Deterministic Reference-Based Protein Structure Verification via Constraint Graph Sheaf Energy

We present SATYA Protein, a deterministic verification engine for protein structures based on constraint graph sheaf energy. Given a candidate structure and a native reference, SATYA constructs a constant cellular sheaf over the residue constraint graph, where six-dimensional stalks encode per-residue spatial displacement (normalized by radius of gyration) and signed backbone dihedral deviation (normalized by 180∘). The binary SAFE/UNSAFE verdict is gated on three independent checks: contact-network preservation (𝑄 ≥ 0.90, native contact fraction), localized damage (per-region constraint severity), and chiral integrity (mean dihedral deviation). Dirichlet energy and spectral gap are reported as continuous diagnostics measuring the smoothness of the deviation field across the constraint graph. Every verification produces a cryptographically signed receipt (Ed25519). The current constant-sheaf implementation uses identity restriction maps, operating as a vector-valued graph Dirichlet-energy verifier; the architecture supports future extension to non-identity maps for obstruction detection beyond the current energy pathway. We validate on a controlled perturbation benchmark of seven proteins spanning 10 to 26,700 residues, verified in 27 seconds total on a single consumer CPU with no GPU, no force field, and no training data. Near-native sensitivity analysis using Gaussian C𝛼 noise at six magnitudes (0.25–6.0 Å) shows zero false positives at thermal scale (0.25 Å), with the SAFE/UNSAFE transition occurring between 0.5 and 1.0 Å noise, corresponding to the onset of native contact disruption. A mirror-image experiment demonstrates that SATYA detects chiral inversions that pairwise distance metrics (including lDDT) mathematically cannot: reflection preserves all C𝛼 distances (lDDT = 1.0), but inverts signed backbone dihedrals, which SATYA detects via the six-dimensional stalk (mean dihedral deviation ∼0.55, verdict UNSAFE). Conformation-specific verification on three fold-switching proteins (KaiB, RfaH, XCL1) shows that SATYA returns UNSAFE and localizes constraint violations to the fold-switching residues, while pLDDT, which is not designed as a conformation-specific reference verifier, does not identify the mismatch. SATYA is complementary to pLDDT and MolProbity, providing independent reference-based ver-ification with per-residue localization and cryptographic audit. Live demo and project materials: https://invariant.pro/protein.