The Geometric Unified Field Framework: Mathematical Proof‑of‑Concept and Predictive Structure

This work presents a mathematical proof‑of‑concept for the Geometric Unified Field Framework (GUFF)—an extension of the Geometric Standard Model (GSM).Starting from a single non‑zero hexagonal seed evolved under the quadratic map (z_{n+1}=z_n^2+c), the framework reproduces the fundamental features of particle physics, gravity, and cosmology through pure geometry and stochastic self‑organization. Linear‑stability analysis, emergent Lorentz invariance, and renormalisation‑group behaviour prove that the discrete dynamics yield a stable, relativistic, and asymptotically safe continuum limit. Derived quantities—including the Higgs vacuum expectation value ((v=246 \mathrm{GeV})), gauge‑coupling hierarchy, and vacuum‑energy density ((\rho_\Lambda\simeq10^{-120}M_{Pl}^4))—match observed magnitudes without free parameters. A new section details geometric formation of Planck‑scale wormholes, hidden‑survivor clusters as dark matter ((\Omega_{\rm DM}/\Omega_b\simeq5.6)), and icosahedral curvature bias as dark energy. Together these results form a coherent “single‑seed” unification of matter, spacetime, and cosmology, accompanied by explicit predictions for gravitational‑wave spectra, fermion mass scaling, and coupling unification.