We present a self-contained analytical framework for deriving the Standard Model lepton mass hierarchy, the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix, and the light neutrino mass scale from first principles within M-theory compactified on a seven-dimensional manifold X7 of G2 holonomy. Our central result is that the infrared (IR) attractor (n1,n2,n3) = (1,207,17), previously identified through GPU-accelerated Markov-Chain Monte Carlo renormalisation-group (RG) flow in the MCMC11D engine, is not a statistical artefact of the string landscape but rather the unique algebraically stable vacuum enforced by the holographic Bekenstein bound on the pregeometric computational substrate. We demonstrate this in four steps. First, we resolve the topological tension between G2 holonomy and chiral fermion generation through the Twisted Connected Sum (tcs) construction of Kovalev, showing that the Atiyah–Patodi–Singer (APS) η-invariant of the codimension-6 boundary link forces Ngen = 3 exactly. Second, we derive the Yukawa amplitudes from M2-membrane instantons wrapping associative 3-cycles Σ, establishing that the volume quantisation of these cycles in the tcs neck geometry produces the ratios mµ/me ≈ 207 and mτ/me ≈ 3477 without fine-tuning. Third, we construct a geometric ansatz for the CKM matrix from codimension 7 intersection angles between quark-localisation loci, reproducing |Vus| ≈ 0.225, |Vcb| ≈ 0.017, and |Vub| ≈ 0.0006 from the same topological integers. Fourth, we identify the gravitational mass gap MX ≈6.3×1013GeV as the gaugino condensation scale of the hidden sector, from which a Type-I seesaw mechanism yields mν ≈ 0.15eV—in remarkable agreement with current cosmological bounds and neutrino oscillation data. Anomaly cancellation is shown to follow from the quantisation of the G4 =dC3 flux, which acts as the intrinsic error-correcting protocol of the tcs vacuum.