Formalization of a Non-linear, Non-Markovian Spectral Structure Based on Direct Mapping Actions without Complexification

This paper formalizes a nonlinear spectral structure in which irreversible crystallization occurs through stage-indexed direct mappings, based on a discrete process indexed by natural number stage n and internal variable x, satisfying non-unitary, non-overlapping, and non-inclusion conditions.The core concept of this study is the Structural Torque, defined as the coupling of a spectral position coordinate and a scalar weight vector. The Structural Torque provides a global invariant expressed as the total conservation of the composite mapping — this constitutes not a physical equilibrium in the conventional sense, but a mathematical equilibrium condition that ensures the consistency of the generated structure.In this study, the initial integral latent structure undergoes structural differentiation induced by the input value, and the differentiated components are rearranged through stage-indexed direct mappings. This rearranging proceeds stage by stage, forming crystallization and Structural Torque, and ultimately generating a spectral shape.The direct mapping defined on this space operates not as a reproduction of an already-given representation space — in contrast to the traditional Gelfand-type spectral correspondence — but as a structure-generating action. That is, the spectrum is not the result of an analytic correspondence, but a structural product crystallized by stage-indexed direct mappings.In particular, irreversibility is not interpreted as information loss; rather, it is formalized as the structural reordering induced by stage-indexed direct mappings, the hierarchical fixation of Structural Torque, and the absence of a left inverse for each direct mapping. Accordingly, as the stage progresses, the structure cannot be restored to its previous state, and the spectrum emerges as the result of cumulative structural rearrangement.Furthermore, this study proposes a fully deterministic single-mapping system in which structure is generated through 'monotonic succession of order' within the environment of stage-indexed direct mappings.The system transforms the structure stage by stage through a pre-structure generation mapping and direct mappings mediated by natural number stage n and internal variable x. Each stage begins from the same reference point, while each individual term is updated independently. After crystallization, the entire set of individual terms transitions from non-closed hierarchical coefficients and polynomials to scalar weight vectors, which then combine with spectral position coordinates to form Structural Torque.The Structural Torque undergoes a gradual transformation process, generating a spectral shape. This shape, starting from the same reference point, is transformed through hierarchical nonlinear selfamplification as the stage progresses and approaches the upper boundary. The Structural Torque at each stage ultimately recovers global sum conservation at the point when all terms are completed.This global sum conservation is not an invariant maintained throughout the entire generation process, but rather a conservation state established after the completion of crystallization. After crystallization, the spectral structure possesses stability in which the internal hierarchical structure is maintained despite stage increases. The final spectrum constitutes a structural equilibrium state that simultaneously satisfies this structural stability and the conservation of the global sum. Therefore, this paper presents the generation mechanism of a nonlinear spectrum in which Structural Torque arises from irreversible crystallization, based on the combination of pre-structure generation mappings and direct mappings, and demonstrates that structural stabilization and global conservation are simultaneously established in the process.