Volume 37 - Anchor-A: Hybrid Chaos-Preserving Tool

This volume establishes the first certified chaos-preserving external tool in the RSI framework. Anchor-A provides a stable, invariant-faithful, notebook-native dynamical oracle built on the Lorenz-63 system. Extending the verification philosophy of Volumes 26–36, this work introduces a hybrid RK4-plus-correction model that maintains one-step accuracy at machine precision while preserving long-horizon chaotic structure, attractor geometry, and physical Lyapunov exponents. The resulting tool is admitted as a fully certified component of the RSI toolchain under a four-gate validation regime. Core Certification Results (Hybrid Regime) One-step fidelity:• Lone,train=2.22×10−9L_{\text{one,train}} = 2.22\times10^{-9}Lone,train=2.22×10−9• Lone,val=3.36×10−9L_{\text{one,val}} = 3.36\times10^{-9}Lone,val=3.36×10−9• Gate: PASS (≪ 10−410^{-4}10−4 threshold) Finite-safe rollouts:• L5=7.30×10−8L_5 = 7.30\times10^{-8}L5=7.30×10−8• L10=2.51×10−7L_{10} = 2.51\times10^{-7}L10=2.51×10−7• L20=8.10×10−7L_{20} = 8.10\times10^{-7}L20=8.10×10−7• Growth ratio L20/L10≈3.2≪20L_{20}/L_{10} \approx 3.2 \ll 20L20/L10≈3.2≪20• Gate: PASS Attractor geometry (MMD²):• Lgeom=7.13×10−3L_{\text{geom}} = 7.13\times10^{-3}Lgeom=7.13×10−3• Threshold: ≤ 1.0• Gate: PASS Physical Lyapunov exponent:• λ1,true=0.72926\lambda_{1,\text{true}} = 0.72926λ1,true=0.72926• λ1,hybrid=0.80881\lambda_{1,\text{hybrid}} = 0.80881λ1,hybrid=0.80881• ∣Δλ1∣=0.07955≤0.8|\Delta\lambda_1| = 0.07955 \le 0.8∣Δλ1∣=0.07955≤0.8• Gate: PASS Tool Status: PASSAnchor-A is admitted as a validated chaos-preserving oracle. 🧮 Canonical Constructs Construct Definition Chaos-Preserving Tool A map that maintains one-step fidelity, attractor geometry, and Lyapunov structure under finite-safe rollouts Hybrid RK4+Δ Model xt+1=RK4(xt)+ΔxΘ(xt)x_{t+1} = \mathrm{RK4}(x_t) + \Delta x_\Theta(x_t)xt+1=RK4(xt)+ΔxΘ(xt) with Δ constrained by correction and radius bounds Geometry Metric RBF-kernel MMD² on clipped, per-dimension z-normalized attractor samples Lyapunov Audit Analytic Jacobian integration verifying λ1>0\lambda_1 > 0λ1>0 and ( Four-Gate Standard {G_one, G_roll, G_geom, G_lambda_phys} jointly determine tool admission ⚙️ Key Modules Module Purpose lorenz63_rk4_step(...) Deterministic external integrator HybridAnchorAModel(...) RK4 base plus learnable correction hybrid_batch_loss(...) Unified loss: one-step, correction magnitude, radius bound generate_hybrid_rollout(...) Long-horizon hybrid evolution compute_mmd2_stable(...) Stable clipped-geometry MMD² estimator estimate_lambda_along_trajectory_analytic(...) Physical Lyapunov exponent from analytic Jacobian evaluate_rollout_mse_finite(...) Finite-safe rollout validation gate 🔬 Findings • Hybrid Anchor-A matches RK4 dynamics at near machine precision• Long-horizon rollouts remain stable, finite, and geometry-aligned• Attractor geometry preserved at O(10⁻²) scale under normalization• Physical Lyapunov exponent preserved within O(10⁻¹) deviation• Correction field converges to a minimal, interpretable residual• Provides first fully certified external oracle for RSI evolution• Strong generalization to unseen segments of validation trajectories 🔁 Continuity V26 → V37: Extends determinant-safety and stability-guard framing toward dynamical tool certification.V27–V36 → V37: Integrates invariance discipline, error-closure methods, and hybridization patterns matured in the internal RSI pathway.Forward: Establishes the foundation for multi-system anchors, invariant-compression tests, and cross-tool validation anticipated in Volumes 38–40. Reproducibility & Provenance • Fully notebook-native; all computations inline• Deterministic environment: Python 3.12.12, NumPy 2.0.2, Torch 2.9.0+cu126• Global random seed: 1337• Hardware-agnostic (CPU execution verified)• Recorded certificate: v37_anchorA_tool_cert.json (SHA-256 stable)• Printouts in Volume_037.ipynb match certificate exactly• This release contains: scientific article, notebook (PDF), source .ipynb, and certificate JSON