The Tri-Quarter Framework: Radial Dual Triangular Lattice Graphs with Exact Bijective Dualities and Equivariant Encodings via the Inversive Hexagonal Dihedral Symmetry Group 𝕋24

The Tri-Quarter framework unleashes a radial dual triangular lattice graph with unified complex-Cartesian-polar coordinates, structured orientation phase pair assignments for directional labeling, and topological zones to build exact bijective mappings without approximations. By establishing combinatorial duality for radial separation, Escher reflective duality for zone swapping, and bijective self-duality for reversible transformations, the discretized framework leverages the lattice graph's order-6 rotational symmetry to natively support angular sectors, modular decompositions, equivariant encodings, and trihexagonal six-coloring for conflict-free parallel algorithms. At this discretized framework's core is the Tri-Quarter Inversive Hexagonal Dihedral Symmetry Group 𝕋24 — the order-24 semidirect product D6 ⋊ ℤ2 — which exploits rotational, reflective, and inversive symmetries to unlock these bijective transformations with exact precision. We provide formal proofs of these dualities, along with numerous step-by-step examples, and demonstrate practical efficiency through benchmarked simulations to achieve ~2x speedups with inversion-based path mirroring via bijections and up to ~6x reductions in symmetry-reduced clustering via rotational orbits. This work advances scalable computations on symmetric structures, with applications in computational geometry, graph traversals, tiling, robotics path planning, multi-agent coordination, lattice-based cryptography, image processing, and signal processing. This work aims to solidify a mathematical and computational foundation for both classical and non-classical computing paradigms — targeting future integrations in complex emergent systems that harness intricate "superposition-like" symmetries to advance symmetry-aware algorithms and data structures across diverse computing architectures.