Mechanism-specific Hamiltonian bias via graph-structured local correlation modeling: If the Hamiltonian energy-conservation bias is applied not globally but as a constraint on a graph neural surrogate where nodes are adsorption sites and edges encode local chemical neighborhood similarity (e.g., shared nearest-neighbor composition motifs), then the method will preferentially exploit structured local correlations unique to heterogeneous binary alloys and achieve ≥20% fewer oracle calls than GP-UCB on high-spread landscapes, while showing no statistically significant difference on randomized-neighborhood controls (same composition, shuffled neighborhoods). Test by ablation: (a) remove graph constraint, (b) shuffle edges, (c) vary neighborhood radius, and compare call counts/regret.

On heterogeneous binary alloy adsorption-site datasets with high energy-landscape spread (defined as coefficient of variation of adsorption energies across sites ≥ 0.4), a graph neural network (GNN) surrogate that enforces Hamiltonian energy-conservation constraints locally over a chemical-neighborhood graph (nodes = adsorption sites, edges = shared nearest-neighbor composition motifs within radius r) will reach a target regret threshold (top-1% of true optimum) using ≥20% fewer oracle (DFT or oracle-surrogate) evaluations than GP-UCB baseline, measured as mean oracle-call count across ≥20 independent optimization runs with matched random seeds and matched acquisition budgets. This advantage will vanish (no statistically significant difference, two-sided Mann-Whitney U test, p > 0.05, effect size |r| < 0.1) when edges are shuffled while preserving node composition (randomized-neighborhood control), confirming the effect is attributable to structured local correlation, not merely to the Hamiltonian constraint or GNN architecture in isolation.

Minimum viable test (MVT): single binary alloy family (e.g., Cu-Pt or Ag-Pd) surface slab dataset with precomputed DFT adsorption energies (using an existing public dataset, e.g., Open Catalyst Project subsets or Catalysis-Hub binary alloy slabs) as oracle ground truth, to avoid live DFT cost.

1. Construct high-spread and low-spread subsets by binning existing systems by empirical CV of adsorption energy.
2. Implement 4 optimizer arms: (a) full method (Hamiltonian-constrained GNN + true neighborhood graph), (b) GNN without Hamiltonian constraint, (c) same GNN+constraint with shuffled edges, (d) GP-UCB baseline (standard Matérn kernel).
3. Run each arm as a Bayesian-optimization-style loop: propose next site via acquisition function on GNN/GP posterior, "query" precomputed oracle value, update surrogate, repeat until regret threshold reached or budget exhausted (cap 500 calls).
4. Repeat 20 seeds × 2 landscape types × 4 arms × 3 radius settings (r=1,2,3) = 480 runs minimum for MVT; scale to 5 alloy families for full validation (2400 runs).
5. Record oracle-call count to reach regret threshold, final regret at budget cap, and wall-clock/compute cost per arm.

# Pseudocode

for alloy in [CuPt, AgPd, NiAu, PdRh, CoFe]:
    sites, energies = load_adsorption_dataset(alloy)
    cv = std(energies) / mean(energies)
    landscape_type = "high_spread" if cv >= 0.4 else "low_spread"

    for radius in [1, 2, 3]:
        graph_true = build_neighborhood_graph(sites, radius, motif="shared_NN_composition")
        graph_shuffled = shuffle_edges(graph_true, preserve_degree=True)

        for arm in ["hamiltonian_gnn+true_graph",
                    "gnn_no_constraint+true_graph",
                    "hamiltonian_gnn+shuffled_graph",
                    "gp_ucb_baseline"]:

            for seed in range(20):
                surrogate = init_surrogate(arm, graph_true if "true" in arm else graph_shuffled)
                observed = []
                budget = 500
                for call in range(budget):
                    next_site = acquisition_ucb(surrogate, unobserved_sites)
                    e_true = oracle_lookup(next_site, energies)   # precomputed DFT value
                    observed.append((next_site, e_true))
                    surrogate.update(observed)
                    if hamiltonian_constraint_active(arm):
                        surrogate.loss += lambda_h * energy_conservation_penalty(surrogate, graph)
                    if regret(observed, energies) <= threshold:
                        record(call_count=call, arm, seed, radius, landscape_type)
                        break
                else:
                    record(call_count=budget, terminal_regret=regret(observed, energies), arm, seed, ...)

analyze: mann_whitney(full_method, gp_ucb); mann_whitney(full_method, shuffled_edge)
report: effect_sizes, CIs, radius_sensitivity_anova