A Bivariate State-Space Model for the Collatz Conjecture: Tracking Structural Decay and 2-adic Potential

The Collatz conjecture (3n+1) needs no introduction; it is complex and has resisted formal proofs dueto the chaotic and unpredictable nature of scalar trajectories. This work proposes a paradigm shift by reframingthe problem as a multivariable dynamical system. A state variable ω (forward potential) is introduced, defined bythe weighted sum of future 2-adic valuations, which acts as a gauge for structural decay and readiness forcollapse. Unlike univariate analyses, this model demonstrates that the 'mass' of a number is not defined solely byits nominal value n, but by its binary inertia. Through the construction of a Lyapunov-like energy function,functional results will be presented. A methodology for numerical testing and validation will be exemplified,demonstrating that ω anticipates the collapse of complex trajectories, suggesting that convergence to the attractoris a consequence of 2-adic potential saturation and that the +1 noise is dissipated by parity information.