We hypothesize that incorporating the proven Hodge Laplacian eigenvalue bounds as a spectral regularization constraint in a simplicial GNN will significantly improve its predictive accuracy for QUBO docking outcomes compared to standard GNNs.

This discovery has a Claude-generated validation package with a full experimental design.

Precise Hypothesis

Spine & Adversarial ReadReady for validation

“This hypothesis tests whether adding a Hodge-Laplacian-eigenvalue-bound-based spectral regularization term to a simplicial GNN's training loss significantly reduces QUBO docking-outcome prediction error compared to an equivalent standard GNN without that regularization.”

• highNo standard public QUBO-docking benchmark exists, so the entire evaluation dataset must be constructed by the researchers themselves, creating high risk of circular validation (the benchmark is built in a way that favors the hypothesis).
  Protocol mitigates via using established docking benchmarks (PDBbind, CASF-2016) as ground truth and only reformulating existing docking scores into QUBO objective form via a documented, published QUBO-docking transformation pipeline rather than a bespoke one; however, no such pipeline is confirmed in the provided search results, so this remains an unresolved gap requiring explicit citation/validation of the QUBO transformation method before the EVP can be considered rigorous.
• highThe claimed novelty (Hodge Laplacian spectral bounds as regularizer) cannot be checked against prior art because no literature search results were available — it is entirely possible closely related spectral-regularized topological GNN work already exists and this claim would need significant narrowing.
  Explicitly acknowledged: EXTERNAL_CONFLICTS and CLOSEST_EXISTING_WORK are empty due to unavailable search tooling, not because prior art was ruled out. A mandatory pre-registration literature search (e.g., searching 'simplicial neural network spectral regularization,' 'Hodge Laplacian GNN molecular,' 'topological deep learning docking') must be completed before claiming novelty; this EVP does not resolve that gap.
• mediumWhy use Hodge Laplacian eigenvalue bounds specifically rather than simpler alternatives (e.g., spectral norm clipping, standard graph Laplacian regularization, or Lipschitz constraints), and why simplicial complexes over standard molecular graphs — the methodology choice itself is not justified against simpler baselines that might achieve the same regularization benefit at lower complexity/cost.
  Partially addressed via the naive-L2 ablation in the protocol, which tests whether the specific theoretical-bound mechanism outperforms a generic penalty. However, the protocol does not include a comparison against simpler non-simplicial spectral regularization (e.g., standard graph Laplacian eigenvalue bounds on the 0-simplex graph alone), which would more directly test whether the higher-order (simplicial) structure is necessary at all versus just spectral regularization in general — this additional ablation should be added before the methodology is considered complete.

Experimental Protocol

ROI Projection

Implementation Sketch

# Pseudocode outline

for complex in dataset:
    simplicial_complex = build_alpha_complex(atoms, radius=6.0)
    L0, L1, L2 = compute_hodge_laplacians(simplicial_complex)
    bounds[complex] = theoretical_eigenvalue_bounds(L0, L1, L2)  # e.g., Horak-Jost bounds

class SimplicialGNN(nn.Module):
    def forward(self, x0, x1, x2, L0, L1, L2):
        h0 = message_pass_0simplex(x0, L0)
        h1 = message_pass_1simplex(x1, L1)
        h2 = message_pass_2simplex(x2, L2)
        return readout(h0, h1, h2)  # -> predicted QUBO objective

def spectral_reg_loss(model, L1, L2, bounds):
    eig1 = eigenvalues(learned_propagation_operator(L1))
    eig2 = eigenvalues(learned_propagation_operator(L2))
    violation = hinge_penalty(eig1, bounds.lower1, bounds.upper1) \
              + hinge_penalty(eig2, bounds.lower2, bounds.upper2)
    return violation

loss = MSE(pred_qubo, true_qubo) + lambda_reg * spectral_reg_loss(...)
optimizer.step(loss)

# Baselines: GNN (0-simplex only, no reg), SGNN (no reg), GNN+naive-L2
# Evaluate: RMSE, Spearman, top-k recall, docking power (CASF protocol)