The First Geometric Definition of Primality: Primes as Cascade Ground States

For 2,300 years, prime numbers have been defined by isolation: each prime stands alone, characterized by what it lacks—no divisors other than one and itself. No geometric object has ever been identified that all primes simultaneously inhabit. No geometric reason has ever been given for why primes, and not other integers, are the atoms from which all integers are built. This paper establishes both.

The foundation is the Bounce Theorem [BT26], which defines the cascade residual cn(σ) = Rn(σ) − 1 for every integer n > 1 and depth σ > 0, where Rn(σ) is the Euler product ratio measuring the offset of n’s cascade trajectory from the Feigenbaum renormalization fixed point g∗.

The central result is:
cn(σ) = 0 for some σ > 0 if and only if n is prime—and in that case cn(σ) = 0 for every σ > 0.

Every prime, without exception, has cascade residual identically zero at every depth. Every prime inhabits the same geometric locus: g∗, the unique stable attractor of the Feigenbaum renormalization operator at σ = 12. This is the first geometric object in the history of mathematics that all prime numbers simultaneously occupy—not a shared property, but a shared address. It is the first proof that all primes are geometrically connected.

The cascade does not merely identify primes. It provides the functional geometry of integer composition: the exact mechanism by which prime factors combine to produce composites, and the precise reason composites cannot occupy the ground state. The AM-GM incommensurability of distinct prime factors forces cn(σ) ̸= 0 at every depth for every composite—and the exact shape of cn(σ) encodes the complete factorization geometry of n. This yields a complete geometric reorganization of the integers: every integer n > 1 belongs to a cascade level (determined by Ω(n), the total prime factor count) and a cascade family (determined by the factorization shape, the exponent multiset). Integers in the same family share the same cascade signature functional form—the same geometric relationship to the floor—regardless of which primes appear. The integers organize into a structure directly analogous to the chemical Periodic Table: primes are the noble gases, in the geometric ground state; every composite occupies a calculable position above the floor, organized into families by the geometry of its prime factor interactions.

Definition G1: An integer n > 1 is prime if and only if cn(σ) = 0 for all σ > 0. This is the first positive geometric definition of primality. The same Feigenbaum floor g∗ organizes Riemann zeros [RH26] and the Yang-Mills mass gap [YM26] through the same universal structure.

The Fundamental Theorem of Arithmetic gives the map: every integer factors uniquely into primes. The cascade gives the geometry behind the map: why primes are the right atoms, how those atoms combine, and what that combination looks like measured from the ground state. Euclid defined what primes are not. The cascade defines what they are and reveals the complete geometry of everything built from them.