De Sitter = Hilbert = Hamiltonian A form-valued Poisson isomorphism and a permanent BRST observable on a C3v Y-junction in real hyperbolic space

What this paper proves, in plain languageSpacetime, the Hilbert space, and the Hamiltonian are the exact same thing.That is the claim of this paper. Not analogous. Not equivalent under a limit. Not related by aduality. The same thing.The framework that led to this is the SEM / UFAP construction with a three-armed (C3v)junction at the proton-scale core of the de Sitter static patch. The work of this paper is to writethe math that proves the equality.We show three things:1. Spacetime, Hilbert space, and Hamiltonian are equal. We prove this as a topologicalisomorphism (a continuous bijection with continuous inverse) on the main symmetry sectorof the junction (the A1 ⊕ E sector). Where physics has always treated these as three thingsto be reconciled, we show they are one object.2. The leftover symmetry sector (A2) is a physical observable, not a defect. Everystandard mechanism that might have removed it — gauge fixing, BPS supersymmetry, CPconservation, the BRST quotient — fails by direct calculation. What remains is real andmeasurable: the magnetic flux threading a small symmetric triangle at the junction vertex.13. This paper proves the equality; a sequel makes it the definition. A Connes-stylespectral triple (A, H, D, J, γ) is to be constructed in which one tuple is the spacetime, is theHilbert space, and is the Hamiltonian, with no map between them. The present paper provesthe equality, exhibiting the map; the sequel makes it definitional.The rest of the paper is the math.