Part IV: The ABCD Coordinate Chart

Part Overview

Parts I–III established the bare-metal foundations of Category τ: a static kernel of five generators and six axioms (Part I), a single generative act producing four orbit rays under an ontic seal (Part II), and a denotational bridge earning internal natural numbers, index arithmetic, and a well-ordered universe (Part III).

With arithmetic now in place — addition, multiplication, exponentiation, and tetration, all earned from the sole operator ρ via the fold chain (Remark [rem:fold-chain-principle]) — we can ask the first pivotal question: can every object of τ be canonically addressed by a finite tuple of typed coordinates?

This Part defines the ABCD coordinate chart: a canonical map Φ(x) = (A, B, C, D) obtained by a greedy peel-off of tower atoms from the prime factorization of x. The forced normal form X = ((A ↑↑ C)^B) · D is the only nesting compatible with the diagonal discipline. The four coordinates correspond to the four orbit channels: A (the π-channel: largest prime), B (the γ-channel: exponent), C (the η-channel: tetration height), D (the α-channel: remainder).

The chapter sequence: the relevant chapter earns internal primes and divisibility from the index arithmetic of Part III. the relevant chapter defines tower atoms and the greedy peel-off algorithm. the relevant chapter constructs the normal form encoding. the relevant chapter defines the ABCD chart itself. the relevant chapter derives the dimension dim_τ = 4 and previews the fibration structure.

Part V will prove that this chart is unique — the Hyperfactorization Theorem, the first of the two hinge theorems that anchor the entire Panta Rhei series.

Chapters

• Chapter 16: Internal Primes and Divisibility
• Chapter 17: Tower Atoms and the Greedy Peel
• Chapter 18: The Normal-Form Address Encoding
• Chapter 19: The ABCD Coordinate Chart
• Chapter 20: Dimension and the Fibration Preview