From Discrete Symmetries to Continuous Couplings: Deriving from the Fano Automorphism Group

In our previous work, Ramos N.E. 2026b, we derived a phase variance parameter ≈ 2from continuous 2 symmetry and discrete quantum walk theory on the causal graph. However, continuous models inherently diverge at the Planck scale, requiring a discrete, instantaneous mechanism to define fundamental couplings. Here, we bridge this gap by identifying the finite automorphism group of the Fano plane, (3,2)(order 168), as the discrete subgroup of the continuous 2 group. By applying the Reserved State Hypothesis, where the 7th Fano point represents computational saturation, we show that active quantum processes are restricted to 168 × (6 7) = 144 usable symmetry channels. This yields a bare fine-structure constant = 1/144, emerging directly from the symmetry density of the voxel's state space. We demonstrate that the continuous ≈ 2arises as the macroscopic renormalization limit of this discrete coupling, unifying our computational finitism framework with exceptional Lie algebra structure. This provides a rigorous, falsifiable derivation of electromagnetic coupling from first principles of discrete geometry.