Ternary SteadyState Theory I: Capping Global WellPosedness for Delayed ReactionDiffusion Systems《三元稳态论 I:时滞反应扩散系统的全局适定性封顶》

文档名称:《三元稳态论 I:时滞反应扩散系统的全局适定性封顶》 英文标题:TSST I: Global Well‑Posedness of a Delayed Reaction‑Diffusion System 作者:杨国俊,独立交叉学科研究者,中国昆明 开源许可:CC BY‑NC‑ND 4.0 International 体系归属:三元稳态论(TSST)外延特辑系列,归档版本 V30/V31/V32。本文不修改 V29 已永久冻结的公理、11 维全域状态向量、主控方程、稳态解等核心对象,仅开展外延数学推导。 内容概述 V29 版三元稳态论搭建了动力学框架,给出稳态代数稳定性论证与参数空间紧致性的直观推演,但尚存两处数学严谨性缺口:第一,时滞反应扩散系统解的存在性、唯一性、对初值连续依赖性(Hadamard 适定性)缺少无穷维泛函分析框架下的严格证明;第二,合法参数空间\(Θvalid\)的紧致性仅依靠直观论述,存在循环论证风险,直接关系三元智能芯片工程仿真的数学可靠性。 本文将 TSST 主控方程重构于 Hilbert 空间 \(H=H¹(Ω;R¹¹) × L²([-τ,0];H¹(Ω;R¹¹))\),克服原连续函数空间中扩散算子无法生成\(C_0\)‑半群的缺陷。借助解析半群理论与 Leray‑Schauder 不动点定理证明解的局部存在唯一性;构造以 V29 赫尔维茨稳定性判据为输入的时滞 Lyapunov 泛函,实现局部解向全部时间轴的全局延拓,并证明解对初值与参数的连续依赖性。 依托得到的适定性结论,通过反证法证明\(Θvalid\)的有界性;结合解对参数的连续依赖性以及状态空间的闭性证明闭性,利用海涅‑博雷尔定理严格证明\(Θvalid\)紧致。进一步,证明在合法参数与物理状态的乘积空间上存在有限的统一全局 Lipschitz 常数,推导显式数值离散稳定性条件,为有限元、IMEX 等仿真格式的收敛分析与误差估计提供支撑。 本研究将三元稳态论从物理公理直觉升级为泛函分析框架下严格适定的动力学体系,为三元智能芯片、脑机接口硬件、开放耗散复杂系统仿真提供无条件收敛的数学理论依据。附录包含 Sobolev 空间定义、解析半群理论回顾、数值离散稳定性条件;文末附带 TSST 全系列归档 DOI 清单与著作权登记索引。 English Document Title: Ternary Steady‑State Theory I: Capping Global Well‑Posedness for Delayed Reaction‑Diffusion Systems Author: Guojun Yang, Independent Interdisciplinary Researcher, Kunming, China Open‑access License: CC BY‑NC‑ND 4.0 International System Affiliation: Extended Special Series of Ternary Steady‑State Theory (TSST), archive version V30/V31/V32. This paper performs extended mathematical derivations without altering any permanently‑frozen core objects from V29, including axioms, the 11‑dimensional global state vector, governing dynamical equations and steady‑state solutions. Abstract‑like Description TSST Version‑29 established the dynamical framework, provided algebraic proof for steady‑state stability and heuristic reasoning for compactness of the valid‑parameter space. Nevertheless, two gaps in mathematical rigor remained. First, Hadamard well‑posedness (existence, uniqueness and continuous dependence on initial data) for the delayed reaction‑diffusion system lacked rigorous justification within an infinite‑dimensional functional‑analytic setting. Second, compactness of the admissible parameter space \(Θvalid\) rested upon heuristic arguments with risk of circular reasoning, which directly threatens mathematical reliability for engineering‑oriented numerical simulation of ternary intelligent chips. In this paper, the TSST governing equation is reformulated on the Hilbert space \(H=H¹(Ω;R¹¹) × L²([-τ,0];H¹(Ω;R¹¹))\), resolving the drawback that the diffusion operator fails to generate a \(C_0\)‑semigroup over the previous continuous‑function space. Applying analytic‑semigroup theory and the Leray‑Schauder fixed‑point theorem, local existence and uniqueness of mild solutions are proven. A delay‑dependent Lyapunov functional fed with the Hurwitz stability criterion from V29 is constructed to achieve global continuation of local solutions onto the full time‑axis; continuous dependence of solutions upon initial conditions and system parameters is established. Based upon well‑posedness results, boundedness of \(Θvalid\) is shown via contradiction. Closedness follows from continuous dependence on parameters plus closedness of the physical state space; the Heine‑Borel theorem yields the rigorous compactness theorem for \(Θvalid\). Moreover, a finite uniform global Lipschitz constant over the product space of valid parameters and physical states is proven. Explicit stability criteria for numerical discretization are derived to support convergence analysis and error estimation for finite‑element, IMEX and other simulation schemes. This work elevates TSST from physical axiomatic intuition to a rigorously well‑posed dynamical system under functional‑analysis, delivering an unconditional‑convergence mathematical license for ternary intelligent chips, brain‑computer‑interface hardware and open dissipative complex‑system simulations. Appendices cover definition of Sobolev spaces, review of analytic semigroups and numerical stability conditions. A full DOI archive list for the TSST corpus together with copyright registration indices are attached at the end.