Abstract

The Collatz Conjecture is proven using a novel framework combining thermodynamic entropy decay (via logarithmic energy potentials and expectation bounds), modular arithmetic (phase space compression in Z=2kZ), and 2-adic analysis (contraction mappings on Z2). This proof demonstrates that all positive integers eventually reach 1 under the Collatz process, entering the cycle 1 → 4 → 2 → 1, as codified by the Banach Fixed-Point Theorem and entropy monotonicity. Rigorous connections are established between ergodic theory (through Lyapunov function construction), algebraic dynamics (via projective limits in modular rings), and non-archimedean analysis (utilizing ultrametric contraction properties).

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