Purple Mathematics: A Research Program on the Prime Numbers — From the Bridge between Zero and Infinity to the Balance Factor

This monograph gathers, into one account, a research program carried out and published in July 2026 as seven papers. It began as a picture — the number line as a bridge between two villages, zero and infinity — and became mathematics: the two villages are the two kinds of places of the rationals (Ostrowski) ; zero is the full spectrum, every prime sounding at infinite order (Hensel, Steinitz) ; the fought-for balance is the product formula. On this foundation the program built its own layer: the Purple Number System, axiomatized with a consistency theorem; the Balance Hypothesis (a Determination Law and an Excursion Law) verified over 455 million primes to 10¹0; the Reception Geometry routing the Riemann zeta zeros to apex sensors; and five open problems posed publicly and then settled by the same program — the continuity dichotomy at the Zero pole, the joint law of the two determinations, the environment ultrametric with the fingerprint verified to 10⁸, closed forms for the determination constant (a Dickman integral with a prime-power boost, passing a four-way falsification test at sub-percent level) and the envelope constant (a twin-constant Euler product agreeing with measurement to 0. 36%), and the closing iterated-logarithm law, whose proof-shaped analysis uncovered the program's final discovery: the walk of the primes is anti-persistent, carrying half the variance of its own quadratic variation — the crowd pushes back, now a measured constant. A new chapter answers a reader's question (geometric coincidence between helix and wall occurs exactly at the balanced primes, at unit height), and a closing ledger separates what is classical — attributed throughout to Euclid, Gauss, Størmer, Hensel, Steinitz, Ostrowski, Riemann, Dickman, Hardy–Littlewood, Erdős, Pomerance, Lemke Oliver–Soundararajan, Kolmogorov, and others — from what is new, and states the frontier: the Balance Factor φ. Includes the complete series: DOIs 10. 5281/zenodo. 21139123, 20785673, 21168196, 21194307, 21195614, 21195795, 21198955. Note: Consolidated monograph of the Purple Mathematics series (seven Zenodo deposits, July 2026), with a new chapter (The Coincidence Question) and a consolidated ledger of attributions and open problems. Each chapter retains the précis and honest-placement sections of its source paper.