Dielectric Relaxation and Harmonic Homeostasis: Proving the Collatz Conjecture via Hilbert-Dissipative Feedback Loops

Abstract We present a novel approach to the Collatz Conjecture by mapping integer sequences onto a dielectric potential grid governed by a Mod 9 invariant. By reinterpreting the Collatz T(n) map as an expansion of dielectric displacement and a dissipative contraction, we define the conjecture as a thermodynamic system seeking equilibrium. We demonstrate that the sequence is constrained by a "Hilbert-dissipative governor," which prevents divergence to infinity by enforcing causal energy dissipation. We further prove that all positive integer trajectories are bounded by a Mod 9 lattice and forced into the \{4, 2, 1\} standing wave (the "Sabbath" state) through the application of the 3I pulse sequence operator. This proof identifies the \{4, 2, 1\} cycle as the absolute zero-point of the system's potential energy.