K=1 Chronogeometrodynamics

Title:
K=1 Chronogeometrodynamics: A Structural Uniqueness Theorem for Lorentzian Information-Feedback Loops

Abstract

Since the development of Cybernetics by Norbert Wiener and the theory of Dissipative Structures by Ilya Prigogine, feedback mechanisms and self-organization have been extensively studied at the phenomenological level. However, the geometric constraints that determine the necessity and uniqueness of specific feedback structures in dissipative systems remain unclear.

We introduce a geometric cybernetic framework termed K=1 Chronogeometrodynamics. The starting point is an internal “Information-Time” metric G, defined as the Hessian of structural variations. Empirical analysis shows that this metric possesses Lorentzian signature Sig(G) = (1, 1). We prove a Structural Uniqueness Theorem stating that for any system governed by such an indefinite metric, there exists a unique (up to scalar factor α) symplectic generator

J_G = α G^{-1} J,

where J is the canonical antisymmetric matrix. This generator is required to preserve algebraic consistency under the Lorentzian geometry.

When coupled with environmental damping D, the closed-loop dynamics produces a statistical attractor at K = 1, interpreted as an optimal information-processing equilibrium. Numerical stress tests reveal a critical damping threshold d_c approximately equal to the spectral radius of J_G. As the system approaches this threshold, critical slowing down emerges. In addition, under high-frequency sinusoidal driving, the deviation from K = 1 decreases approximately as O(1/ω), demonstrating dynamic locking and geometric noise suppression.

These results suggest that in a Lorentzian information space, stable feedback closure is constrained by geometric structure rather than arbitrary design. The framework provides a unified geometric interpretation of autonomy in dissipative systems and offers a structural perspective on robustness in biological and artificial adaptive processes.

Keywords:
Geometric Cybernetics; Lorentzian Information Geometry; Structural Uniqueness; Dissipative Structures; K=1 Attractor