Book II: Categorical Holomorphy
Finite Readouts of Infinity
Parts
From Kernel to Interior
Book I closed with the Bridge Theorem (I.T34): Category τ is complete. Five generators, one operator, seven axioms (K0–K6) have built a full mathematical…
Interior Points and the τ³
Part I defines the point set of τ³ and reveals the fibration structure τ³ = τ¹ ×_f T². The approach is coordinate-first: Book I earned the ABCD chart…
Local Domains: Cylinders as Prefix Predicates
Part II constructs the local domains and proves the central inversion: {holomorphic implies continuous}. Four chapters build the topological…
Topology and Global Shape
Part III establishes the global topological shape of τ. Six chapters prove that τ is a Stone space, derive topology as an invariant of canonical…
Geometry: The Tarski Program
Part IV executes the Tarski program: deriving Euclidean geometry from ultrametric foundations. The two-readout principle (II.D18a, Chapter…
Earned Transcendentals: Lines, Circles, and the Constants π, e, j
Part V earns the transcendental constants π, e, j, and ιτ from purely countable discrete structure. Six chapters bridge from ABCD refinement rays to…
Local Hartogs and the Holomorphic Interior
Part VI is {pivotal}: split-complex holomorphy becomes load-bearing. Eight chapters build the Local Hartogs extension theory, prove the central…
Regularity and Mutual Determination
Part VII executes the {regularity program}, proving that {holomorphic = idempotent-supported}. Five chapters build a three-lemma chain…
Self-Enrichment, Yoneda, and Higher Categories
Part VIII proves that τ enriches over itself, yielding Yoneda as a theorem. Five chapters build self-enrichment and the enrichment ladder. Chapter…
The Central Theorem and Categoricity
Part IX is the climax: the {Central Theorem} and categoricity. Six chapters assemble boundary-to-interior correspondence and prove uniqueness.…
Closure: Synthesis and Bridge to Book III
Part X is the closure: synthesis and the bridge to Book III. Seven chapters audit the complete dependency chain, earn τ-manifold structure from the…
The Fork — Category τ versus Orthodox Mathematics
Part XI is the fork: a systematic comparison of Category τ against orthodox mathematics. Books I and II have built an alternative foundation from seven…
Reading edition
This is the Corpus projection of Book II. It exposes the monograph in Book → Part → Chapter order as part of the constructed research body.
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About This Volume
Book I closed with the Bridge Theorem (I.T34): Category τ is complete. Five generators, one operator, seven axioms (K0–K6) have built a full mathematical kernel—numbers, coordinates, sets, topology (implicit), categories, a topos, holomorphic transformers, and Global Hartogs Extension proving that boundary data determines interior structure.
Book II begins where Book I ended: at the boundary-interior threshold. The organizing paradigm is boundary-first—the inversion of classical complex analysis. Orthodox SCV builds from interior to boundary (holomorphic functions extended to boundary values, Cauchy integrals encoding interior data); τ inverts this by building from boundary to interior (boundary data determines interior structure via Global Hartogs).
Three chapters set the stage. the relevant chapter articulates the paradigm shift and the five exports from Book I. the relevant chapter explains why the elliptic complex numbers ℂ fail (Laplacian glue, no canonical propagation, unipolar structure) and why split-complex H_τ with j² = +1 is forced (wave-type glue, bipolar idempotents, prime polarity). the relevant chapter maps the eleven-Part roadmap and the inverted dependency chain: holomorphy ⇒ continuity ⇒ topology ⇒ geometry ⇒ transcendentals ⇒ holomorphic interior — exact inversion of orthodox order.
A note on terminology: throughout this book, τ³ means “τ viewed through its geometric coordinate chart.” The superscript 3 is a structural index naming the three-factor fibered product τ¹ ×_f T². It is not a dimension count (the interior is four-dimensional), not a power (τ³ ≠ τ × τ × τ), and not an independent construction (τ³ is τ, read geometrically). The base τ¹ and fiber T² are coordinate projections, not standalone entities. This distinction—between the ontic unity of τ and the geometric readout τ³—runs through every chapter that follows.
From here, the ten Parts of the book deploy the interior engine. Parts I–IV earn the topological and geometric scaffolding (interior points, cylinders, topology, Tarski geometry). Part V earns the transcendental constants π, e, j, ιτ from countable discrete structure. Part VI makes split-complex holomorphy load-bearing (Mutual Determination, the five-way equivalence). Part VII proves holomorphic = idempotent-supported (the three-lemma chain). Part VIII proves Yoneda as a theorem (τ enriches over itself). Part IX proves the Central Theorem O(τ³) ≅ A_spec(𝕃)—boundary determines interior, interior encodes boundary. Part X closes the circle and bridges to Book III.
Welcome to the holomorphic interior.
Linked publication and verification artifacts
- Registry: 68 chapters mapped to registry objects
- Dashboard: Formalization status and dependency graph
- Formalization: TauLib BookII — Lean 4 verification