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

Abstract

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‑framework barriers, this paper establishes an entirely new discrete dynamical paradigm derived from Status‑Relational Entropy (SRE) Dynamics combined with multidimensional metric‑scaling manifold reconstruction. Continuous fluid media are reformulated as statistical‑information networks governed purely by local topological invariants and causal correlations among massive discrete microscopic states. This manuscript belongs to the SRE underlying 0‑State pure‑dimensionless ontological‑layer formulation; mapping toward SI engineering units is not performed herein, and calibration against real‑world fluid experiments is reserved for follow‑up research. The framework is driven by a composite functorial chain of three primary operators: (1) The Local Graph Expansion Operator ($\mathcal{G}_{n \to n+1}$), which expands the system via single‑step increment‑wise structural equations, enforcing read‑only subspace inheritance and yielding a universal diagonal path‑interaction invariant. (2) The Local Metric and Probabilistic Pruning Operator ($\mathcal{M}_\chi \circ \mathcal{E}_{\text{local}}$), which leverages spectral radii of historical sub‑graphs to decouple parameter dead‑locks, and imposes maximum‑entropy Boltzmann pruning probabilities under the strictly‑discrete Forced Spin‑1 rule. (3) Final Allocation Operator No. 3, which introduces a 5‑node non‑homogeneous pentagonal lattice for parity‑symmetry breaking, spontaneously instantiating universal NAND logic and achieving full system‑wide Turing‑completeness. Furthermore, a rigorous Chapman‑Enskog asymptotic expansion is constructed, proving that the algebraic master‑equation of this discrete framework precisely degenerates into standard Navier‑Stokes equations in the continuum limit. Numerical empirical simulations demonstrate that under zero artificial constraints, the macroscopic coherence order‑parameter $\Phi(N)$ remains strictly confined within the robust time‑delay Lyapunov attractor envelope $[0.75, 1.00]$, departing from the $0.5$ disordered thermal baseline. Multidimensional‑scaling manifold reconstructions visually verify that un‑pruned chiral core paths spontaneously condense into highly‑connected bounded toroidal attractor loops representing topologically‑confined vortex‑filament cores, centripetally surrounded by diffuse dissipative turbulent shells. This methodology furnishes an axiomatic mathematical foundation for investigating complex fluid behaviours originating purely from discrete causal‑information networks. 经典纳维‑斯托克斯方程在连续介质框架下描述流体动力学,在极端湍流条件下会固有地遭遇到数学奇异性(爆破)与发散难题。为克服连续框架带来的这类障碍,本文基于状态‑关系熵(SRE)动力学结合多维度量标度流形重构,建立一套全新的离散动力学范式。将连续流体介质重新表述为完全由海量离散微观状态之间局域拓扑不变量与因果关联所支配的统计信息网络。本文属于SRE底层0‑State纯无量纲本体层表述,本文不执行向SI工程单位的映射,面向真实流体实验的标定工作留待后续研究开展。 本框架由三大核心算子构成复合函子链: (1)局域图扩张算子($\mathcal{G}_{n \to n+1}$):通过单步增量式结构方程完成系统维度扩张,保障只读子空间继承性,并导出通用对角路径交互不变量; (2)局域度量与概率剪枝算子($\mathcal{M}_\chi \circ \mathcal{E}_{\text{local}}$):利用历史子图谱半径解除参数循环死锁,在严格离散的强制自旋‑1规则下实现最大熵玻尔兹曼剪枝概率; (3)第三分配算子:引入五节点非齐次五边形晶格实现宇称对称性破缺,自发生成通用与非(NAND)逻辑,达成完备的图灵完备性。 进一步构建严格的查普曼‑恩斯科格渐近展开,证明该离散框架的代数主方程在连续极限条件下可以精确退化为标准纳维‑斯托克斯方程。数值仿真实证表明:在无任何人为约束条件下,宏观相干序参量$\Phi(N)$被严格约束在鲁棒的时延李雅普诺夫吸引子包络区间$[0.75,1.00]$,脱离$0.5$的无序热平衡基线。多维标度流形重构可视化证实,未被剪枝的手性核心路径会自发凝聚成高连通、有界环面吸引子环路,对应拓扑束缚的涡丝核心,外围向心包裹弥散混沌耗散湍流外壳。该方法为研究完全源自离散因果信息网络的复杂流体行为,提供公理式数学基础。