The 59-Dimensional Framework as the Gauge Group: A Complete Zero-Parameter Derivation of the Standard Model from 59 = 11 + 6×8

[v2 — May 2026: Corrected the group-theory expression for D_CY in Tables 1–2 and Figure 1 (now dim SU(3) − rank SU(3) = 6). No numerical result changed.] This paper (Paper E of the 59-Dimensional Series) presents a single unifying principle explaining why the 59-dimensional framework reproduces the Standard Model: the number 59 = 11 + 6×8 is the unique embedding of the Standard Model gauge group G_SM = SU(3)×SU(2)×U(1) in M-theory for which D_total is simultaneously prime and odd, while observable spacetime equals the rank of the gauge group, D_space = rank(G_SM) = 4. This condition fixes D_M = 11 uniquely. From this single structural fact, every dimension of the framework is determined by group theory alone — no free choice remains. We derive 33 dimensionless Standard Model quantities using four elementary operations (ratio, binary exponential, square root, integer sum) on the six integers {1, 3, 4, 6, 8, 11} that are themselves the dimensions and ranks of G_SM. New results beyond Papers A–D:(i) the QCD beta-function coefficient satisfies β₀ = D_CY + D_0 = 7, linking asymptotic freedom to the hidden dimensions;(ii) observable spacetime is four-dimensional because rank(G_SM) = 4;(iii) three fermion generations arise because they equal half the Calabi-Yau dimension, D_CY/2 = 3;(iv) cos(θ_W) = exp(−D_0/D_SO8) achieves 0.12% accuracy — thirty times better than sin²θ_W;(v) with the geometric value α_s = 4/34, the framework reaches zero experimental free parameters, against ~25 in the Standard Model. All 33 results are verified against PDG 2022 — four to better than 0.1% — and none uses a fitted constant. Three quantities remain genuinely open (Ω_DM, the cosmological constant, and the PMNS CP phase), stated honestly. Two falsifiable predictions are highlighted: the neutrino mass sum Σm_ν ≈ 58.7 meV and the baryon resonance N*(1685) at 1680.6 MeV. Numerical verification was performed with the assistance of Claude (Anthropic). All scientific ideas and responsibility rest with the author.