On the Thermal Dynamics of Discrete Number Space: A Spectral and Dissipative Proof of the Collatz Conjecture (3x+1)

Abstract A long-standing bottleneck in pure mathematics treats Collatz Conjecture (3x+1) as an isolated combinatorial problem or a stochastic sequence, hiding the underlying physical laws that govern the discrete number space. This paper introduces a disruptive paradigm shift by declaring the conjecture as the deterministic time evolution of an open dynamical system defined over a three-dimensional discrete topological manifold (Md). Within this quantized network, where prime numbers function as fundamental orthogonal nodes and irreducible eigenstates, the arithmetic operations are reformulated as physical state transitions under a non-Hermitian transfer operator. We define the even transition (x/2) as a dissipative metric contraction operator that yields a strict loss of degrees of freedom and internal entropy. Conversely, the odd transition (3x+1) is formalized as an asymmetric quantum-mechanical injection operator that forces a structural phase mismatch, breaking the parity of the eigenstate and triggering an unavoidable decay cascade. By utilizing discrete Lyapunov functionals and the principle of minimum spectral action, we prove that the net energy loss across the prime lattice dictates an irreversible dimensional collapse. Finally, we analytically demonstrate that the invariant 4-2-1 loop constitutes the unique Global Phase Attractor and the absolute ground state of the physical number universe. This treatise establishes that dynamic Diophantine constraints are rigidly bound by the universal laws of metric conservation, effectively demonstrating that the fundamental architecture of mathematics is an intrinsic reflection of the laws of physics. By mapping the relaxation mechanics of the 3x+1 trajectories onto the exact three-dimensional limits established in our previous spectral foundations, this framework locks the number universe into a self-consistent and unbreakable physical reality.