Two Paths to the Fine-Structure Constant — A Comparative Synthesis of Chain Theory and G-MaTT Time-Braids

Abstract This note presents a comparative analysis of two independent frameworks that address the origin of the fine-structure constant, α ≈ 1/137.035999: Chain Theory (Wissner-Gross, 2026) and the G-MaTT phase-resonance relational framework (ref 2 to ref 4). Chain Theory approaches the problem via discrete quantum-information networks, deriving α from a combinatorial count over entangled qudits, verified in Lean. G-MaTT derives α via continuous topological geometry, originating from the non-associative instability of the primordial S⁶ phase manifold, which undergoes Topological Natural Selection (TNS) to crystallize into the exceptional Lie group cascade (E₈ → F₄ → G₂), yielding the precursor constant C_G₂ = 35/416. We demonstrate that while the two frameworks use fundamentally different mathematical toolkits (discrete bits vs. continuous geometry), they converge on a shared ontological rejection of smooth spacetime and point-like particles. Crucially, G-MaTT elevates α to a structural threshold (W = √α) that directly explains the 68:27:5 cosmic energy distribution via Same Time-Braids, boundary collisions, and smooth bulk overflow.