The Bounce Theorem: Primality as Cascade Floor-Touch in the Feigenbaum Universality Class

We establish a geometric characterization of prime numbers within the Feigenbaum universality class framework. The cascade floor σ = 1⁄2, proven to be the unique attractor of the Feigenbaum renormalization flow [Paper 43], exhibits a fundamental primality discrimination property: the cascade trajectory of an integer n reaches the floor if and only if n is prime. For prime p, the renormalization flow descends symmetrically from both sides of the critical line σ = 1⁄2, touching the floor in perfect synchrony — a direct consequence of the functional equation symmetry and the absence of internal compositeness degrees of freedom. For composite n, the factorization structure breaks this symmetry, generating a restoring force that produces a turning point σₙ > 1⁄2. The cascade trajectory bounces before reaching the floor. We call this the Bounce Theorem (Theorem B2).

We prove two central results beyond the Bounce Theorem. First, Theorem C2 (Structural Independence): the algebraic witness of primality (Z/nZ is a field) and the geometric witness (T_n^σ reaches the cascade floor) are structurally incompatible — the former requires the additive ring structure of Z/nZ, the latter operates in a purely multiplicative function space. No natural homomorphism translates between them. Second, Theorem M3 (the Meta-Theorem): both witnesses detect the same underlying atom property of n through the semigroup homomorphism φ: n ↦ T_n from (N, ×) to the renormalization semigroup. Their agreement is atom preservation under φ. The Euler product is the explicit bridge. The Linearization Lemma establishes that the cascade residual vanishes exactly for primes — c_p = 0 — and is bounded away from zero for composites — |c_n| ≥ 2C/ln(n) > 0 — characterizing primality geometrically through the projection onto the unstable eigenvector of the Feigenbaum renormalization operator.