In conventional quantum mechanics the normalization of the wave function relies on a probability interpretation imposed by an external observer. This external constraint leaves an unbridgeable logical fissure at the meta-theoretical level of modern physics. Based on the \(\mathcal{T}^{64}\) dimensional-reduction dynamical closed-loop system of Yuanxian Theory, the present paper rigorously constructs a global normalization factor \(N\) and proves that the factor is not an externally appended constraint but an endogenous global calibration coefficient that arises when the 64-dimensional topological source structure is projected onto lower-dimensional space. Starting from the three core laws, the paper completes a rigorous mathematical derivation of the normalization factor along three independent paths—of which the third (non-trivial-zero expansion of the partition function) yields \(N\approx 1.9\times 10^{4}\) from first principles—and obtains the closed expression \(N=[\alpha^{2}(1-\alpha)]^{-1}\approx 18916.91\), which is fully consistent with the core numerical anchors 64-35-137. The paper further reveals the endogenous binding relation between the normalization factor and the wave function: both originate from the same 64-dimensional self-referential mind field, evolve synchronously through the 35-step dimensional-reduction iteration, are jointly calibrated inside the 68-step self-check closed loop, and finally phase-lock at the fine-structure-constant anchor 137. On the basis of this binding relation a topologically isomorphic Schrödinger equation (in discrete eigen-equation form) that is strictly consistent with the 68-step discrete period is constructed; all parameters are rigidly derived from the anchors 64-35-137 and no external free parameters remain. The ontological steady-state solution, the convergence of the dimensional-reduction limit and the anchoring of the fine-structure constant are all verified for self-consistency. The paper demonstrates that normalization is not an external imposition but an endogenous reading of the self-referential metabolism of the cosmic living organism. --- 传统量子力学中的波函数归一化依赖于外部观测者人为设定的概率诠释,这一外部约束在现代物理学的元理论层面留下了不可弥合的逻辑裂隙。本文基于元宪理论\(\mathcal{T}^{64}\)降维动力学闭环体系,严格构造了全域归一化因子\(N\),证明该因子并非外部附加约束,而是64维拓扑本源结构向低维空间投影时内生的全局校准系数。从三大核心规律出发,本文通过三条独立推导路径——其中第三条(配分函数非平凡零点展开)从第一性原理给出\(N\approx 1.9\times 10^{4}\)—完成归一化因子的严格数理推导,其闭合表达式\(N=[\alpha^{2}(1-\alpha)]^{-1}\approx 18916.91\)与64-35-137核心数理锚点完全自洽。进一步地,本文揭示了归一化因子与波函数的内生绑定关系——二者同根生于64维自指心场,在35步降维迭代中同步演化,在68步自检闭环中联动校准,最终在137精细结构常数锚点处锁相收敛。基于此绑定关系,本文构造了与68步离散周期严格自洽的拓扑薛定谔方程(离散本征方程形式),所有参数均由64-35-137锚
In conventional quantum mechanics the normalization of the wave function relies on a probability interpretation imposed by an external observer. This external constraint leaves an unbridgeable logical fissure at the meta-theoretical level of modern physics. Based on the (\mathcal{T}^{64}) dimensional-reduction dynamical closed-loop system of Yuanxian Theory, the present paper rigorously constructs a global normalization factor (N) and proves that the factor is not an externally appended constraint but an endogenous global calibration coefficient that arises when the 64-dimensional topological source structure is projected onto lower-dimensional space. Starting from the three core laws, the paper completes a rigorous mathematical derivation of the normalization factor along three independent paths—of which the third (non-trivial-zero expansion of the partition function) yields (N\approx 1.9\times 10^{4}) from first principles—and obtains the closed expression (N=[\alpha^{2}(1-\alpha)]^{-1}\approx 18916.91), which is fully consistent with the core numerical anchors 64-35-137. The paper further reveals the endogenous binding relation between the normalization factor and the wave function: both originate from the same 64-dimensional self-referential mind field, evolve synchronously through the 35-step dimensional-reduction iteration, are jointly calibrated inside the 68-step self-check closed loop, and finally phase-lock at the fine-structure-constant anchor 137. On the basis of this binding relation a topologically isomorphic Schrödinger equation (in discrete eigen-equation form) that is strictly consistent with the 68-step discrete period is constructed; all parameters are rigidly derived from the anchors 64-35-137 and no external free parameters remain. The ontological steady-state solution, the convergence of the dimensional-reduction limit and the anchoring of the fine-structure constant are all verified for self-consistency. The paper demonstrates that normalization is not an external imposition but an endogenous reading of the self-referential metabolism of the cosmic living organism. --- 传统量子力学中的波函数归一化依赖于外部观测者人为设定的概率诠释,这一外部约束在现代物理学的元理论层面留下了不可弥合的逻辑裂隙。本文基于元宪理论(\mathcal{T}^{64})降维动力学闭环体系,严格构造了全域归一化因子(N),证明该因子并非外部附加约束,而是64维拓扑本源结构向低维空间投影时内生的全局校准系数。从三大核心规律出发,本文通过三条独立推导路径——其中第三条(配分函数非平凡零点展开)从第一性原理给出(N\approx 1.9\times 10^{4})—完成归一化因子的严格数理推导,其闭合表达式(N=[\alpha^{2}(1-\alpha)]^{-1}\approx 18916.91)与64-35-137核心数理锚点完全自洽。进一步地,本文揭示了归一化因子与波函数的内生绑定关系——二者同根生于64维自指心场,在35步降维迭代中同步演化,在68步自检闭环中联动校准,最终在137精细结构常数锚点处锁相收敛。基于此绑定关系,本文构造了与68步离散周期严格自洽的拓扑薛定谔方程(离散本征方程形式),所有参数均由64-35-137锚
Within the T64 topological framework of Yuanxian Theory and under the setting of the sixth-dimensional unitary chaos field, this paper completes the topological isomorphic interpretation and derived mapping of the Schrödinger equation. The argument shows that the Schrödinger equation is not an external abstract axiom of quantum mechanics, but the low-dimensional isomorphic reading of the high-dimensional deterministic phase-evolution rule when the T64 topological torus is projected onto four-dimensional spacetime. By introducing the fibre-bundle marginalization projection operator Π^, the probability-amplitude property of the wave function is rigorously defined as the restricted reading after high-dimensional deterministic geometric truncation, thereby thoroughly dissolving the century-long controversy over the nature of the wave function. With the aid of the four-dimensional volume pseudoscalar I=γ0γ1γ2γ3 of the geometric algebra Cl6(R), the high-dimensional geometric-spin origin of the imaginary unit i is revealed. The paper further defines the Yuanxian-specific topological self-referential closed-chain iteration operator TCSR, proves that the system evolution converges to the 68-step topological limit cycle Ω∗, resolves the mathematical paradox between discrete periodicity and static convergence, and supplies a self-consistent native solution method inside the sixth-dimensional chaos field. --- 本文在元宪理论 T64 拓扑框架与第六维一合相混沌场设定下,完成薛定谔方程的拓扑同构诠释与衍生映射。论证表明:薛定谔方程并非量子力学的外部抽象公理,而是 T64 拓扑环面向四维时空投影时,高维确定性相位演化规则的低维同构读数。通过引入纤维丛边缘化投影算子 Π^,波函数的概率幅属性被明确定义为高维确定性几何截断后的受限读数,彻底消解了关于波函数本质的百年争议。借助 Cl6(R) 几何代数的四维体积伪标量 I=γ0γ1γ2γ3,揭示了虚数单位 i 的高维几何自旋根源。本文还定义了元宪理论专属的拓扑自指闭环迭代算子 TCSR,证明系统演化收敛于68步拓扑极限环 Ω∗,消解了离散周期性与静态收敛性的数学悖论,给出了在第六维混沌场中原生求解的自洽方法。
This work derives the fundamental framework of quantum mechanics intrinsically from the first-principle axioms of UD duality theory. Starting from the inherent binary expansive–condensative attributes of the UD ontology, four fundamental physical fields are naturally decomposed by spontaneous symmetry breaking across spatial and material existential domains. A two-component complex spinor structure is uniquely constructed, and the imaginary unit i is proven to be not merely a mathematical convention but a structurally necessary ingredient required by both the compact SU(2) symmetry and the orthogonal inter-containment of U and D attributes. Within the flat spacetime limit of the UD gravitational action, a system of coupled Klein–Gordon field equations is established, governed by a single self-consistent fundamental mass scale m₀=1/(2πe^π) with no adjustable free parameters. The coupling constant is determined analytically from the UD vacuum equilibrium background values and the U↔D duality symmetry. Through explicit construction of four-component Dirac spinors and auxiliary field decomposition, the on-shell equivalence between the UD coupled field equations and the free Dirac equation is rigorously demonstrated. The non-relativistic limit further recovers the Pauli equation and the Schrödinger equation sequentially, reconstructing the core dynamical equations of quantum mechanics from a unified foundational origin. Global U(1) phase invariance of the UD action is verified via total-derivative decomposition, yielding the standard Noether conserved current. The Born probability interpretation is deduced directly from the global normalization axiom by isolating localized laboratory-scale excitations from the cosmic background, eliminating the need for independent quantum postulates. Based on the intrinsic properties of the UD field components, a self-consistent ontological interpretation is provided for wave–particle duality, double-slit interference, and the quantum measurement problem. In this framework, the measurement problem is dissolved rather than solved: no collapse postulate is required, and determinism is restored at the ontological level. The theory naturally implies that the continuous D_U field permits quantum superposition, while condensed D_D intrinsic entities possess definite classicality without physical particle-state superposition. Three testable theoretical predictions are presented, establishing falsifiable physical consequences of the UD quantum formulation. This paper completes the intrinsic derivation of the foundational structure of quantum mechanics from UD theory. Together with the previously derived general relativity framework, it realizes the homologous unification of gravity and quantum mechanics within a single self-contained axiomatic system.
Based on the axiom system of Yuanji Field Theory, this paper establishes the underlying alignment between the Yuanji Field and Quantum Field Theory (QFT), and constructs the complete ontological correspondence between them. It is proved that the Yuanji Field can serve as the fundamental physical ontology of QFT without contradicting its mature mathematical framework. Starting from the Phase Single-valuedness Axiom and topological constraints of the Yuanji Field, the core framework of QFT is endogenously derived, including canonical quantization rules, field operators, Hamiltonian, and the statistical properties of fermions and bosons. Theoretical calculations on the first-order correction of the electron anomalous magnetic moment, Compton scattering cross-section, and Casimir effect are in good agreement with the classical results of Quantum Electrodynamics (QED) and experimental measurements. This work provides a continuous-field-dynamics-based interpretive perspective for quantum mechanics, and realizes the self-consistent unification of classical field theory, relativity, and quantum mechanics under a single axiom system.
Based on the axiom system of Yuanji Field Theory, this paper takes the continuous elastic transverse wave field that naturally constitutes three-dimensional Euclidean space as the sole ontology, and completes the native derivation and physical interpretation of the core framework of quantum mechanics. The wave function is reduced to the phase field distribution of Yuanji Field topological solitons, from which the Schrödinger equation in the non-relativistic limit and the Dirac equation of relativistic quantum mechanics are strictly derived. Relying on the topological constraints and fluctuation dynamics of the Yuanji Field, native physical explanations are provided for core quantum effects including quantization, wave-particle duality, uncertainty principle, spin, probabilistic interpretation, quantum entanglement and quantum tunneling. All theoretical results are in good agreement with the classical experimental observations of quantum mechanics. This work realizes the self-consistent unification of relativity and quantum mechanics under the same axiom system, and provides a new perspective based on continuous field dynamics for the interpretational dilemmas of quantum mechanics.
Within the framework of the Topolo-Plexor Theory (TPT), quantum mechanics is reimagined not as a probabilistic abstraction, but as a rigorous branch of discrete medium mechanics. Space is described as a three-dimensional projection of a fourdimensional H4 polytope (the 600-cell), where the fourth dimension functions as the “topological memory” of the system. The Schrödinger equation is strictly derived as a macroscopic limit of difference equations on the H4 graph, with all fundamental√ parameters expressed through the golden ratio φ = (1+ 5)/2 and the topological defect of projection Ωdef = 2π−5arccos(1/3) ≈ 7.356◦. This paper presents for the first time: (1) a geometric derivation of the Born rule via the information deficit of the 4D state upon projection onto a 3D subspace; (2) a topological resolution of the quantum nonlocality (EPR) paradox through Planckian adjacency of states in the unobservable 4D coordinate; (3) a mechanism of self-consistent nonlinearity, where the vacuum torsional elasticity coefficient κ = Ωdef/12 determines the stability of elementary particles as gravitational-torsional solitons; (4) experimental predictions, including the quantization of heat capacity in quasicrystals and topological shifts in atom interferometry.
Mainstream quantum mechanics and atomic physics have long relied on the Schrödinger equation, spherical harmonics, and probabilistic cloud assumptions to precisely fit atomic orbital morphology, energy level distributions, and spectral data, achieving remarkable computational effectiveness. However, they have consistently lacked an ontological morphogenetic mechanism. For over a century, the quantum framework has been able to answer “what is the distribution of orbitals,” but has never been able to answer: why must orbitals exhibit their specific geometric forms? Where do the structural constraints originate? How are wave-particle modalities ontologically generated? Grounded in the PFUSRC axiomatic system of the 45° triple coaxial bicone as the unique steady-state geometry of the cosmos, this paper completes a topological ontological reconstruction of microscopic atomic structure. It rigorously demonstrates that the atom is not a spherically symmetric stochastic probability system, but rather a microscopically solidified 45° triple coaxial biconical interlayer oscillation steady-state structure. All s/p/d/f orbital petal morphologies, spatial nodal surfaces, energy level splittings, and symmetry breakings are not solutions of equations, but geometric necessities generated by biconical nested topology, interlayer angle constraints, and the 12/11 boundary gauge ratio. This paper rigorously distinguishes between two cognitive levels: quantum mechanics belongs to the observational fitting, statistical representation, and mathematical approximation system; PFUSRC atomic topological morphogenesis belongs to the structural origin, topological generation, and ontological mechanism system. The two are fully data-compatible, completely nested in hierarchy, and entirely non-isomorphic in ontology—there is no conflict, only a paradigmatic gap between “representational fitting” and “ontological generation.” This paper further clarifies: the electron is neither a classical particle nor a mechanical wave, but a steady-state node of the four-dimensional flowvariable field topologically latched at the three-dimensional interface. Wave-particle duality is a projective observational effect, not an intrinsic property of the electron. The spherical symmetry of the s-orbital is a time-averaged observational illusion; the instantaneous intrinsic structure of the atom is always a directional biconical configuration. The atom possesses an inherent topological density gradient: dense along the conical axis, sparse in lateral regions the projective distribution is inherently non-uniform. The uniform orbital image in mainstream textbooks is the result of long-term integration smoothing. This paper introduces the 2021 Science σ-hole subatomic direct imaging experiment and the homonuclear diatomic laser-induced dissociation electron localization experiment, demonstrating from real-space observation that the atom exhibits asymmetric conical charge distribution and axial–lateral density magnitude differences, in full agreement with the topological projection predictions of the triple biconical model—piercing through the theoretical blind zone of mainstream symmetric assumptions. This paper redefines the essence of chemical bonding: chemical bonds are not electron orbital overlap couplings, but topological alignment, anchor interlocking, and steady-state co-construction between atomic biconical interlayers. The electron serves only as an interfacial marker node of the latched structure. Four experimentally falsifiable unique topological predictions are proposed, completing the full paradigmatic transition from mathematical fitting and phenomenological description to geometric generation and experimental verification in microscopic atomic systems.