I.
Prologue
Motivation and reading guide: the foundations problem, why a structure-first approach, and how Book I sets up Book II.
II.
The Core Axioms of τ
Signature and axioms of Tτ; canonical model; generators/orbits and the basic categorical universe.
III.
Label-Independence
Objects defined by behavior, not naming; rigidity and categoricity results; interpretive preview toward Book II.
IV.
Internal Set Theory
Sets inside τ: membership from divisibility, decomposition trees, Cayley-graph structure, and Boolean fragments.
V.
Metric Geometry
Full word-metric development; Cayley-graph theorems; canonical representations and categoricity as geometry.
VI.
Total Order
From metric to total order; effective comparison procedures; decidable core predicates and computability framing.
VII.
The Tarski Program
Euclidean geometry via τ: primitives, betweenness/congruence, verification of Tarski's axioms, and geometric consequences.
VIII.
Topos & Self-Enrichment
Yoneda and presheaves; topos viewpoint; internal logic; self-enrichment and countability considerations.
IX.
Internal Arithmetic
Arithmetic tower inside τ: ℕτ, ℤτ, ℚτ, ℝτ; computable reals and the number-system progression.
X.
The Canonical Calibration
Master invariant ιτ; emergence of constants and calibration dictionary; complex/quaternionic extensions and bridge forward.
XI.
Topological Foundations
Ultrametric and solenoidal topologies; τ as Stone space; 0D/1D/2D hierarchy and links to geometry/physics.
XII.
τ-Computation
τ-Tower Machine, observation-finiteness, τ-complexity classes, and the Interface Width Principle.
XIII.
Foundations
Comparisons to classical foundations; constructive/decidable scope; diagnostics, limitations, and research program framing.
XIV.
Bridge to Book II
Preview of τ “in the large”: τ³ fibration, compactification/lemniscate boundary, and τ-holomorphy themes.
