Emergent Relativistic Spacetime and Lorentz Invariance via the Universal Kaleidoscopic Filter on Integer Partitions

This paper establishes a rigorous, comprehensive theoretical framework wherein continuous relativistic spacetime, causal structure, and Lorentz invariance are derived not as fundamental, irreducible axioms, but as emergent thermodynamic limits of the discrete combinatorics of purified integer partitions. By executing the Kaleidoscopic Filter Theorem, we prove that the systematic application of alternating Weyl reflection operators to unrestricted partition lattices entirely annihilates lower-dimensional topological boundary noise and metric singularities. This mathematical sieve yields an interior log-concave core characterized by strict structural rigidity. The macroscopic speed of light c is rigorously derived as a maximum algorithmic processing rate governing state transitions between contiguous Weyl chambers. Furthermore, we provide a complete treatment of high-energy Lorentz symmetry breaking at the Planck scale, deriving a modified dispersion relation and introducing a Hopf-algebraic deformation of the Poincar\'e algebra to show how causality is preserved under non-linear representations. We extend the formalism to prove that Gauge symmetries, the Yang-Mills mass gap, the Spin-Statistics theorem, the Geometric Origin of Inertia, and the Einstein Field Equations are strictly geometric consequences of this discrete architecture. We conclude with precise, testable cosmological predictions regarding photon dispersion and the Dark Sector, rendering this unified model strictly falsifiable.