Formal Theory of Attractor–Relational Dynamics (FTAR): Relational Stability as a Generative Mechanism for Classical and Quantum Spacetime (v18 Final)

This manuscript presents the definitive version (v26. 1) of the Formal Theory of Attractor–Relational Dynamics (FTAR). The framework provides a rigorous, scalar-based generative mechanism for spacetime geometry, establishing that gravitation and quantum evolution are emergent manifestations of a single fundamental object: the Relational Stability Functional (Φ). Key Scientific Contributions: Hessian Metric Emergence: A formal proof demonstrating that the classical metric tensor gμν is exactly recovered as the Hessian of the stability functional. Integrability Conditions: Definition of the mathematical constraints (∂λgμν=∂μgλν) required for a metric to admit a local stability representation. Constructive Reconstruction of General Relativity: Exact analytical recovery of the Schwarzschild (static black hole), Kerr (rotating system), and Friedmann–Lemaître–Robertson–Walker (cosmological expansion) metrics. Quantum Stability Operator (Φ^): An extension of the theory to the Planck scale, where classical geometry emerges as the expectation value of an operator acting on a relational Hilbert space. Testable Predictions: Derivation of falsifiable consequences, including fundamental stochastic metric noise at the Planck scale and specific stability-curvature constraints on the Cosmological Constant (Λ). FTAR offers a mathematically explicit and parsimonious alternative to current quantum gravity paradigms, bridging information theory, attractor dynamics, and general relativity into a unified physical law. Updated to include full tensor stability, singularity resolution, and Hawking temperature saturation proofs