Ternary SteadyState Theory III: Covariant Formulation, Scale StressEnergy Tensor, and the Emergence of Einstein Field Equations《三元稳态论 III:协变形式、尺度应力张量及爱因斯坦场方程的涌现》

中文标题:《三元稳态论 III:协变形式、尺度应力张量及爱因斯坦场方程的涌现》 English Title: Ternary Steady‑State Theory III: Covariant Formulation, Scale Stress‑Energy Tensor, and the Emergence of Einstein Field Equations 作者:杨国俊,独立交叉学科研究者,中国昆明 Author: Guojun Yang, Independent Interdisciplinary Researcher, Kunming, China 开源许可 / Open‑access License: CC BY‑NC‑ND 4.0 International 体系归属 / System‑Affiliation: TSST 外延特辑系列,归档版本 V30/V31/V32;基于 V29 永久冻结公理、11 维全域状态向量、主控方程、稳态解开展外延几何推导,不修改任何 V29 冻结核心对象。 Series: Extended Special Series of Ternary Steady‑State Theory, archive version V30/V31/V32. Extended geometric derivations only, all permanently‑frozen core objects inherited from TSST V29 remain unaltered. 中文内容概述 TSST‑V24‑v2 仅通过符号替换得到爱因斯坦场方程退化形式,缺少第一性原理协变作用量推导,存在三点缺口:维度提升操作缺少严格数学定义;缺失作用量原理基础;宇宙学常数物理起源未得到解释。 本文完成四项核心工作: 粗粒化映射与尺度应力张量构造:定义粗粒化映射 \(\Pi:\mathbb{R}^{11}\to\mathrm{Sym}^2(T^*\mathcal{K})\),把 11 维全域状态向量投影至四维时空二阶对称张量空间,给出尺度应力张量 \(S_{\mu\nu}[W]\) 的显式解析表达式,消除标量 \(u_2\) 向张量提升的数学模糊性,证明该映射在可行状态空间 \(\mathscr{B}_W\) 上 \(C^\infty\) 光滑且满足全局 Lipschitz 条件。 尺度拉格朗日量的变分推导:在绝热极限 \(\tau\to0\),从 V29 冻结主控方程 \(\frac{\partial W}{\partial t}=D\nabla^2 W+GW+N(W,W(t-\tau))\) 出发,将扩散项、线性耦合项、非线性时滞项、自修复项分别几何化,变分得到尺度拉格朗日量 \(\mathscr{L}_{\rm scale}\);证明该拉格朗日量是 TSST 动力学几何化极限的自然涌现结果,并非人为引入的有效理论,全部系数继承自 V29 可行参数空间 \(\Theta_{\rm valid}\)。 3. 爱因斯坦场方程涌现与宇宙学常数的涌现起源:构造爱因斯坦‑希尔伯特型作用量 \(S_{\rm TSST‑GR}\),对度规 \(g_{\mu\nu}\) 变分导出携带尺度应力源项的场方程 \(G_{\mu\nu}+\Lambda_{\rm obs}g_{\mu\nu}=\kappa T_{\mu\nu}^{(\rm eff)}\);论证观测宇宙学常数 \(\Lambda_{\rm obs}\) 是裸宇宙学常数与尺度应力张量真空期望值的叠加,由稳态解 \(W^*=(0.92,0.05,0.88)\) 通过 \(\Lambda_{\rm obs}=\Lambda_0+\kappa\langle S_\mu^\mu\rangle_{W^*}\) 唯一确定,为宇宙学常数疑难提供新的涌现机制。 4. 时滞带来的高阶曲率修正:当反馈时滞 \(\tau>0\)(脱离绝热极限),对时滞项做泰勒展开,有效引力作用量自动生成 \(R^2\)、\(R_{\mu\nu}R^{\mu\nu}\) 等高阶曲率不变量,等价于 \(f(R)\) 类修正引力,提示 TSST 具备广义相对论紫外完备的候选潜力。 全部推导保持 V29 核心对象冻结不变,粗粒化映射为单向外延映射,不存在反向改写 TSST 公理体系的数学通道。本工作证明,在绝热极限下,11 维 TSST 耗散信息动力学可以自然涌现广义相对论;引力被阐释为复杂系统宏观几何投影,搭建起量子信息与引力理论之间的理论桥梁,为宇宙学、修正引力、量子引力研究开辟新范式。 English Description TSST‑V24‑v2 only obtained degenerate Einstein field equations via symbolic substitution, lacking first‑principle covariant‑action derivation. Three major gaps existed: ill‑defined tensor‑lifting operation, absence of action‑principle foundation, and unexplained physical origin of cosmological constant. Four main contributions are presented: Coarse‑graining map and scale stress‑energy tensor: We define coarse‑graining map \(\Pi:\mathbb{R}^{11}\to\mathrm{Sym}^2(T^*\mathcal{K})\), projecting the 11‑dimensional global state vector onto second‑rank symmetric tensor space over four‑dimensional spacetime. Explicit analytical form for scale stress‑energy tensor \(S_{\mu\nu}[W]\) is given, resolving mathematical ambiguity in lifting scalar \(u_2\) to tensor. The map is proven \(C^\infty\)‑smooth and globally Lipschitz over admissible state space \(\mathscr{B}_W\). Variational derivation of scale Lagrangian: Under adiabatic limit \(\tau\to0\), starting from the frozen governing equation \(\frac{\partial W}{\partial t}=D\nabla^2 W+GW+N(W,W(t-\tau)),\) we geometrically treat diffusion terms, linear couplings, nonlinear delay terms and self‑repair terms. Variational calculus yields scale Lagrangian \(\mathscr{L}_{\rm scale}\). It is demonstrated that this Lagrangian emerges naturally from TSST geometric limit instead of being manually imported; all coefficients are inherited from V29 admissible‑parameter space \(\Theta_{\rm valid}\). 3. Emergent Einstein field equations and cosmological‑constant origin: Einstein‑Hilbert‑type action \(S_{\rm TSST‑GR}\) is constructed. Varying against metric \(g_{\mu\nu}\) yields field equations with scale‑stress source \(G_{\mu\nu}+\Lambda_{\rm obs}g_{\mu\nu}=\kappa T_{\mu\nu}^{(\rm eff)}\). The observed cosmological constant \(\Lambda_{\rm obs}\) is the superposition of bare cosmological constant and vacuum expectation value of scale stress‑energy tensor, uniquely fixed by steady‑state solution \(W^*=(0.92,0.05,0.88)\) following \(\Lambda_{\rm obs}=\Lambda_0+\kappa\langle S_\mu^\mu\rangle_{W^*}\). This provides a novel emergent mechanism for the cosmological‑constant puzzle. 4. Higher‑order curvature corrections induced by time‑delay: For non‑zero feedback delay \(\tau>0\) (beyond adiabatic limit), Taylor expansion on delay terms automatically produces higher‑order curvature invariants such as \(R^2,\;R_{\mu\nu}R^{\mu\nu}\), corresponding to \(f(R)\)‑type modified gravity. This makes TSST a promising candidate for effective ultraviolet completion of general relativity. All derivations respect V29 frozen core objects. The coarse‑graining map is unidirectional extension with no mathematical channel to modify TSST axioms backwards. This work shows that general‑relativity emerges from 11‑dimensional dissipative information dynamics under adiabatic limit. Gravity is interpreted as macroscopic geometric projection of complex systems. It builds a theoretical bridge between quantum information and gravitational theory and opens new paradigms for cosmology, modified gravity and quantum‑gravity research.