A Mathematical Unification of the Helix–Light–Vortex (HLV) Framework: Discrete Geometry, Spiral Time, Unified Lagrangians, and TOE-Level Structure

We present a unified mathematical framework underlying the Helix--Light--Vortex (HLV) approach, formulated as a geometric and information-theoretic extension of relativistic quantum field theory. The construction is based on four core elements: (i) a triadic spiral-time operator $\psi(t)=t+i\phi(t)+j\chi(t)$ encoding coherent, memory, and feedback components of temporal evolution; (ii) a discrete golden-ratio quasicrystal lattice (G-lattice) serving as the spacetime substrate; (iii) a symbolic operator grammar $G_{\rm HLV}$ constraining admissible field, lattice, and information processes; and (iv) a master effective Lagrangian unifying continuum and discrete dynamics. Within this framework, we construct the HLV Laplacian and propagators, derive modified dispersion relations, and demonstrate how standard relativistic field equations arise in controlled recovery limits. Topological phase effects and entanglement invariants emerge naturally from the triadic temporal structure. We show that in the limits $\varepsilon\!\to\!0$, $\dot{\phi}\!\to\!0$, $a\!\to\!0$, and $U_3\!\to\!0$, the framework consistently reduces to conventional quantum field theory and General Relativity, while HLV-specific corrections remain as well-defined, testable deviations. These deviations provide concrete targets for precision experiments, including $g{-}2$ measurements, King-plot nonlinearity, quark--gluon plasma observables, and quantum information tests.