TMDD: Oscillators, Invariants, Boundary. Part IV: Double Oscillators

This paper investigates an elementary oscillator system within the Tensor Model of Dis-crete Dynamics (TMDD): a pair of interacting θ-cells. The action functional and canonicalequations of motion are derived for this system in both parametric and spectral forms,describing the dynamics as a gradient flow with respect to the internal consistency cycleparameter τ. A comprehensive stability analysis is conducted, including the study of phaseportraits near the boundaries of the information volume Vmin and Vmax, where the dynamicsqualitatively change: from informational fragility at the lower boundary to stabilization andpreparation for structural reorganization (θ-splitting) at the upper boundary. Generalizationto a linear chain of N cells demonstrates the emergence of a critical number Ncrit, defin-ing the minimum size of a stable collective capable of generating emergent spacetime. It issuggested that stability and the nature of the dynamics (subcritical, critical, supercriticalregimes) may depend on the balance between the internal informational tension of the cellsand the strength of their non-local interaction.