Robust Interval Estimation and Error Accumulation Bounds in Shortest Path Algorithms via Johnson's Reweighting

We introduce a formalized mathematical bridge between deterministic graph theory and
robust optimization under stochastic uncertainty. By applying principles of interval arith
metic to Johnson’s reweighting algorithm for sparse graphs, we demonstrate that shortest
path optimality is strictly preserved under bounded uncertainty. Moving beyond trivial
width preservation, we establish mathematically verified bounds for path error accumula
tion, proving that deterministic robust intervals maintain a strictly linear additive variance.
Furthermore, we prove that arbitrary heuristic potentials derived from the lower asymptotic
bound guarantees topological validity across the entire uncertainty spectrum. The theorems
have been machine-verified natively in Lean 4 without omitted declarations.