The Kakeya Conjecture and the Primordial Potential in G-MaTT: Emergence of 3D Space from Directional Completeness

Abstract We explore the relationship between the three-dimensional Kakeya conjecture and Generalized Mass as Twisted Time (G-MaTT), showing that the conjecture's 3D dimension bound provides a mathematical foundation for the primordial potential \(\mathcal{M}_\mu\). In G-MaTT, \(\mathcal{M}_\mu\) is pre-geometric and dimensionless, with phase evolution generating time and torsional gradients generating mass/space. The Kakeya conjecture — proven in 2025 — states that any set in 3D Euclidean space containing a unit line segment in every direction must have Minkowski dimension at least 3. This maps to the minimal directional coverage required for \(\mathcal{M}_\mu\) to span all phase directions without lower-dimensional gaps, forcing emergent space to be 3D and aligning with the three stable braid generations in \(\mathcal{B}_3\). We derive that \(\mathcal{M}_\mu\) can be formally defined as a Kakeya-like set in phase space, bounding its "volume" and refining the coherence margin to α⁻¹ = 137.035999206, where the 0.035999 correction arises from the volume deficit of the minimal Kakeya set in emergent space. This connection strengthens G-MaTT's pre-geometric ontology and predicts testable signatures in neutron interferometry and cosmic void statistics.