Chapter 1: Where Does Physics Live?

Books I and II constructed a complete mathematical universe from five generators and seven axioms, culminating in the Central Theorem O(τ³) ≅ A_spec(𝕃). But this universe does not look like the three-dimensional Cartesian space in which physics is formulated. The fibered product τ³ = τ¹ ×_f T² has four coordinates, all solenoidal—one-sided, angular, with no canonical ± symmetry. Quantum field theory and general relativity, by contrast, live in ℝ^{1,3}, whose spatial coordinates are two-sided, linear, and Cartesian. This chapter identifies the gap and resolves it: the three-dimensional Cartesian space of physics arises as the Hartogs bulk of the fiber T². The central task of Book III is then to show that these local bulk projections glue into a single globally coherent space—and that this gluing is precisely the content of the self-enrichment layer E₁. Physics is E₁.