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.

This paper constitutes the mathematical foundation paper (Paper A) of the Transrelational Universal Grammar framework. The prior art for the conceptual foundations of this work traces to Potgieter (2001, 2012, 2018) as documented in Table 1 of the paper. Subsequent papers will develop the full multi-node generalisation and cross-domain applications.Note on terminology: Despite the use of the term “Universal Grammar,” this work is not related to linguistic theory. It defines a physical relational dynamical framework formulated via a Lagrangian. The term “grammar” is used in a structural sense to describe invariant relational rules governing system dynamics. Correspondence: gericke@artifexknowledge.co.za.

Version 2 Changes: Fixed quasi-conservation variation across the document Removed redundant reference to the leapfrog implementation in Section 10 Replaced the Schrödinger declaration with a research question to avoid overclaiming Added a paragraph in Section 1.3, positioning this paper as the first in a planned series and explaining why the philosophical sections are foundational to the series rather than incidental to this paper.