OG-II-01 — X — Exact Geometry and Singular Phases of an E₈ Cut-and-Project Model

We analyze a four-dimensional E₈ cut-and-project model whose acceptance window is the internal projection of the parent Voronoi cell. Exact arithmetic and completeness arguments determine a window with 720 vertices, 3,120 edges and 1,200 facets, uniform carrier bounds, and all 25,800 radius-two event hyperplanes. The associated boundary-displacement graph reduces to the 600-cell graph. Its rational first homology is five regular binary-icosahedral representations plus one trivial representation; an invariant-quotient argument excludes the analogous integral decomposition. We prove stability of the central 120-cell, uniqueness of an unordered shortest pair at any closed global phase, and sharp phase–midpoint separation bounds. Pair-containing exceptional realizations lie outside the nonsingular hull. At a selected pair, a complete thirty-wall atlas has 5,570 chambers and 546 Boolean patterns. Its internal nerve is homotopy equivalent to S³, whereas the physical sites form an icosahedral prism. The unrefined vertex-identity map is obstructed; finite refinement permits vertex-preserving maps of every integer degree. For a specified finite-collar cut energy, an exact continuous-phase path has zero positive barrier relative to its singular starting phase. These results provide quantitative geometry, arithmetic and topology with reproducible finite certificates. They distinguish static phase changes from a physical evolution law and identify the additional map and coupling data needed for a source interpretation. Finite-index colour classes have equal positive-density frequencies, while their fixed-offset correlations exclude a specified independent twenty-channel reading.