The stability of hierarchical structures in nature, such as atomic electron shells, has long been explained by quantum mechanics as a consequence of electron orbitals and the Pauli exclusion principle. However, the Pauli exclusion is a phenomenological rule, not a fundamental cause. This paper proposes a novel perspective: the existence of a Minimal Closure Scale, denoted τ (tau), which constitutes a necessary condition for the self-consistent closure of a system. This concept is independent of quantum postulates and Hilbert space formalism. We articulate the origin of hierarchical stability and closed-loop incompatibility through the incompressibility of a fundamental difference Δ₀ and a framework termed Minimal Path Closure Theory (MPCT). We further suggest pathways for approximate validation using known physical constants and data
The stability of hierarchical structures in nature, such as atomic electron shells, has long been explained by quantum mechanics as a consequence of electron orbitals and the Pauli exclusion principle. However, the Pauli exclusion is a phenomenological rule, not a fundamental cause. This paper proposes a novel perspective: the existence of a Minimal Closure Scale, denoted τ (tau), which constitutes a necessary condition for the self-consistent closure of a system. This concept is independent of quantum postulates and Hilbert space formalism. We articulate the origin of hierarchical stability and closed-loop incompatibility through the incompressibility of a fundamental difference Δ₀ and a framework termed Minimal Path Closure Theory (MPCT). We further suggest pathways for approximate validation using known physical constants and data
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 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 introduces Axiom 0, a minimal foundational principle for physical reality: the impossibility of any globally closed description of the total correlational structure. Rather than postulating space, time, dynamics, locality, or physical laws as primitive, the framework assumes only: (i) the existence of a correlational structure, (ii) the structural impossibility of complete self-closure, and (iii) consistency under partial comparison. From these ingredients, the paper shows how: physical laws emerge as stabilizing regularities of incomplete descriptions, time, dynamics, and geometry arise as effective, regime-dependent structures, irreversibility follows from projection and information loss, and why no description can be both exact and globally complete. Non-closure is treated as an ontological constraint, not as epistemic ignorance or observer dependence. Physical reality exists precisely because complete closure is impossible, independently of observation or modeling. The paper does not attempt to replace existing physical theories or derive numerical constants. Instead, it provides a structural foundation within which effective physical theories (quantum, classical, relativistic, and open-system regimes) can be consistently understood as emergent and domain-limited. This work is intended as a conceptual and structural foundation for correlational and relational approaches to fundamental physics, clarifying the origin of laws, dynamics, and physical regularity without invoking additional ontological postulates.
Foundational problems in physics—including quantum measurement, spacetime emergence, gravitational dynamics, nonlocality, the arrow of time, and the status of physical existence—are typically addressed in isolation, via model-dependent extensions or interpretive assumptions. This work proposes a different approach. We introduce a constraint-based framework, the Closure Admissibility Principle (CAP), which specifies minimal structural conditions any physical description must satisfy in order to be globally consistent. CAP is formulated in operational terms, using admissibility of states, recoverability under refinement, and stability of closed records, without assuming spacetime, dynamics, or ontology a priori. We show that enforcing these constraints induces a unifying closure structure on law-space, from which several persistent foundational puzzles are resolved as structural consequences rather than postulates. In particular, collapse is reclassified as record selection, spacetime and gravity emerge only in compressible regimes, General Relativity appears as an infrared universality class rather than a fundamental law, singularities correspond to failures of admissible description, and null (“nothing”) models are excluded as inadmissible fixed points. The framework is explicitly falsifiable via operational and weak-field tests, and its scope and limits are stated precisely. The results suggest that many foundational paradoxes arise not from incomplete dynamics, but from admitting globally incoherent physical descriptions.
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.
Atomic orbitals are traditionally described through a set of quantum numbers arising from solutions to the Schrödinger equation under Coulombic boundary conditions. While these quantum numbers are well established and empirically validated, they are typically treated as independent formal labels rather than as manifestations of a unified structural principle.This paper proposes that atomic orbital structure is governed by three distinct but harmonically unified classes of boundary constraint: radial recursion, angular recursion, and closure (phase/spin) recursion. These constraints correspond respectively to the principal, angular, and spin quantum numbers, and together form a triadic harmonic structure that determines admissible orbital configurations.This interpretation does not modify quantum mechanics or introduce new mechanisms; rather, it reframes existing quantum numbers as structural descriptors of standing-wave closure under constrained recursion. The triadic structure is shown to arise naturally from minimal requirements for stable, self-consistent wave confinement and offers a unifying perspective on orbital families, degeneracy, and constraint hierarchy.