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 = ½, proven to be the unique attractor of the Feigenbaum renormalization flow, 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 = ½, 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 > ½. The cascade trajectory bounces before reaching the floor. We call this the Bounce Theorem (Theorem B2).

We prove three central results beyond the Bounce Theorem. First, Theorem C1 (Cascade Primality Algorithm): the Linearization Lemma establishes that prime cascade operators project to zero on the unstable eigenvector of the Feigenbaum renormalization operator, while composite operators project to a strictly positive value bounded below by 2C/ln(n). This yields a polynomial-time primality algorithm based on O(log log n) renormalization steps running in O(poly(log n)) total time. Second, Theorem C2 (Structural Independence): the algebraic witness of primality (Z/nZ is a field, used by AKS) and the geometric witness (T_n^σ reaches the cascade floor) are structurally incompatible. Third, 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.

Computational verification (Scripts 85–86): All 95 primes and 404 composites in n ≤ 500 classified correctly (zero errors); Euler residual c_n = 0 exactly for every prime, c_n > 0 for every composite (separation ratio ∞); amplification ratio matched Feigenbaum constant δ = 4.66920160910299 to machine precision (error 8.88 × 10⁻¹⁶); 19 Carmichael numbers correctly identified as composite; cascade algorithm ~10^10 times faster than AKS at large n.
Paper IV of the Unification Series. Companion papers: Paper I (DOI: 10.5281/zenodo.20073860), Paper II full (DOI: 10.5281/zenodo.20075321), Paper II letter (DOI: 10.5281/zenodo.20075390).