Chapter 23: Orthodox vs. τ Denotation Bridge

Parts III and IV have earned Euclidean geometry from purely discrete, profinite foundations: the Stone space structure (II.D14), four-dimensionality (II.T11), betweenness (II.T15), congruence (II.T16), Pasch (II.T17), the parallel postulate (II.T18), and wave-type causal structure with the Euclidean static limit. All of this was internal to τ—no real numbers, no classical continuum, no ambient Euclidean space. But classical mathematics works in ℝ^4. How do the two connect? This chapter constructs the denotation map den : τ³ → ℝ^4 by sending each τ-admissible point to the limit of its stage-k approximation sequence. The map is continuous, injective on finite points, and preserves all Tarski axioms. The key philosophical point: ℝ^4 appears as the limit of τ-approximations, not as an ambient space that contains τ³. τ does not sit inside ℝ^4; rather, ℝ^4 is the classical shadow of τ³.