Recursive-Expansive Dynamics Frameworks Formalization with Negation Isolation and Inverse Zero Operators

This enhanced framework presents a unified theory of recursive-expansive dynamics through prime-adelic foundations, introducing two novel operators: the Negation Isolation Operator (NIO) and Inverse Zero Operator (IZO). The theory proposes that spacetime emerges from prime arithmetic constraints, with physical constants arising as eigenvalues of fractal-geometric operators. The framework resolves fundamental problems in quantum gravity through recursive stabilization mechanisms and makes testable predictions across gravitational wave physics, black hole thermodynamics, and quantum tunneling phenomena. We posit that all physical dynamics are rooted in arithmetic over primes, unified through adelic structures. For example, the cosmological constant Lambda is derived via: Prime-Adelic Foundations IZO represents the true cosmological constant through: IZO(s)=∑n=1∞1ns⋅∏p prime(1−p−s)−1\text{IZO}(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \cdot \prod_{p \text{ prime}} \left(1 - p^{-s}\right)^{-1}IZO(s)=n=1∑∞ns1⋅p prime∏(1−p−s)−1 This iterative structure generates Lambda through the recursive application of zeta regularization, where each iteration refines the cosmological constant's value through prime-modulated feedback loops. Physical Manifestation IZO manifests as Lambda through the zeta function regularization of vacuum energy. The cosmological constant emerges not as a static parameter but as the dynamical output of iterative zeta operations acting on the quantum vacuum: ΛIZO=lim⁡n→∞∑k=1nζ(−1,k)⋅ϕ(k)\Lambda_{\text{IZO}} = \lim_{n \to \infty} \sum_{k=1}^{n} \zeta(-1, k) \cdot \phi(k)ΛIZO=n→∞limk=1∑nζ(−1,k)⋅ϕ(k) where ϕ(k)\phi(k)ϕ(k) is Euler's totient function, encoding the prime structure underlying spacetime's recursive geometry.