Chapter 26: Three Perspectives on π

Chapters and earned lines and circles from the primorial tower: ℝ as the Archimedean shadow of the radial α-ray (II.T20), and S¹ as the Archimedean shadow of the solenoidal circle (II.T21). Both objects are now available within Category τ—as limit objects, not as imported continua. This chapter earns the first transcendental constant: π = 3.14159…. Three independent constructions produce π, and their agreement is a theorem (the relevant theorem, II.T22): enumerate -[(T)] Topological π: the half-period of the lemniscate 𝕃 (I.T05, Book I), inherited from Prime Polarity’s bipolar structure. -[(G)] Geometric π: the circumference-to-diameter ratio of the earned circle S¹, computed via the Archimedes polygon sequence (the relevant definition, II.D29). -[(S)] Spectral π: the spectral radius of the solenoidal B-channel (I.D19, Book I), computed from character values on the boundary ring. enumerate All three yield the same number 3.14159…, confirming that τ’s π is the classical π.