The Reach Function: An Arithmetic Framework on the Integers (Purple Mathematics)

Purple Mathematics is the study of a single function on the integers — the *reach*, ρ(n) = the largest prime factor of n² − 1 — and the structures revealed by its normalisations, its companion, and its differences. Each integer is examined through the local operation that produces the reach (the *dig*): factor the two neighbours n − 1 and n + 1 and read their largest prime factors; the reach is the larger of the two. Read as one function, the reach has strata, and the strata are the framework's findings: the primes are its ceiling (ρ(n) = n + 1 exactly when n + 1 is prime); the twin primes are the ceiling of its companion (the smaller factor); a heavy-tailed prime-driven walk lives on one harmonic band; and a finite anchored domain sits on its floor (49 integers, by Størmer's theorem).

This monograph consolidates the framework's standalone chapters into one account organised around the reach, with an identity card and two calibrations, a harmonic spectrum, and the associated Dig-ID walk.

It is expository and explicit about novelty: every component is classical or elementary and flagged as such, and no new theorem is claimed. The single genuine research frontier — an excursion law for the Dig-ID walk — is stated as a precise conditional theorem with one open decorrelation estimate and a maximal inequality.

The contribution is unification: several familiar phenomena seen as strata of one carefully chosen function.

Consolidated, reach-centred edition of the Purple Mathematics framework; supersedes earlier framework drafts. Companion technical note (the Dig-ID walk and its open problem) published separately — see related works.