Abstract This paper proposes an algebra-topology unified framework based on Chronogroup to address non-perturbative challenges in gauge theory and quantum gravity. By constructing the time-varying group law (∗t) and spectral temporal cohomology (Hk t(G,E)), we prove: 1. Isomorphism between first-level cohomology H1 t of SU(N) gauge group and Chern number (Theorem 3.1) 2. AdS/CFTdualityasalgebraic homomorphism Φ between bulk-boundary chronogroups (Theorem 4.2) 3. Equivalence between mass gap existence and triviality of H1 t at low energy scales (Theorem 5.1) 4. Quark confinement as emergent topological obstruction at energy scales ΛQCD < E <∆(Corollary 5.3) This framework provides intrinsic mathematical formulations for gauge theory, mass gap, and holographic duality, eliminating phenomenological dependencies