Retrieval-Warmed Energy-Based Reasoning: A Five-Arm Ablation Methodology for Diffusion-as-Inference on Structured Reasoning Tasks

Warm-started diffusion samplers accelerate iterative inference, but it is rarely clear which part of the pipeline carries the gain. We study $\mathbf{\text{retrieval-warmed}}\mathbf{\text{energy-based}}\mathbf{\text{reasoning}}\mathbf{\text{(RW-EBR)}}$ – an IRED energy-based diffusion model \cite{du2024ired} augmented with a Modern Hopfield trajectory memory – and contribute a $\mathbf{\text{five-arm}}\mathbf{\text{ablation}}\mathbf{\text{methodology}}$ (oracle, best-constant, per-query-random, shuffled, aligned) that separates three confounded effects: class-prior bias shift, stochastic warm-starting, and graph-aligned value reuse. The diagnostic decomposition is adapted from LLM-RAG evaluation \cite{ru2024ragchecker}. On $\mathbf{\text{connectivity-2}}$ (Erdős–Rényi all-pairs reachability), the aligned-vs-shuffled-oracle swing reaches $+35\mathbf{\text{pp}}$ balanced accuracy on a fixed 1{,}000-graph validation-set diagnostic, with value distribution and retrieval mechanics fixed, only per-graph alignment destroyed, while per-query random initialisation falls below cold – per-graph alignment, not bias shift or stochasticity, dominates. Yet the \emph{deployable} cold-prediction pipeline misses the acceptance gate at stored-value quality. The same diagnostic logic, stopped at the key-quality screen, applied to $\mathbf{\text{Sudoku}}$ with a task-specific key encoder produces a clean negative at a \emph{different} component – key quality, under the current setup. The decomposition names the first blocking component on each task. The setting – graph reachability refined by an iterative diffusion sampler, with explainability of failure modes as the lens – places the work within structured and spatio-temporal reasoning.