I.
Prologue
Sets the stage for categorical holomorphy: what changes from Book I, key questions, roadmap, and prerequisites.
II.
The τ³ Fibration
The geometric heart of the book: τ³ as a fibration with base and fiber; coordinates, metrics, and structural role.
III.
The Lemniscate
Why compactification is needed; the canonical boundary shape; the “lemniscate” L = S¹ ∨ S¹ and its role at infinity.
IV.
The Spectral Algebra
Characters on the boundary; CR-compatible modes; ring/convolution structure; two-lobe decomposition and finiteness.
V.
τ-CR Equations & Holomorphy
τ-analogue of Cauchy–Riemann; discrete Fueter operator; τ-holomorphic functions and the central holomorphy criterion.
VI.
Hartogs & Liouville
Extension and boundedness principles in τ³: Hartogs-type extension, Liouville-type statements, and maximum principles.
VII.
Advanced Holomorphy
Residues, poles, Laurent expansions, meromorphic functions, analytic continuation, and a sheaf-theoretic viewpoint.
VIII.
Number Theory
Spectral zeta connection; modular forms and L-functions in τ-language; arithmetic applications and open problems.
IX.
Emergent Physics
Physical interpretation: τ³ as a spacetime-like arena; holomorphic functions as fields; boundary observables and symmetries.
X.
Categoricity
Uniqueness results for the holomorphic structure; universal properties; why the τ³ profile is structurally forced within the framework.
XI.
τ-Manifold Geometry
Generalization to τ-manifolds: τ-calculus, sheaves/cohomology, connections/curvature, Hodge theory, and gauge structures.
XII.
Synthesis and Bridge
Summary of major results, open questions, and the bridge forward from holomorphy to the spectral program of Book III.
