Transrelational Universal Grammar (TUG): Emergent Dynamics from a Relational Lagrangian

We present a relational dynamical framework in which motion emerges from internal coupling between radial and angular degrees of freedom on a helical manifold, without externally applied torque. A general Lagrangian is derived from six irreducible axioms governing a nested recursive matrix of relational nodes. Through principled reduction to the single-node case, a canonical structure is obtained in which radial breathing oscillations modulate the moment of inertia and generate a net positive angular velocity. The associated Noether charge Q_helix is shown to be approximately conserved with drift bounded by the symmetry-breaking terms. Numerical integration confirms sustained helical traversal, quasi-periodic breathing dynamics, and quasi-conservation of Q_helix with variation below 6%. Three falsifiable predictions follow from the reduced model: frequency independence of the drift rate, quadratic amplitude scaling, and linear coupling threshold. All three are confirmed analytically and numerically. The reduced case is a principled derivation from the general framework; the full multi-node generalisation is the subject of subsequent work. Simulation code is included.