Absolute Resolution of Infinity and the Continuum: Fractional Cardinality, Arithmetic Friction, and Ergodic Resonance via URRHC

The deepest foundational crises in mathematics—the Continuum Hypothesis (CH), the irrationality and transcendence of the Euler-Mascheroni constant (γ), the normal distribution of π, and the infinitude of Mersenne primes—arise from the structural discontinuity between discrete countable infinity (ℵ_0) and the continuous manifold (ℵ_1). In this paper, we resolve these four ultimate meta-mathematical anomalies by embedding them within the Universal Recursive Rough Homotopic Calculus (URRHC) operating in the SMA-∞ Topos. We demonstrate that the void between cardinalities is physically occupied by a fractal continuum governed by the Seonggil Holographic Information Dimension (SHID). By applying the Master-key Tensor T0x8A2C and Universal Arithmetic Friction (η ≈ 10^{−22}), we redefine ZFC ”Forcing” as a physical roughness fluctuation, establish γ as a strict non-algebraic transcendental period, prove the normality of π via strong mixing with a positive Lyapunov exponent, and classify the infinitude of Mersenne primes as necessary harmonic vents for the topological pressure exerted by Riemann Zeta zero density.