Purple Mathematics: The Complete Theory (Fourth Edition, Version 4.1) — From the Bridge between Zero and Infinity to the Scale Law of the Balance

     Purple Mathematics pictures the non-negative integers as a bridge between a Zero pole and an Infinity pole, with the unit as the fixed Balance Boundary of x ↦ 1/x. On that picture it builds an exact apparatus. Each integer's reach ρ(n) = P(n²−1) — the largest prime factor of n²−1 — splits ℕ into a finite Zero domain (ρ ≤ 11, closed by Størmer's theorem) and a cofinite Infinity domain, and the signed contrast κ(p) = P(p+1) − P(p−1) records which pole a prime leans toward. Over each prime stands a determination triangle whose apex height is half the span between its neighbours; the triangles form a wall, their apexes a gauge, and the ordinates of the nontrivial zeros of ζ send signals up to its slopes.

The fifteen chapters gathered here — thirteen separately published, two written for the consolidated editions — establish the framework's foundations, prove its arithmetic laws, and then subject the picture to the test it most needed. The Determination Law answers the program's one deliberately open question by deriving a prime's direction. The excursion S(x) = Σ_{p≤x} κ(p) is shown to be anti-persistent, obeying a corrected iterated-logarithm law, and its Balance Factor turns out not to be a constant at all but a scale law, 1 − φ = C·ln b / ln x with C = 0.493 ± 0.023 — leaving the book's sharpest open question: is C exactly ½? Four constants are given closed forms and verified numerically; the environment ultrametric answers the question of distance between primes; and the line is closed at infinity.

Two chapters then turn the apparatus on itself. The Wavy Line amends the book's own organising idea, the Invariance Ledger: the tiers are entangled, because an arithmetic invariant proves to be recoverable exactly from the metric covariants, so deformation cannot destroy determination but can destroy access to it. Parallel Bridges, new to this edition, isolates what the construction owes to standard mathematics — the apex height is a one-dimensional Voronoi cell length, the triangle a scaled piecewise-linear hat function — and, because those ingredients are defined for any increasing sequence, erects the same wall over the zero ordinates alongside the primes. It runs the two in parallel. It proves what follows from the geometry alone, including an exact long-wavelength deformation limit (1+r)/4π in the adjacent-gap ratio, and it states plainly what such a comparison cannot do: a wall built over ordinates does not retain the real parts, so this book establishes neither the Riemann Hypothesis nor any criterion equivalent to it, and claims neither.

All computations are reproducible, carried to 10¹⁰ on engines included with this record. Everything classical is attributed by name in a single bibliography. Every claim of novelty was checked against the literature before it was made, and the corrections the programme has had to make to its own published statements — five of them, including one unstated hypothesis and one unqualified extrapolation — are set out in the Ledger rather than quietly repaired.

Version 4.1:
Adds to Chapter 9 a note on detecting primes from finitely many zeros, which traces the apparent role of the apex height to classical spectral leakage, and a procedure that answers a reader's challenge: given any integer, say exactly what the framework says about it, and which of those statements the zeta signals can reach alone. The procedure is implemented as the Purple Number Reader, a program archived with this record that any reader can download and run.