Ternary SteadyState Theory II: Topological Construction of a Survival Invariant and the Geometric Criterion of LifeDeath《三元稳态论 II:生存不变量的拓扑构造与生死的几何判据》

三元稳态论(TSST)V24‑v2 借助物理底线公理与相空间拓扑约束重构自创生生命理论,提出候选生存守恒量 \(Fₛᵤᵣᵥᵢᵥₐₗ\),但该量的严格数学定义、守恒律证明以及与耗散动力学的兼容性尚未完整解决。本文在 V29 永久冻结的公理体系与 11 维全域状态向量之上,严格定义生命系统闭合生存流形 \(Σ\) 为 TSST 相空间 \(P¹¹\) 内紧致无边界辛子流形;该流形在耗散流 \(Φ_t\) 下满足单模不变性,受限刘维尔测度得以守恒,该性质由 V29 自修复项 \(δ·(1-u_1)· u_3\) 在稳态吸引子上的非平衡定态行为保障。 本文将诺特定理推广至耗散系统,引入共动对称性:生存流形上与动力学流可交换的微分同胚群。构造生存不变量 \(Fₛᵤᵣᵥᵢᵥₐₗ\) 作为共动对称性对应的矩映射,利用嘉当魔公式在 \(Σ\) 上严格证明轨迹守恒,补全泊松括号关系 \(\{Fₛᵤᵣᵥᵢᵥₐₗ, H_Σ\}_Σ=0\) 的代数证明,填补 V24‑v2 留待核验的缺口。 将死亡定义为拓扑分岔:当底线存续指数 \(u_3\) 跌破临界阈值 \(u_3ᶜʳⁱᵗ\),生存流形 \(Σ\) 丧失不变性、共动对称性破缺,\(Fₛᵤᵣᵥᵢᵥₐₗ\) 不再守恒,系统轨迹不可逆逃离生存流形。该判据与 V29 赫尔维茨全局稳定性定理严格对接:\(Fₛᵤᵣᵥᵢᵥₐₗ>Fcrit\) 对应全部雅可比特征根实部为负的稳态吸引域;\(Fₛᵤᵣᵥᵢᵥₐₗ≤ Fcrit\) 对应至少一个特征根穿越虚轴的失稳区间。本文进一步给出跨尺度生存代理指标的统一构造原则,证明从单细胞膜电位振荡到全球文明网络连通性的生死判定可纳入同一套拓扑守恒框架。全部推导为纯数理演绎,无经验拟合参数,与 TSST 全套冻结体系严格自洽。 本文超越现象学生物学描述,在微分拓扑层面给出 “生命” 与 “死亡” 的严格数学定义:生命是相空间上一类特定拓扑结构,死亡是该拓扑结构发生不可逆分岔。该结果将生命概念从碳基化学载体中剥离,升华为几何‑信息属性,为人工生命、量子生命的界定,临床医学生死判据、脑机接口神经功能评估提供普适量化标准。本文不改动 V29 冻结核心对象,属于 TSST 外延特辑归档手稿。 English Ternary Steady‑State Theory (TSST) V24‑v2 reconstructed autopoietic life theory via physical bottom‑line axioms and phase‑space topological constraints and proposed a candidate survival conserved quantity \(Fₛᵤᵣᵥᵢᵥₐₗ\). Nevertheless, its rigorous mathematical definition, conservation‑law proof and compatibility with dissipative dynamics remained unresolved. Built upon the permanently‑frozen axiomatic system and the 11‑dimensional global state vector of TSST Version 29, this paper rigorously defines the closed survival manifold \(Σ\) of living systems as a compact, boundary‑free symplectic submanifold inside the TSST phase space \(P¹¹\). It obeys unimodal invariance under dissipative flow \(Φ_t\), such that the restricted Liouville measure is conserved. This property is guaranteed by the non‑equilibrium steady‑state behaviour of the frozen self‑repair term \(δ·(1-u_1)· u_3\) on steady‑state attractors. This work generalizes Noether’s theorem to dissipative systems and introduces the concept of co‑moving symmetry: a diffeomorphism group commuting with dynamical flow on the survival manifold. The survival invariant \(Fₛᵤᵣᵥᵢᵥₐₗ\) is constructed as the moment map associated with co‑moving symmetries. Applying Cartan’s magic formula on \(Σ\), we rigorously prove its conservation along trajectories and complete the algebraic verification of the Poisson‑bracket relation \(\{Fₛᵤᵣᵥᵢᵥₐₗ, H_Σ\}_Σ=0\), filling the mathematical‑verification gap left in TSST V24‑v2. Death is defined herein as a topological bifurcation: once the bottom‑line index \(u_3\) falls below the critical threshold \(u_3ᶜʳⁱᵗ\), the survival manifold \(Σ\) loses invariance, co‑moving symmetry breaks, \(Fₛᵤᵣᵥᵢᵥₐₗ\) ceases to be conserved, and system trajectories escape irreversibly from \(Σ\). This criterion precisely interfaces with the Hurwitz global‑stability theorem of V29: \(Fₛᵤᵣᵥᵢᵥₐₗ>Fcrit\) corresponds to the steady‑state basin where all Jacobian eigenvalues possess negative real parts; \(Fₛᵤᵣᵥᵢᵥₐₗ≤ Fcrit\) corresponds to the destabilization regime where at least one eigenvalue crosses the imaginary axis. Unified construction principles for cross‑scale surrogate survival indicators are presented. We demonstrate that life‑death judgements ranging from single‑cell membrane‑potential oscillations to global civilizational‑network connectivity fit within this single topological‑conservation framework. All derivations are purely mathematical without empirically‑fitted parameters and are strictly consistent with the full frozen TSST formalism. Going beyond phenomenological biological descriptions, this paper delivers rigorous differential‑topology definitions for “life” and “death”: life denotes a specific topological structure in phase space, while death stands for its irreversible bifurcation. This emancipates the concept of life from carbon‑chemistry substrates and frames it as a geometric‑informational property. It furnishes universal quantitative criteria for defining artificial‑life / quantum‑life, clinical life‑death judgement and neural‑function evaluation in brain‑computer‑interfaces. No frozen core components from V29 are modified; this manuscript belongs to the TSST extended special‑series archive.