Prologue: Where Does Physics Live?
The Central Theorem is proved and the holomorphic machinery is in hand. But τ³ = τ¹ ×_f T² does not look like Cartesian three-dimensional space. Where, then, does physics live?
The naïve answer—take three solenoidal coordinates (π, γ, η) and call them spatial dimensions—fails: solenoidal coordinates are one-sided rays, not two-sided axes, and they carry no canonical linear structure. The correct answer is subtler. The fibre T² at each worldline point is a surface—the two-dimensional boundary of a solid torus. The Hartogs extension projects boundary data from this surface into the three-dimensional bulk interior, and the interior coordinates are genuinely linear. Perceived three-dimensional Cartesian space is this Hartogs-projected bulk.
The central task of Book III is to show that these local bulk projections—one per worldline—glue together into a single, globally coherent three-dimensional space. Each of the seven Clay Millennium Problems, together with the Langlands programme, provides a specific piece of this gluing guarantee. The self-enrichment layer E₁ is precisely the statement that local spatial structures cohere globally. Physics is E₁.