Emergence Inevitability and Algebraic Computational Methods of Turbulence Based on Discrete Microscopic Causal Statistics and Multidimensional Manifold Reconstruction

The classical Navier-Stokes equations formulate fluid dynamics within a continuous continuum medium, which inherently encounters mathematical singularities (blow-up) and divergence difficulties under extreme turbulent conditions. To bypass these continuous barriers, this paper establishes an entirely new discrete dynamical paradigm derived from Status-Relational Entropy (SRE) dynamics and multidimensional metric scaling. We reconstruct continuous fluid media as a statistical information network governed purely by local topological invariants and causal correlations among massive discrete microscopic states. The framework is driven by a composite functorial chain of three primary operators: (1) The Local Graph Expansion Operator (G₍ ₍+₁) that expands the system via a single-step, increment-wise structural equation while securing read-only subspace inheritance and a universal diagonal path-interaction invariant. (2) The Local Metric and Probabilistic Pruning Operator (M_ E₋₎₂₀₋) that utilizes the historical sub-graph spectral radius to decouple parameter deadlocks and enforces a maximum-entropy Boltzmann pruning probability under the strictly discrete Forced Spin-1 rule. (3) The Final Allocation Operator No. 3 that introduces a 5-node non-homogeneous pentagonal lattice to break parity symmetry, spontaneously instantiating universal NAND logic to achieve complete Turing completeness. Furthermore, we establish a rigorous Chapman-Enskog asymptotic expansion, proving that our algebraic master equation precisely degenerates into the standard Navier-Stokes equations under the continuum limit. Numerical empirical simulations demonstrate that under zero artificial constraints, the macroscopic coherence order parameter (N) is strictly locked within a robust time-delay Lyapunov attractor envelope 0. 75, 1. 00, breaking away from the 0. 5 disordered baseline. Multidimensional scaling reconstructions visually confirm that un-pruned chiral core paths spontaneously condense into a highly connected, bounded toroidal attractor loop representing the topological manifold confinement of a vortex filament, centripetally enveloped by a diffuse chaotic shell. This methodology provides a solid, axiomatic mathematical foundation for computing complex fluid behaviors entirely from discrete networks.