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 and inherently encounter mathematical singularities (blow-up) and divergence difficulties under extreme turbulent conditions. To bypass these barriers, this work establishes a discrete dynamical paradigm derived from Status–Relational Entropy (SRE) dynamics combined with multidimensional metric-scaling manifold reconstruction: a continuous fluid medium is reformulated as a statistical-information network 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; no mapping onto SI engineering units is performed, 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 Gn→ n+1, which expands the system dimension through 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 M_χlocal, which leverages spectral radii of historical sub-graphs to decouple parameter dead-locks, explicitly separates the exogenous dissipation coefficient Λ from the endogenous adaptive gain λ(n) as two comparable modes, and imposes maximum-entropy Boltzmann pruning probabilities under the strictly-discrete Forced Spin‑1 (Paradigm B) 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 system-wide Turing completeness. A rigorous Chapman–Enskog asymptotic expansion is further constructed, proving that the algebraic master equation of this discrete framework degenerates into the standard Navier–Stokes equations in the continuum limit, with the kinematic viscosity written formally as ν=ζ²Λ. With no externally fitted parameters and no superimposed analytical formulae, numerical experiments ($N=60$, five random seeds, all results reproducible from the reference implementation) yield two testable conclusions: the structure density serving as macroscopic coherence order parameter decreases monotonically from 0.491 to 0.003 as Λ grows, forming a smooth transition spanning about 1.5 orders of magnitude (Λ∈[0.08,\,2.5]) rather than a unique critical point; and the one-dimensional power spectrum of the chiral field along causal order exhibits a stable power-law scaling range with slope near −1.85 on the high-Reynolds side — a proxy for the cascade rather than the three-dimensional energy spectrum $E(k)$, and therefore not a collapse onto Kolmogorov $-5/3$. Core–shell separation is retained as an algebraic consequence of Operator 3, but its "centripetal toroidal attractor" geometric image is judged an artefact of the metric-reconstruction criterion and is not claimed as an empirical result, pending verification by embedding-independent topological indicators such as persistent homology and intrinsic dimension. An Oseen-vortex injection experiment is also reported and repositioned within the chain of evidence. A two-point correlation coupling matrix conforming to the Lamb–Oseen Gaussian distribution with an asymmetric unidirectional rotational velocity field is injected as the initial value for the first 30 steps, after which the network self-organises freely to $N=100$ at Λ=0.05; under the phase diagram above this working point lies at the upper edge of the structure–dissipation transition region (structure density about 0.44, spectral slope about −1.83), not at an extreme Reynolds number as previously claimed. Two original observations are reinterpreted: the order parameter Φ≡1.0000 is equivalent to all frontier channels surviving at every step, i.e. the zero-dissipation limit with pruning probability approximately zero — trivial by construction, reproduced by any initial value, and therefore unable to support the pseudo-spectral bound conjecture; the MDS reconstruction residual 8.93×10⁻¹ is an embedding error measuring the non-representability of the pairwise dissimilarity matrix in three-dimensional Euclidean space, with no established quantitative relation to intrinsic non-Euclidean curvature. What this experiment supports is that an Oseen-type initial value can be injected into the discrete operator chain and retains its chiral character through subsequent evolution; what it cannot support is any proof of the pseudo-spectral bound, topological noise-immune locking, intrinsic non-Euclidean curvature, or the emergence of a one-dimensional rigid central axis. This methodology furnishes an axiomatic mathematical foundation — verifiable rather than merely narratable — 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,在极端湍流条件下固有地遭遇数学奇异性(爆破)与发散困难。为绕过连续框架的这类障碍,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:连续流体介质被重新表述为完全由海量离散微观状态之间的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,不执行向 SI 工程单位的映射,面向真实流体实验的标定留待后续研究。框架由一条复合函子链驱动:(1) 局域图扩张算子 Gn→ n+1,以单步增量式结构方程完成维度扩张,施行只读子空间继承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 M_χlocal,利用历史子图谱半径解除参数死锁,将外生耗散系数 Λ 与内生自适应增益 λ(n) 显式分离为可对照的两种模式,并在严格离散的强制自旋‑1(Paradigm B)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)逻辑并达成系统级图灵完备性。进一步构建严格的 Chapman–Enskog 渐近展开,证明该离散框架的代数主方程在连续极限下退化为标准纳维–斯托克斯方程,运动粘度形式地写作 ν=ζ²Λ。在无任何外部拟合参数、无解析公式叠加的条件下,数值实证($N=60$,5 个随机种子,全部结果可由规范实现复现)给出两项可检验结论:作为宏观相干序参量的结构密度随 Λ 增大由 0.491 单调降至 0.003,形成跨越约 1.5 个数量级的平滑过渡区(Λ∈[0.08,\,2.5]),而非唯一临界点;手性场沿因果序的一维功率谱在高雷诺数侧呈现稳定的幂律标度区,斜率约 −1.85,该谱为级联的代理指标而非三维能谱 $E(k)$,故不与 Kolmogorov $-5/3$ 合拢。核–壳分层作为算子 3 的代数结论予以保留,但其"向心环面吸引子"几何图像被判定为度规重构准则的产物,不作为实证结论,须由持续同调、本征维数等嵌入无关的拓扑指标另行验证。本文同时报告奥森涡初值注入实验,并在统一核验后对该实验在证据链中的位置作出定位修正:以符合兰姆–奥森涡高斯分布的两点关联耦合方阵与非对称单向旋转速度场作为前 30 步初值,随后移交网络自组织生长至 $N=100$,取 Λ=0.05;按上述相图定标,该工况位于结构–耗散过渡区上沿(结构密度约 0.44,谱斜率约 −1.83),并非极端高雷诺数。两项原始观测被重新解读:序参量 Φ≡1.0000 等价于每一步全部前沿通道存活,即剪枝概率近似为零的零耗散极限,是构造使然的平凡结果,换用任意初值同样得到 Φ≡1,故不能支持拟谱界猜测的任何形式证明;MDS 重构残差 8.93×10⁻¹ 是嵌入误差,度量二元相异度矩阵在三维欧氏空间中的不可表示程度,与"内蕴非欧曲率"之间无已建立的定量关系。据此,本实验能够支持的是:奥森涡型初值可在离散算子链中注入并在后续演化中保留手性特征;不能支持的是:拟谱界猜测的证明、拓扑抗噪锁死、内蕴非欧曲率、以及一维刚性中轴的涌现。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,提供了一个可被检验而非仅可被叙述的公理式数学基础。