Quantum-Event Reachability Certificates, dissipative Mixing, and formally verified Coherence Benchmarks

Can a quantum event be certified without simulating the full quantum state? This study develops rigorous, event-specific probability bounds for constrained, finite-dimensional open quantum systems described by Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) dynamics. For initially unexcited targets, locally bounded Hamiltonian transitions that change at most one target occupation, and dissipation that preserves or lowers these occupations, it proves the sharp joint-excitation envelope pA(t) ≤ ∏j∈A sin²(min{π/2, ∫₀ᵗ uj(s) ds}). Independent resonant drives attain this bound within the stated model class. Complementary rational-polynomial certificates, interaction-dependent estimates, and bounds under pumping, coherent leakage, and guaranteed loss provide checkable alternatives for finite-time and all-time reachability questions. The analysis also examines how much information a particular event actually requires, distinguishing exact observable reductions and scalar trajectories from the exponentially larger ambient state description. Positive-operator barriers and operator-Schmidt-rank constraints clarify the resources and limitations of compact certificates. For independent generalized-amplitude-damping channels, the work establishes an ancilla-stable dissipative mixing bound: ‖⊗ⱼ Φⱼ,ₜ − ⊗ⱼ Rⱼ‖⋄ ≤ min{2, ∑ⱼ(rⱼ + rⱼ²)} ≤ 2N exp(−γ*t), where rⱼ = exp(−λⱼt/2) and γ* = minⱼ(λⱼ/2). The arbitrary-finite-ancilla hybrid inequality and resulting product-channel envelope are supported by kernel-checked Lean theorems; the leading long-time diamond-norm coefficient is established analytically, not formally verified. A separate formal development checks gauge-fixed Fibonacci fusion and braiding coherence identities, including pentagon and hexagon relations, and provides an ideal braid benchmark with conditional bounds for noisy encoded implementations. Taken together, the results offer a framework for auditable quantum-event certificates, model-specific noise benchmarks, and experimentally distinguishable predictions, while keeping their scope explicit: the claims depend on specified dynamics and controls and do not establish a new conservation law, microscopic topological protection, or universal computational advantage. Natural next steps include broader machine-checked dynamical certificates and validated interfaces from experimental models to formal proof objects.