ECF III: Self-Consistency Gradient Flow Dynamics of Self-Constrained Systems

Abstract (v3.0 Updated)This is an updated version (v3.0) of ECF III, originally deposited in 2026 (v2.0). The core mathematical framework—the gradient flow dynamics and the global asymptotic stability of λ=1/2—remains unchanged. The revisions align this paper with the upgraded philosophical foundations of ECF I (v2.0) and enhance the precision of its argumentative tier labeling.Key updates in this version:1. Self-consistency as the constitutive condition of existence. The paper now explicitly adopts the ECF I (v2.0) formulation: self-consistency is not a property of any single dimension but the constitutive condition of "existence" itself, inherited by P1 as contradiction exclusion and by P2 as tension resolution. The gradient flow dynamics is the execution of the tension-resolution process.2. The continuum as Dedekind–MacNeille completion. The construction of the continuum is reformulated as the Dedekind–MacNeille completion of the strength preorder (B-tier structural necessity), replacing the earlier "metricization operation" that introduced the Archimedean and continuity conditions as separate C-tier construct choices. The completion automatically satisfies both conditions.3. Anchoring of the fixed point. The fixed point λ=1/2 is now explicitly anchored by the definition of self-consistency together with self-dual symmetry. The gradient flow does not "discover" the fixed point; it executes the approach toward it.4. Two-level structure of directionality. The existence of an evolutionary direction is a B-tier necessary consequence of the self-consistency condition. The concrete gradient flow form dλ/dτ = -S'(λ) remains a C-tier construct choice.5. Finite-step inaccessibility and the origin of time's arrow. A new corollary proves that λ=1/2 can never be attained in finite steps, providing the dynamical basis for the irreversibility of time's arrow—a topic absent from v2.0.6. Comparison with the second law of thermodynamics. A new discussion clarifies the relationship between ECF gradient flow dynamics and thermodynamic evolution.All B-tier theorems (uniqueness of gradient flow, global asymptotic stability, finite-step inaccessibility) are preserved from v2.0. The C-tier configuration hypothesis and D-tier open problems remain unchanged.