TMDD:Oscillators, invariants, boundary. Part I: Elementary Oscillators

Within the framework of the Tensor Model of Discrete Dynamics (TMDD), a systematicanalysis of the dynamics of an isolated θ-cell is carried out, treating it as a six-dimensionalnonlinear oscillator. Canonical equations of motion are derived in spectral variables, reveal-ing three coupled yet physically interpretable sectors: coherent (Q, C), geometric (K, G),and informational (Σ, X). Three qualitatively distinct regimes of internal dynamics areinvestigated under the condition of constant information volume det(Iˆ) = V0 = const —termed ”breathing”: (1) near the minimum boundary Vmin (degenerate vacuum with sad-dle instability), (2) far from boundaries (stable hyperbolic oscillations, balancing structureand fluxes), and (3) near the maximum boundary Vmax (information saturation regime withcritical slowing down and super-stability). It is shown that the boundaries Vmin and Vmaxare not imposed externally but emerge dynamically from the structure of the hybrid actionfunctional. The concept of effective inertia Meff(V ) is introduced as a measure of dynamiccomplexity and self-regulation. Three fundamental invariants of motion are discovered andinterpreted, determining the structural stability of the θ-cell. The results demonstrate thateven an isolated elementary unit of TMDD possesses rich internal dynamics, fundamentallydifferent from classical oscillators, and contains in embryonic form the necessary prerequisitesfor the emergent emergence of space, time, and collective behavior.