March 3, 2026Open Access

Um Modelo de Espaço de Estados Bivariado para a Conjectura de Collatz: Rastreando o Decaimento Estrutural e o Potencial 2-ádico

Key Points

The proposed model reveals that the structural decay of trajectories is tied to future 2-adic valuations and binary inertia.
Key evidence shows that this multivariable approach anticipates collapse, underscoring a shift from univariate to bivariate analysis.
Analysis using a state variable and Lyapunov-like energy function offers a new lens through which to view the Collatz conjecture's complexities.
Highlights the potential of 2-adic saturation in aiding convergence, indicating significant implications for dynamical systems within number theory.

Abstract

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.

Um Modelo de Espaço de Estados Bivariado para a Conjectura de Collatz: Rastreando o Decaimento Estrutural e o Potencial 2-ádico

Key Points

Abstract

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