T₃拓扑弹性场论: 尺度动力学、标度辐射与宇宙学演化的统一数学推导 T₃ Topological Elastic Field Theory: Unified Mathematical Derivation of Scale Dynamics, Scale Radiation and Cosmological Evolution

Abstract Conventional physics confines motion to positional variation within three spatial coordinates and treats scale as an auxiliary parameter of coordinates. This forces cosmology to introduce ad hoc entities such as the inflaton and dark energy when confronting inflation and Hubble tension. This report works within the T₃ topological elastic field theory. The scalar amplitude in the T₃ hyper-Euler map is defined as an independent dynamical degree of freedom — scale motion — on an equal footing with position and rotation . We systematically derive the scale rate , scale acceleration , scale momentum and scale radiation power , and prove the principle of scale relativity and the topological compatibility theorem. We then perform a hard matching between scale dynamics and the FLRW cosmological metric. We demonstrate that the cosmological Hubble parameter is intrinsically the scale rate . Hubble tension arises from the partial deconfinement of the scale degree of freedom in the early universe. Inflation corresponds to an extreme burst of scale acceleration, while reheating (the formation of the hot fireball) originates from energy conversion via scale radiation . Without introducing independent dynamical scalar fields such as inflaton or dark energy, this theory yields algebraically self-consistent solutions for inflation, reheating and Hubble tension solely from the scale dynamics of T₃ algebra. Introduction: The Illusion of Geodesics and the Awakening of Scale Motion 1.1 No "God’s Ruler": The Dependence of Translation Physics has long sought an absolutely objective geometric stage, yet there exists no transcendent "God’s ruler" in the universe immune to spacetime distortion. The geodesic — what we call a "straight line" — is essentially the path traced by light. If the spacetime background itself is highly topologically tangled and drastically deformed, light paths will twist along with it. Under such circumstances, we possess no physical means to distinguish between two scenarios: (1) An object travels along a true straight line within a flat background; (2) The background itself is contorted, light follows this contortion, and we mistake this twisted optical path for the absolute standard of a straight line. Conclusion: When light trajectories serve as the sole yardstick, the very concept of translation (motion from point A to point B) depends on the background topology. Translation is an appearance built upon a stable metric and background calibration, rather than the most fundamental primordial motion of the universe. 1.2 Redefining Motion: Rest in Position Does Not Imply Dynamical Freezing Conventional physics, including classical mechanics and relativity, adopts a binary classification of motion: translation + rotation. This carries a hidden premise: a stable manifold of positions must exist, and motion means shifting coordinate locations. This is merely an approximation valid only under locally mild topology. This theory proposes a third primordial form of motion — scale motion. It requires no displacement from A to B and entails no change in positional coordinates. In the language of the T₃ framework, an object may remain fixed in coordinates and free of rotation, while its intrinsic scale degree of freedom evolves, stretches or contracts, stores energy, and emits scale radiation. It is stationary in the translation and rotation channels yet undergoes continuous dynamical motion within the scale channel. Classical view: No positional shift and no rotation = rest. T₃ theory: Stationarity in position only means two channels are frozen; the entire system is not dynamically frozen. 1.3 Fundamental Demarcation from General Relativity This theory is easily conflated with General Relativity (GR). GR also acknowledges that geodesics are defined by light and that spacetime can be strongly curved. However, GR takes a different path: it abandons an external flat ruler, treats geodesics as intrinsic definitions, and still defines motion as coordinate displacement of points on the manifold. GR accepts that rulers are bent by spacetime, yet it retains the core premise that motion equals positional transformation on the manifold. This theory advances one critical step further: if the yardstick that defines position, straightness and distance can itself be a dynamical variable, then the evolution of this yardstick constitutes a fundamental category of motion independent of positional displacement. • GR: The metric is a property of spacetime. • T₃: The metric/scale itself is an independently evolving dynamical channel endowed with momentum, acceleration and radiative capacity. Summary: The intuitive premise of conventional physics holds that "motion means changing location". This theory rewrites the foundational definition of motion: Motion refers to the evolution over proper time of any one of the three orthogonal T₃ channels: translation, rotation and scale. Translation and rotation are the two channels readily observable to us under ordinary low‑energy conditions. The scale channel is normally confined deep within a topological potential well (binder confinement). When topological stress grows sufficiently large, the scale channel deconfines, and intense fundamental motion proceeds even if the object does not shift coordinates or spin. 摘要 传统物理学将运动局限于三维坐标内的位置变化,将尺度视为坐标的附属参数,导致宇宙学在面临暴胀、哈勃张力等问题时,不得不引入“暴胀子”、“暗能量”等未知动力学场。 本报告基于 T₃ 拓扑弹性场论,将 T₃ 超欧拉映射 中的标量振幅 定义为与位置 、旋转 完全平权的独立动力学自由度——尺度运动。报告系统推导了标度速率 、标度加速度 、标度动量 、标度辐射功率 ,证明了尺度相对性原理与拓扑相容性定理。 随后,本报告将尺度动力学与 FLRW 宇宙学度规硬对接。证明宇宙学哈勃参数 本质上就是标度速率 ;哈勃张力源于早期宇宙尺度自由度的半解禁;暴胀是标度加速度的极端爆发,而重加热(火球诞生)则是由标度辐射 转化而来。本理论无需引入暴胀子、暗能量等独立的动力学标量场,仅依托T₃尺度自由度动力学,给出暴胀、重加热与哈勃张力的代数自洽解。 引言:测地线的幻觉与尺度运动的觉醒 1.1 没有“上帝直尺”:平移的依附性 物理学界一直在寻找一种绝对客观的几何舞台,但宇宙中根本不存在一把超脱其外、不受时空扭曲影响的“上帝直尺”。 我们定义的测地线(我们口中的“直线”),本质上是光走出来的路径。如果时空背景本身高度拓扑缠绕、剧烈形变,那么光的路径也会跟着一起扭曲。此时,我们没有任何物理手段去区分两件事: (1)物体沿着平直背景里的真直线在走; (2)背景本身拧成一团,光跟着拧,我们把这条扭曲的光路,当成了绝对的直线标准。 结论:以光路作为唯一标尺时,“平移(A点→B点)”这个概念本身,就是依附于背景拓扑的。平移是建立在一套稳定度量与背景标尺之上的表象,并非宇宙最基础的原生运动。 1.2 重新定义运动:位置静止不等于动力学冻结 传统物理(包括经典力学与相对论)对运动的二分法是:平移 + 旋转。其隐含前提是:必须存在一个稳定的位置流形,移动就是更换坐标位置。但这只是在局部拓扑平缓时的近似。 本理论提出第三类原生运动——尺度运动()。它不需要“从A到B”,不需要位置坐标发生变化。 在 T₃ 的语言里,一个物体坐标固定不动、也不旋转,但它的内禀尺度自由度 可以在演化、伸缩、储能、释放标度辐射。它在平移与旋转通道上是静止的,但在尺度通道上却处于持续的动力学运动中。 经典物理认为:位置不变、不旋转 = 静止; T₃ 理论认为:位置静止只是其中两个通道冻结,不代表整个系统的动力学冻结。 1.3 与广义相对论的终极划界 很多人容易将本理论与广义相对论(GR)混淆。GR同样承认测地线是光决定的、时空可以高度弯曲。但GR的选择是:放弃外部平直标尺,把测地线当作内禀定义,运动依旧定义为流形上点的坐标迁移。 GR承认尺子被时空掰弯了,但依然坚持“运动=流形上的位置变换”。 而本理论的选择是更进一步:既然连定义“位置、直、距离”的标尺本身都可以作为动力学变量,那标尺自身的演化,本身就是独立于位置迁移的一类基础运动。 • GR:度量是时空的属性。 • T₃:度量/尺度,本身就是一个可以独立演化、有动量、有加速度、会辐射的动力学通道。 总结:传统物理的直觉是“运动=换地方”。本理论重新改写了“运动”的底层定义:运动 = T₃ 三个正交通道(平移、旋转、尺度)中,任意一个分量随本征时间的演化。 平移和旋转,是我们在日常低能下最容易看见的两个通道。而尺度通道,平时被紧紧藏在拓扑势阱(结合子禁闭)里面。当拓扑应力足够大,尺度通道解禁,哪怕物体不换坐标、不转圈,它依然在发生基础层面的剧烈运动。