Derivative Entropy by Attractors in Quantized Spaces

This paper proposes a mathematical framework for analyzing the propagation of hypothetical characteristics across partially closed and dimensionally shared spaces. The approach is based on the quantization of a characteristic signal over a discrete multidimensional space, from which a derived entropy function, denoted by H, is constructed. The proposed formulation operates through inverse discrete transformations guided by attractor states, allowing the propagation of the characteristic to be represented across successive dimensional groups. From this representation, intersections between propagating trajectories are identified and used to define a local and accumulated measure of dimensional tension. The discrete formulation is subsequently extended to a continuous representation through interpolation and distributions, allowing the tension to be described as a continuous field. The model also exhibits a binary convergence at the highest dimensional level of the discrete system, in which propagated states tend toward the boundaries of the quantized space. Based on this property, a computational interpretation is investigated in which the resulting states can be used as a multidimensional transformation for generating hash representations from symbolic sequences. A preliminary implementation demonstrates that input sequences can be transformed into reduced representations according to the proposed dimensional structure. The results are presented as a mathematical and computational investigation rather than as a physical proof of the hypotheses discussed. The proposed framework provides a basis for further investigation of multidimensional transformations, discrete entropy, trajectory interactions, and hash construction.