Cyclotomic Fields as Recursive Time Structures

This work introduces Hypatian Mathematics, a novel framework integrating number theory, fractal geometry, and adelic mathematics to model spacetime as a recursive, fractal structure. The approach employs cyclotomic fields and Euler’s totient function to encode spacetime’s modular symmetry and recursive time loops. The Hausdorff dimension is corrected by the totient function, and adelic mathematics is used to link real and p-adic components of spacetime. A central feature is the Hypatian zeta function, defined as ζφ(s)=ζ(s)/ζ(s+1)ζφ(s)=ζ(s)/ζ(s+1), which supports retrocausal information flow and introduces a shifted critical line for zeta zeros. The framework also introduces the Inverse Zero Operator to regularize divergences in gravitational and quantum spacetime models, leading to predictions such as gravitational wave echoes with frequencies linked to the golden ratio, specific anomalies in the cosmic microwave background, and modifications to geodesic paths and memory effects in spacetime. The Bekenstein-Hawking entropy formula is revised, predicting a 4.5% entropy suppression. Hypatian Mathematics lays the groundwork for nine new mathematical fields, including prime-encoded quantum cosmology and adelic holography, aiming to unify aspects of quantum gravity, cosmology, and number theory. The framework provides empirical predictions testable with current and future gravitational wave and cosmological observations.