Emergent Spacetime, Dark Sectors, and the Boundaries of Algebraic Derivation in Computational Finitism

This is the third paper in a three-part series on the electron in Computational Finitism. The first two papers [Ramos N.E. (2026) The Electron in Computational Finitism and Derivation of the Electron] derived five properties of the electron from a specific 9-dimensional Fano-TRB algebra over ℤ/9ℤ: the fine-structure constant α = 1/137, the 12-vertex icosahedral shell, the 13-fold twist, spin-1/2, and the Zitterbewegung frequency. That paper left four items unresolved. This paper addresses them. We report four results. First, the icosahedral group A₅ does not embed into the Fano automorphism group PSL(2,7), by Lagrange's theorem, the two structures coexist as independent branches of the substrate, each handling different aspects of the electron. Second, the "continuum limit" is not a limit at all: the substrate remains strictly discrete at the Planck scale at every resolution. The continuum is an emergent description that works because the Fano topology is uniquely optimized to hide its own discreteness, as established in the coarse-graining papers. Third, the 1-loop anomalous magnetic moment is derivable from the algebra as α × 7/44, matching observation to 0.11% and QED 1-loop to 0.04%; the higher-order coefficients of QED involve transcendental constants that a finite algebra over ℤ/9ℤ cannot reproduce, so the Finitism series is its own object, agreeing with QED at current precision and diverging at higher order by terms involving ζ(3), π² ln 2, and so on. Fourth, the second chiral solution G, with t² = −1 instead of t² = 0, contains stable elements that annihilate the Fano sector, candidates for dark matter. But their mass scale requires a discrete loop expansion in a non-associative algebra, which is defined here as the future program. The paper closes with an honest ledger of what Finitism derives and what it does not. Four rigorous results from the first paper, four results here (two rigorous, one partial, one framework), and one future program. The substrate is discrete. The continuum is emergent. The computation is finite.