Scalable Contact-Gated Stochastic Operators and Universal Reduction Laws for Macromolecular Protein Dynamics, Allostery, and Kinetics

Constructing continuous, thermodynamically reversible reduced equations of motion from atomistic molecular dynamics (MD) that scale from fast-folding miniproteins to multi-domain allosteric enzymes in explicit water remains a foundational problem in computational biophysics. Across 15 molecular systems (890,000 SHA-256-verified frames, 0.89 us cumulative sampling) spanning 13 structurally diverse proteins from Chignolin (d=18) to explicit-solvent Adenylate Kinase (d=426, 40,110 particles) and Maltose-Binding Protein (d=738, 49,215 particles), plus two analytical verification peptides (d=2, 4), we formulate a contact-gated, GENERIC-regularized stochastic operator on the backbone torus T^d. We show that collective slow modes compress sub-linearly across two orders of magnitude in chain length - with spectral participation ratio scaling as PR(d)=1.061 d^0.494 (95% CI [0.332, 0.666], Delta AICc=-18.15) and short-time diffusion rank as r_D^90(d)=0.799 d^0.930 (Delta AICc=-14.39) - whereas cumulative 90% equilibrium variance in large native-basin enzymes transitions to extensive proportionality (10.0% blind extrapolation error vs. 35.7% for small-protein power laws). Consequently, macromolecular tractability above d>200 requires off-diagonal sparsity rather than global PCA truncation alone: universal peptide-plane crankshaft anti-correlation (mean R(psi_i, phi_i+1)=-0.603 +/- 0.054) and sparse 3D tertiary contact gating (mean k_eff=5.97, reaching 99.3% sparsity at d=738) enable a sparse directed acyclic contact-graph Cholesky estimator scaling empirically as O(d^0.92) in time and O(d^1.06) in memory (65.8x memory compression, 99.6x faster than Graphical LASSO, and higher blind held-out log-likelihood +0.303/d vs. dense +0.205/d on MBP). Biophysically, relative solvent-accessible surface area (rSASA_i) unifies dry linear equipartition in protein cores (alpha_core=1.00, 12.33% blind thermal error) with viscosity-coupled loops (alpha_loop=1.28) and peptides (alpha_peptide=1.57); parameter-free kinematic projection reproduces solution NMR Lipari-Szabo S^2 order parameters (mean r_s=+0.483 across four BMRB enzymes, Ubiquitin Pearson r=+0.921) and X-ray B-factors (mean r_s=+0.276); and 13-fold zero-shot Leave-One-Protein-Out transfer predicts unseen protein operators directly from static PDB coordinates (r_s=+0.706, R^2=+0.359 ridge; r_s=+0.721, R^2=+0.410 GBDT). Applied to functional discovery, explicit-solvent apo toroidal operators localize mechanical domain-closure hinges in AdK (PR-AUC=0.348) and MBP (AUROC=0.690 vs. 0.446 ANM), resolve distal Met20 FG catalytic loop coupling in E. coli DHFR (71.6% recovery at 93.2% sparsity) and beta1-beta2 Ile44 routing in Ubiquitin (7.1x amplification) on blind trajectories, and quantify sub-diffusive Mori-Zwanzig reaction-coordinate memory (mean gamma_M=0.593 +/- 0.126). A validation-tuned head-to-head benchmark against Markov State Models, TICA, Smoluchowski diffusion, Elastic Network Models, and unconstrained SDEs across all 13 proteins maps the physical regimes where continuous contact-gated GENERIC operators (lowest Frobenius error 0.288, exact 0.000 spurious entropy vs. median 65.79 kB/ns/d, IQR 53.76-104.66, mean 346.17) and discrete or harmonic baselines each excel.