Patient-selection-style analogue for catalysis: site-selection biomarker for when Hamiltonian-RL is beneficial: A computable 'heterogeneity biomarker' (e.g., variance of Bader charge across surface atoms, distributional entropy of local coordination environments, or KL-divergence between site-descriptor clusters) predicts whether Hamiltonian entropy-weighted RL will outperform GP-UCB by ≥20% in oracle-call reduction. Hypothesis: selecting only surfaces with biomarker above threshold yields a statistically significant improvement vs applying HRL indiscriminately. Test by training a classifier/regressor on simulation datasets to predict HRL-vs-GP-UCB advantage and validating prospectively on held-out alloy systems.

There exists a computable "surface heterogeneity biomarker" H(S) — defined as a scalar or low-dimensional function of (a) variance of Bader charge across symmetry-inequivalent surface atoms, (b) Shannon entropy of local coordination-number/generalized-coordination-number distributions, and/or (c) KL-divergence between clusters of site-level descriptor vectors (e.g. d-band center, coordination, strain) — such that: for catalyst surfaces S with H(S) > τ (a fixed, pre-registered threshold), Hamiltonian entropy-weighted reinforcement learning (HRL) reduces the number of oracle calls (DFT single-points) required to reach a fixed regret/accuracy target by ≥20% relative to GP-UCB, with statistical significance (paired one-sided test, α=0.05, effect size Cohen's d≥0.5) across a held-out set of alloy surfaces; and for S with H(S) ≤ τ, HRL shows no significant advantage (mean improvement <10%, or GP-UCB non-inferior). The claim is specifically that H(S) is predictive (AUC ≥ 0.75 for the binary "HRL beats GP-UCB by ≥20%" classification task), not merely correlated in-sample.

Design: Simulation-based supervised learning + prospective held-out validation, structured as a train/validate/test split over alloy systems (not just data points), to avoid leakage across chemically similar surfaces.

• Training corpus: N₁ = 40–60 alloy surface systems (binary alloys × 3–4 facets × 2–3 adsorbates = ~300–500 optimization "runs"), each run producing (H(S) features, HRL-vs-GP-UCB oracle-call advantage %).
• Validation corpus: N₂ = 10–15 held-out alloy systems, same protocol, used for threshold calibration and early stopping.
• Prospective test corpus: N₃ = 8–12 completely held-out alloy systems (different metal pairs, ideally including at least 2 ternary or unusual-facet systems to test extrapolation) — biomarker + classifier applied blind, HRL vs GP-UCB run to completion, actual advantage measured and compared to prediction.
• Baselines per run: GP-UCB (standard exploration-exploitation, Matérn kernel), HRL (Hamiltonian entropy-weighted variant as defined in the parent paper), and a naive random-sampling baseline as a sanity floor.
• Repeats: Each (surface, method) pair run with ≥10 random seeds to estimate variance in oracle-call-to-target.

# Phase 0: Feature + label generation
for alloy in training_systems:
    slab = generate_slabs(alloy, facets=[111,100,211])
    relaxed = surrogate_relax(slab)  # OC20 GNN
    bader_var = compute_bader_charge_variance(relaxed)
    coord_entropy = shannon_entropy(coordination_numbers(relaxed))
    kl_div = kl_divergence(cluster(site_descriptors(relaxed)))
    H = feature_vector(bader_var, coord_entropy, kl_div)

    for seed in range(10):
        calls_gpucb = run_GPUCB(surrogate_oracle, slab, target_eps=0.02, seed)
        calls_hrl   = run_HamiltonianRL(surrogate_oracle, slab, target_eps=0.02, seed)
    advantage_pct = mean_reduction(calls_gpucb, calls_hrl)
    dataset.append((H, advantage_pct))

# Phase 1: Model fitting
X, y = dataset.features, dataset.advantage
clf = GradientBoostedTrees()  # binary: advantage>=20%
reg = GradientBoostedTrees()  # continuous advantage%
nested_cv(clf, reg, X, y, outer_k=5, inner_k=3)
tau = youden_threshold(clf, validation_set)

# Phase 2: Prospective test (frozen model)
for alloy in held_out_test_systems:
    H_test = compute_biomarker(alloy)
    predicted_advantage = reg.predict(H_test)
    predicted_class = clf.predict(H_test) > tau

    actual_advantage = run_full_comparison(alloy)  # real HRL vs GP-UCB
    log(predicted_advantage, actual_advantage, predicted_class)

evaluate(AUC, R2, calibration_curve, paired_significance_test)