Grassmannian Dynamics: Topological Stability, the Arrow of Time, and Testable Predictions

Building upon the Grassmannian framework for fundamental physics established in Papers I–IV of this series, we derive the dynamical equations governing transitions between stable Grassmannian manifolds. We prove three central results: (1) proton stability arises from a Stiefel-Whitney class obstruction between Gr(3,5) and Gr(2,5); (2) dark matter stability emerges from the division-algebra gap at k = 5,6, which induces a conserved Z₂ charge without ad hoc assumptions; and (3) the arrow of time corresponds to Grassmannian descent, with the Second Law of Thermodynamics following from the entropy difference between structured matter and radiation. We derive the effective potential V(k,n,T) = αR(k,n) − T S(k,n) + V_top(k,n) from first principles using Riemannian geometry, statistical mechanics, and topological constraints. The framework makes specific, falsifiable predictions including proton lifetime τ_p = ∞ (not 10³⁵ years), dark matter mass m_DM ∈ [90, 250] GeV, spin-independent cross section σ_SI ∼ 10⁻⁴⁵ to 10⁻⁴⁷ cm², and the absence of a fourth fermion generation. We outline experimental tests at current and next-generation facilities.