zone-boundary doubler of the same chirality only together with the sign (−1)^{n_ω}; and the see-saw inversion, which is not a symmetry of H but the DFF inversion conjugating H into K.Hyperbolic picture: the window is the radial direction of AdS₂, and the see-saw inversion is a reflection in a geodesic with a fixed circle at the see-saw scale, so that the reflected quotient is reflection positive, whereas the fixed-point-free antipodal quotient of de Sitter space is not.Entanglement of the window field across the see-saw wall: a c = 1 logarithm, no area term, and no contribution to the induced Newton constant.The Goldstone phase of the condensate along the axis: isotropic at the speed of light in the coordinate ω/√κ, with a tree-level lattice anisotropy of −0.18 (E/E_Pl)² on the ω-uniform Planck lattice.Tori and the modular group: the framework's mirror and three-beat generate the modular group acting on the space of torus shapes.Every statement is graded as computed, restated from the main text, toy model, or open. The companion derives no new physical prediction; it is a structural account of a symmetry the main text already uses.