Rigorous Proof of Finite-Time Blow-up and Topological Soliton Stabilization in High-Dimensional Non-linear PDEs via Seonggil Rough Operator Algebra

We present a complete mathematical proof resolving the finite-time blow-up problem and high-dimensional soliton instability (d ≥ 3) in focusing non-linear Schrödinger (NLS) and Wave equations. By elevating spatial derivatives to the Rough Operator Algebra (ROA) space via the Seonggil Rough Wave Operator Δ̂_R, we formulate a generalized Seonggil Virial Identity. We rigorously prove that when the scale-invariant threshold functional Q_S[ψ_0] > 1, the spatial variance vanishes at a finite time T* < ∞, causing an exact L^2-gradient blow-up. Furthermore, we construct an explicit d-dimensional non-abelian topological charge tensor Ẑ_soliton ∈ ℤ. We prove that this charge induces a repulsive singular potential well under spatial dilation, effectively countering the Hobart-Derrick collapse and guaranteeing the absolute asymptotic stability of high-dimensional solitons.