This paper proposes and validates a revolutionary foundational physics framework, demonstrating for the first time that quantum mechanics can naturally emerge from pure principles of geometric constraints. We establish a two-level unified theory: at the top level is the Relation Matrix R_{ij} describing the abstract relational network of the universe, and at the bottom level is the Minimal Phase Closed Triangle (MPCT) network serving as the physical implementation unit. By introducing a curvature constraint action, we rigorously derive the Schrödinger equation, quantization conditions, and Planck's constant, and for the first time directly calculate the fine-structure constant from geometric principles. Core Innovations: Proposed the new paradigm "Existence is relation, relation is geometry," defining fundamental physical entities as relations rather than particles. Established an exact mathematical mapping between the Relation Matrix R and the MPCT network. Derived quantum mechanics from the curvature constraint action \mathcal{I} = \int [L_c + \frac{\lambda}{2}(\kappa - \kappa_R)^2]dt. Discovered the geometric compression factor \eta(n,l,m) and its scaling law \eta \propto n^{0.67}. Implemented strict curvature calculations based on the relation matrix, verifying \eta \propto (\text{Var}(\kappa_R))^{0.15}. Numerical simulations of the hydrogen atom system confirmed the quantized distribution of MPCT areas, discovering the new scaling law \Delta A \propto n^{3.55}. Theoretical predictions agree with numerical results within 3.4% accuracy, with the fine-structure constant calculation error at 15.2%. This work provides a novel geometric perspective for understanding the essence of quantum phenomena and indicates a possible path toward unifying quantum mechanics and gravity. Keywords: Foundations of quantum mechanics, Geometric constraints, Relation matrix, Minimal Phase Closed Triangle, Fine-structure constant, Hydrogen atom, Scaling laws, Emergent phenomena
This paper proposes and validates a revolutionary foundational physics framework, demonstrating for the first time that quantum mechanics can naturally emerge from pure principles of geometric constraints. We establish a two-level unified theory: at the top level is the Relation Matrix R_{ij} describing the abstract relational network of the universe, and at the bottom level is the Minimal Phase Closed Triangle (MPCT) network serving as the physical implementation unit. By introducing a curvature constraint action, we rigorously derive the Schrödinger equation, quantization conditions, and Planck's constant, and for the first time directly calculate the fine-structure constant from geometric principles. Core Innovations: Proposed the new paradigm "Existence is relation, relation is geometry," defining fundamental physical entities as relations rather than particles. Established an exact mathematical mapping between the Relation Matrix R and the MPCT network. Derived quantum mechanics from the curvature constraint action \mathcal{I} = \int [L_c + \frac{\lambda}{2}(\kappa - \kappa_R)^2]dt. Discovered the geometric compression factor \eta(n,l,m) and its scaling law \eta \propto n^{0.67}. Implemented strict curvature calculations based on the relation matrix, verifying \eta \propto (\text{Var}(\kappa_R))^{0.15}. Numerical simulations of the hydrogen atom system confirmed the quantized distribution of MPCT areas, discovering the new scaling law \Delta A \propto n^{3.55}. Theoretical predictions agree with numerical results within 3.4% accuracy, with the fine-structure constant calculation error at 15.2%. This work provides a novel geometric perspective for understanding the essence of quantum phenomena and indicates a possible path toward unifying quantum mechanics and gravity. Keywords: Foundations of quantum mechanics, Geometric constraints, Relation matrix, Minimal Phase Closed Triangle, Fine-structure constant, Hydrogen atom, Scaling laws, Emergent phenomena
This paper presents, for the first time, a rigorous derivation of the Schrödinger equation of quantum mechanics from the eight ontological axioms of the Minimal Phase Closure Triangle (MPCT). We prove: 1) The continuum limit of an MPCT network naturally produces a complex wavefunction field ψ(x,t); 2) The variational principle of MPCT phase dynamics is equivalent to the Schrödinger action principle; 3) The quantum of action ħ emerges from the minimal phase residue of MPCT closure constraints; 4) The mass parameter m is determined by the coupling strength K of the MPCT network. This derivation provides a geometric and topological foundation for quantum mechanics and establishes a profound connection between ontology and quantum theory. Keywords: Minimal Phase Closure Triangle; Schrödinger equation; first principles; foundations of quantum mechanics; emergence theory
This paper presents a novel computational framework, MPCT-τ, that demonstrates theemergence of hydrogen atom quantum orbitals from first principles using relational matrixevolution. Unlike traditional quantum mechanical approaches that start with predefinedwavefunctions, our method begins with a relational (density) matrix and evolves it throughMinimal Phase Closure Triangles (MPCT) operations. The framework successfully reproduces key quantum features: radial probability distributions with correct nodal structures(n−l−1nodes), peak positions following the n2 rule with less than 5% average error, andwavefunction similarities exceeding 99.9% compared to exact solutions. The MPCT closure mechanism provides a physical interpretation of quantum coherence formation, whiletarget guidance terms accelerate convergence. This work provides computational evidencesupporting relational interpretations of quantum mechanics, where quantum states emergefrom fundamental relational principles rather than being primitive entities
This paper presents a novel computational framework for electron cloud deter-mination based on the Minimal Phase Closure Triangle (MPCT) method,fundamentally distinct from traditional quantum mechanical approaches. Unlikethe Schr¨odinger equation-based methods that rely on wavefunctions and probabil-ity amplitudes, our approach derives electron distributions directly from the geo-metric properties of phase space trajectories. We introduce a two-level theoreticalfoundation: at the abstract level, a relational matrix describes quantum systemsthrough their fundamental interrelationships; at the implementation level, MPCTnetworks provide concrete geometric realizations. The method successfully com-putes hydrogen atom electron clouds with remarkable accuracy: Bohr radius de-termination within 1.9% error, n2 law verification within 0.6% error, and spectralline prediction with 0.1% accuracy. This framework offers a geometric-first alter-native to conventional quantum computation, potentially opening new avenues forquantum system analysis and material design.Keywords: Electron cloud computation; Minimal Phase Closure Triangle; Phasespace geometry; Relational quantum mechanics; Geometric quantum computation;MPCT method
This work presents a unified theoretical framework that aims to bridge relativity and quantum mechanics through a single structural principle: delayed realization. Instead of assuming spacetime, time evolution, or quantum states as primitive entities, the framework introduces a pre-geometric definability network in which relational differences are initially undefined and become stabilized through nonlinear dynamics. The emergence of spacetime is then reinterpreted as the stabilization of definable differences, while the present is not given a priori but selected via delayed consistency over histories. Within this formulation, quantum phenomena arise naturally when multiple realization candidates coexist, producing interference structures governed by a variational principle. In the low-defect limit, the framework recovers effective relativistic geometry and field-theoretic behavior, including emergent gravitational curvature and gauge structures. A central contribution of this work is the formulation of a unified definability action, combining stabilization, curvature minimization, delayed consistency, and informational structure into a single variational framework. Under appropriate coarse-graining, this action yields structures consistent with Einstein field equations, quantum amplitudes, and gauge invariance as different regimes of the same underlying mechanism. This paper does not claim a fully closed mathematical unification of all physical sectors. Instead, it provides a coherent structural framework in which spacetime emergence, present realization, quantum probability, and effective field dynamics are organized as manifestations of a single principle. The framework is intended as a foundational step toward a deeper unification of physics, where time, geometry, and probability are not assumed, but emerge from the stabilization and realization of definable relational structures.
The quantization of action, discrete energy levels, and finite phase-space resolution are among the most fundamental yet conceptually opaque features of quantum physics. In standard theory, the universal constant is introduced empirically, and canonical quantization rules are postulated without deeper ontological justification. In this work we present a first-principles derivation of quantization based on the Structural Unified Field (SUF) framework. By treating physical entities as layered “existence objects” characterized by a boundary parameter and an existence function , we argue that structural identity requires a minimal nonzero layer separation . This structural irreducibility implies a universal minimal distinguishability in all attribute domains, which, when projected into phase space, yields a minimal phase-space cell and thus a universal minimal action unit. Upon mapping to physical units, this structural minimal action corresponds to Planck’s constant . The SUF interpretation thereby promotes quantization from an empirical postulate to a necessary consequence of the ontological geometry of existence. The resulting framework naturally accounts for discrete spectra, uncertainty relations, action quantization, canonical commutation relations, topological quantization phenomena, and the universality of . We discuss broader implications for the foundations of quantum theory, the quantum–gravity interface, and potential avenues for testing structural deviations.
This paper proposes a fundamental framework based on information geometry that unifies quantum mechanics, thermodynamics, and spacetime geometry into a single foundational theory. Starting from three elementary axioms regarding information as the primary physical ontology, we derive the universal information metric, construct the information action functional, and show that all known physical laws emerge as extremal conditions of this functional. The theory recovers standard quantum mechanics, classical thermodynamics, and general relativity in appropriate limits, while predicting new relations between quantum entanglement, gravitational curvature, and entropy. We conclude that all physical phenomena—matter, energy, spacetime, and interactions—are equivalent to geometric structures on the space of quantum information states.