Closed Spherical System: A 9:3:1
Closed Spherical System: A 9:3:1 Proportion Calibration Framework for Gravity–Quantum Unification
Abstract
This framework proposes that the universe is a closed, conservative spherical system governed by a converging 9:3:1 energy proportion rule, unifying general relativity and quantum mechanics on the common foundation of dynamic tension–proportion calibration. Dark energy is the macroscopic manifestation of global boundary tension, dark matter forms the structural skeleton sustained by this tension, and ordinary matter arises from ordered excitations of fields under hierarchical constraints. This approach eliminates artificial assumptions including the fixed cosmological constant and infinitely divisible spacetime, achieves logical consistency between the two theories, matches all confirmed observational results, and provides quantitatively testable predictions.
1. Fundamental Axioms
1. Closed Conservation: The universe is a finite closed system with fixed total energy and topological constraints; energy only transforms between forms and scales, with no creation, destruction, or net gain or loss.
2. Spherical Steady State: Under Poincaré topological constraints, the system naturally evolves toward a spherical equilibrium state, with energy hierarchies scaling by a factor of 3, forming a fractal nested structure of 27:9:3:1.
3. Proportion Convergence: The global energy density fractions will eventually converge to the steady-state ratio dark energy : dark matter : ordinary matter = 9:3:1. Current observational values reflect intermediate states during the ongoing calibration process.
4. Irreversible Causality: Structure precedes matter: boundary tension → skeletal topology → field stratification → particle generation → macroscopic structure. Causality proceeds from global to local and large to small scales, and cannot be reversed.
1. General Relativity: Dynamic Cosmological Constant and Proportion Calibration
2.1 Limitations of the Standard Field Equation
The standard Einstein field equation is written as:
G_{\mu\nu} = 8\pi G,T_{\mu\nu} + \Lambda_{\text{const}},g_{\mu\nu}
This formulation treats the cosmological constant \Lambda as a fixed global parameter, decoupling the evolution of large-scale structure from local physical conditions. It fails to resolve tensions such as the Hubble discrepancy and evolving component ratios, and exhibits divergences in extreme regimes.
2.2 Core Revision: Dynamic Proportion–Tension Equation
The fixed cosmological constant is replaced by a dynamic tension term responsive to deviations from the steady-state proportion:
\boldsymbol{G_{\mu\nu} = 8\pi G,T_{\mu\nu} + \Lambda_{\text{dynamic}}(r,t),g_{\mu\nu}}
where the dynamic cosmological constant is defined as:
\boldsymbol{\Lambda_{\text{dynamic}}(r,t) = \Lambda_0 \cdot \frac{\mathcal{T}{\text{global}}(t)}{\sqrt{\left(\frac{\Omega\Lambda}{\Omega_{\Lambda,\text{ss}}}-1\right)^2 + \left(\frac{\Omega_m}{\Omega_{m,\text{ss}}}-1\right)^2 + \left(\frac{\Omega_b}{\Omega_{b,\text{ss}}}-1\right)^2} + \epsilon}}
Notation and Physical Meaning
• G_{\mu\nu}: Einstein tensor; T_{\mu\nu}: energy–momentum tensor; g_{\mu\nu}: metric tensor; G: Newtonian gravitational constant; \Lambda_0: current observed baseline value of the cosmological constant;
• \Omega_\Lambda,\Omega_m,\Omega_b: current energy density fractions of dark energy, dark matter, and ordinary baryonic matter, respectively;
• \Omega_{\Lambda,\text{ss}}:\Omega_{m,\text{ss}}:\Omega_{b,\text{ss}} = 9:3:1, the steady-state proportion;
• \mathcal{T}_{\text{global}}(t): global boundary tension of the closed system as a function of evolutionary stage;
• \epsilon \ll 1: small numerical stabilizer to prevent division by zero.
Physically, greater deviation of local proportions from the steady state produces a stronger response in \Lambda_{\text{dynamic}}, accelerating calibration. As proportions converge to equilibrium, \Lambda_{\text{dynamic}} settles to a stable value without requiring any fixed constant.
1. Quantum Mechanics: Natural Hierarchy and Global Coupling
3.1 Minimum Calibration Scale: Automatic UV Regularization
The assumption of infinitely divisible spacetime is abandoned. The 9:3:1 topological constraint naturally defines a minimum resolvable length scale:
\boldsymbol{l_{\text{min}}(t)^2 = l_P^2 \cdot \frac{k(t)}{13},\quad l_P = \sqrt{\frac{\hbar G}{c^3}}}
• l_P: Planck length; \hbar: reduced Planck constant; c: speed of light in vacuum;
• k(t): time-dependent topological coefficient (early universe k(t)\approx1, converging to unity in the final steady state);
• All quantum integrals are naturally truncated at l_{\text{min}}(t), eliminating ultraviolet divergences without extra dimensions, supersymmetry, or ad hoc renormalization.
3.2 Hierarchical State Counting: Consistency with Pauli Exclusion
Under spherical symmetry and proportion constraints, the number of allowed quantum states scales as a power of 3:
\boldsymbol{N_n \propto 3^n \quad (n=0,1,2,\dots)}
This yields the sequence N=1,3,9,27,\dots, consistent with the Pauli exclusion principle and the three-layer skeleton of dark matter structure.
3.3 Global Tension Coupling Term
A unified calibration coupling is added to the Standard Model Lagrangian to connect microscopic dynamics to macroscopic evolution:
\boldsymbol{\mathcal{L}{\text{total}} = \mathcal{L}{\text{SM}} + \alpha \cdot \mathcal{T}{\text{global}}(t) \cdot \mathcal{F}{\text{prop}}(r,t)}
• \mathcal{L}_{\text{SM}}: Standard Model Lagrangian;
• \alpha: dimensionless coupling constant; \mathcal{F}_{\text{prop}}(r,t): function describing local deviation from the steady-state proportion;
• Quantum fluctuations are no longer purely random, but arise as local coherent responses to global system calibration.
1. Core Unification of Gravity and Quantum Mechanics
General relativity and quantum mechanics are linked through the ratio of scales and tension:
\boldsymbol{\frac{G_{\text{eff}}(r,t)}{G} = \frac{l_{\text{min}}(t)}{\lambda_{\text{quantum}}} \cdot \frac{\mathcal{T}{\text{local}}(r,t)}{\mathcal{T}{\text{global}}(t)}}
Physical Interpretation
• Macroscopic gravity is the gradient of balance between global boundary tension and local structural pressure, an emergent effect of deformation in the cosmic skeleton;
• Microscopic quantum interactions are the transmission and confinement of energy along the tension skeleton, a manifestation of local hierarchical alignment;
• Both interactions originate from the same underlying principles, differing only in characteristic scale and hierarchical level, and can be solved self-consistently within one framework.
1. Testable Predictions
1. Dark energy equation of state: w \approx -0.97 \pm 0.02, not fixed at -1, with small time-dependent fluctuations as calibration proceeds;
2. Mass gap: The abundance of compact objects between 2.5 and 5 solar masses is significantly suppressed, as this range corresponds to an unstable transition zone in proportion alignment;
3. Early supermassive black holes: At high redshift (z>10), the fraction of supermassive black holes is higher than predicted by the standard \LambdaCDM model;
4. Proportion near black holes: Immediately outside the event horizon, the ratio of dark matter to ordinary matter approaches 1:1, rising gradually toward the asymptotic 3:1 at large distances;
5. Cosmic curvature and Hubble tension: The universe has a small net positive curvature. The Hubble tension arises from the difference in calibration progress between the local and global universe, and will gradually diminish over time.
1. Compatibility with Established Physics
• All weak-field and stationary-state tests of general relativity are fully preserved, including gravitational lensing, gravitational redshift, and Solar System constraints;
• All confirmed results of quantum mechanics and the Standard Model remain valid, including particle interactions, interference effects, and quantum statistics;
• Only incorrect underlying assumptions are corrected, without altering established computational methods, so the framework can be directly integrated into existing numerical simulations and observational pipelines.