Axiomatic System and Core Constants of Discrete Variable Lattice Dissipative Functionals Well-Posedness Proof of Discrete Dissipative Helmholtz Problem on Finite Hierarchical Fractal Lattice

Abstract

Abstract This paper constructs an axiomatic mathematical framework for discrete variable lattice dissipative functionals. Utilizing four mutually independent fundamental axioms as a minimal complete basis, it systematically investigates the operator properties, steady-state structures, and scale characteristics of fractal hierarchical lattice families. The text strictly distinguishes between axiomatic assumptions and theorem corollaries, comprehensively correcting all mathematical defects in previous versions regarding the definitions of difference operators, weight recursion logic, conjugate dissipation multiplier calculations, and normalization symbols. Gradient, divergence, and curl operators are defined within a three-dimensional orthogonal discrete lattice space, and the second-order uniform convergence of discrete operators to continuous differential operators is rigorously proven via fourth-order Taylor expansion. A discrete Gauss-Stokes integral identity is established to construct a well-defined discrete-continuous averaging mapping operator. Based on fixed-point constraints and hierarchical self-similar mappings, it is rigorously proven that the non-trivial steady-state scale parameter of the fractal lattice family corresponds uniquely to the golden ratio, and that intra-level field conduction and cross-level energy transport share the same dissipation constraint, with a self-consistent and unified global conjugate decay rule. Furthermore, the existence, uniqueness, and stability of the discrete dissipative Helmholtz boundary value problem are fully proven based on the Lax-Milgram theorem, providing a second-order error upper bound and pointwise error estimation in the energy norm. The axiomatic system presented herein features clear boundaries, step-by-step rigorous derivations, and complete algebraic self-consistency. All core constants are strict corollaries of the axiomatic system, providing a theoretical foundation for the modeling and numerical computation of multi-scale dissipative systems. Keywords: Discrete Lattice Space; Fractal Hierarchical Lattice Family; Scale Self-Automorphism; Fixed-Point Theorem; Dissipative Helmholtz Equation; Lax-Milgram Theorem; Second-Order Convergence Error Estimation 摘要 本文构建离散可变格耗散泛函的公理化数学框架,以四条相互独立的基础公理为最小完备基底,系统研究分形分层格族的算子性质、稳态结构与尺度特征。全文严格区分公理假设与定理推论,完整修正前期版本中差分算子定义、权重递推逻辑、共轭耗散倍率计算、归一符号等全部数学缺陷;定义三维正交离散格空间的梯度、散度、旋度算子,通过四阶泰勒展开严格证明离散算子对连续微分算子的二阶一致收敛性;建立离散高斯-斯托克斯积分恒等式,构造良定义的离散-连续平均映射算子;依托不动点约束与层级自相似映射,严格证明分形格族的非平凡稳态尺度参数唯一对应黄金分割比,且同层级场传导与跨层级能量传输共享同一耗散约束,全域共轭衰减规则自洽统一;基于Lax-Milgram定理完整证明离散耗散亥姆霍兹边值问题的存在性、唯一性与稳定性,给出能量范数下的二阶误差上界与逐点误差估计。本文体系公理边界清晰、推导无跳步、代数全自洽,所有核心常数均为公理体系的严格推论,可为多尺度耗散系统的建模与数值计算提供理论基础。关键词:离散格空间;分形分层格族;尺度自同构;不动点定理;耗散亥姆霍兹方程;Lax-Milgram定理;二阶收敛误差估计