Residual Frontier Refinement: Exact Shortest-Path Propagation with Adaptive Partial Ordering

Residual Frontier Refinement (RFR) is an exact shortest-path propagation method that replaces unconditional global priority-queue ordering with adaptive partial ordering over residual distance bands. The method processes frontier bands in batches only when conservative safety checks prove that doing so preserves Dijkstra-equivalent shortest-path distances; ambiguous bands are split or resolved with local exact fallback. Version 1.1.1 corrects the band safety relation of the general-graph solver. Previous versions checked two safety conditions (outside precedence and outside candidate creation) but omitted a necessary third: internal closure — no within-band edge may still be able to lower its target. The omission could finalise a band member at a stale label via an improvement chain running entirely inside the band. The counterexample was found by a machine-checked (Lean) implementation of the safety relation; the fix adds the internal-closure check, with unsafe bands splitting or falling back to local exact processing as before. The correction affects the general-graph solver only, at band widths exceeding the minimum edge weight. The invariant-band field solver is unaffected by construction — band width at most the minimum step cost makes within-band edges impossible — and all v1.1.0 benchmark results stand. The whitepaper's safety section, batch lemma, and an erratum section document the change; the regression suite pins the counterexample across band widths. Version 1.1 added an invariant-band execution model for cost fields: when band width does not exceed the minimum possible step cost, every band is unconditionally safe, eliminating runtime safety checks. Settling and relaxing whole bands as vectorized NumPy operations then converts the proven independence into wall-clock performance. On full-field solves the invariant-band solver computes exact Dijkstra-identical distance fields 1.7×–5.5× faster than hand-written grid Dijkstra at 512²–2048² points, with the advantage growing with scale, reproduced across two independent benchmark environments. This release contains the RFR whitepaper (v1.1.1), Python reference implementation, tests, benchmark artefacts, independent-review harness, and topographic routing simulator. The work validates exactness against Dijkstra across tested graph families and demonstrates how structured frontiers can reduce global ordering work while exposing useful diagnostics such as safe batches, split pressure, local fallback, and residual frontier ambiguity. The current implementation is a research prototype. The general-graph Python implementation remains slower than Dijkstra on pre-built graph benchmarks; single point-to-point queries still favour A*; invariant-band wall-clock gains require full-field computation on roughly 10⁵ points or more. These limits are stated explicitly in the whitepaper.