DOI: 10.5281/zenodo.17010049
title: Singularity as Foundation : The Geometry of Existence

description:  Singularity as Foundation: Fundamental Laws of the Observed Universe

Author: Maksym Marnov  ORCID: [0009-0000-0832-9597](https://orcid.org/0009-0000-0832-9597)  GitHub: MaksymMarnov](https://github.com/MaksymMarnov)  Repository: [Singularitat-als-Grundlage](https://github.com/MaksymMarnov/Singularitat-als-Grundlage)  License: CC BY 4.0  Version: 1.0  Date:  August 29, 2025  

Abstract:This work presents a structured corpus of fundamental laws governing physical phenomena in the observed universe, from quantum scales to cosmological structures. The corpus integrates mathematical formulations with conceptual definitions, providing a unified framework for understanding singularity, stability, quantization, and gravitational retention.

 Constants:- \( c \): speed of light in vacuum- \( G \): gravitational constant- \( h \): Planck constant- \( \hbar \): reduced Planck constant (h-bar)- \( k_B \): Boltzmann constant

Fundamental Laws:

 1. Law of the Point (Singularity)Definition: A point defines zero volume measure and acts as a limiting boundary condition/source (delta distribution) for fields and potentials.

Mathematical Formulation:- \( \nabla^2 \Phi(\mathbf{r}) = 4\pi G M \delta^3(\mathbf{r}) \)- \( \Phi(r) = -\frac{GM}{r} \)- \( \lim_{V \to 0} \int_V \rho  dV = M_{\text{point}} \)

Notes:Point ↔ δ-function; core of field theory formulations and Newtonian/Poisson gravity.

 2. Law of Minimal Stable FormDefinition: The minimally rigid configuration in 2D is a triangle; in 3D it is a tetrahedron. These are the foundations of minimal stability for nodes/lattices.

Mathematical Formulation:- Planar rigidity (Laman): \( m = 2n - 3 \) (minimal rigidity); \( n = 3 \Rightarrow m = 3 \) (triangle)- Spatial rigidity (Maxwell): \( m \geq 3n - 6 \); \( n = 4 \Rightarrow m = 6 \) (tetrahedron)- Tetrahedron properties: \( \{F, E, V\} = \{4, 6, 4\} \)

References: Laman (2D minimal rigidity), Maxwell's rule (3D DOF count).

 3. Law of Energy QuantizationDefinition: Oscillator energy is quantized; the quantum is the minimal excitation portion.

Mathematical Formulation:- Planck-Einstein: \( E = h f \)- Harmonic oscillator: \( E_n = \left(n + \frac{1}{2}\right) \hbar \omega \), \( n \in \mathbb{N}_0 \)- de Broglie: \( \lambda = \frac{h}{p} \)- Frequency from tension: \( \omega = \sqrt{\frac{k}{m}} \)

Notes: Wave-particle duality follows from standing/traveling modes and spectrum discreteness.

 4. Law of Similarity (Self-Similarity)Definition: Structures preserve invariants under scaling; properties follow power laws.

Mathematical Formulation:- Box-counting dimension: \( N(\epsilon) \sim \epsilon^{-D} \Rightarrow D = -\lim_{\epsilon \to 0} \frac{\ln N(\epsilon)}{\ln \epsilon} \)- Scaling law: \( F(\alpha x) = \alpha^{-\beta} F(x) \)- Renormalization group hint: \( g(\ell) \) transforms under RG flow as scale \( \ell \) changes

Notes: Fractal dimension \( D \) and scaling laws describe cascades from micro to macro.

 5. Law of Energy Retention (Gravity)Definition: Gravity is a consequence of energy/mass curving spacetime; retention = negative binding energy.

Mathematical Formulation:- Newton-Poisson: \( \nabla^2 \Phi = 4\pi G \rho \)- Einstein field equations: \( G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \)- Binding energy (uniform sphere): \( U \approx -\frac{3GM^2}{5R} \)- Virial theorem: \( 2\langle K \rangle + \langle U \rangle = 0 \) (stationary bound state)

Notes: Binding energy \( U < 0 \) is a quantitative measure of gravitational "retention".

6. Law of Planetary Formation (Stable Resonant Node)Definition: Planets form via gravitational instability and accretion in disks: compression, coagulation, pebble/planetesimal growth, anchored by stability and angular momentum.

Mathematical Formulation:- Jeans length: \( \lambda_J = c_s \sqrt{\frac{\pi}{G\rho}} \)- Toomre Q parameter: \( Q = \frac{c_s \kappa}{\pi G \Sigma} < 1 \) ⇒ gravitational disk instability- Hill radius: \( r_H = a \left( \frac{M_p}{3M_*} \right)^{1/3} \)- Accretion rate: \( \frac{dM_p}{dt} \approx \pi R_{\text{eff}}^2 \Sigma v_{\text{rel}} \)- Angular momentum: \( L = m r^2 \Omega \) (conservation during accretion)

References: Jeans (gravitational instability), Toomre Q (disk instability), Hill (sphere of influence).

 7. Law of the Retention Limit (Black Hole)Definition: At \( r \leq r_s \), the retention field becomes absolute: \( v_{\text{esc}} \geq c \); an event horizon forms.

Mathematical Formulation:- Schwarzschild radius: \( r_s = \frac{2GM}{c^2} \)- Collapse condition: \( v_{\text{esc}} = \sqrt{\frac{2GM}{r}} \geq c \Rightarrow r \leq r_s \)- Chandrasekhar limit: \( M_{\text{Ch}} \approx 1.44 M_\odot \) (limit for degenerate electron gas)- TOV limit: \( M_{\text{TOV}} \sim 2-3 M_\odot \) (neutron star limit, model-dependent)- Bekenstein-Hawking entropy: \( S = \frac{k_B c^3 A}{4G\hbar} \), \( A = 4\pi r_s^2 \)- Hawking temperature: \( T_H = \frac{\hbar c^3}{8\pi G M k_B} \)

References: Schwarzschild (r_s), Chandrasekhar (limit), TOV, Bekenstein-Hawking (entropy and temperature).

 Chain of UnificationThe laws form a coherent narrative across scales:1. Point-source (δ) sets field boundary conditions2. Minimal form (△/tetrahedron) provides minimal nodal/lattice rigidity3. Node oscillations → frequencies; energy quantization (Planck/oscillator)4. Self-similarity scales nodes → fractal ensembles5. Energy retention (U < 0) is geometrized as curvature (Einstein)6. In disks: instabilities (Jeans/Toomre) + accretion → planets7. Retention limit at r ≤ r_s → black holes with area-entropy and Hawking temperature

 LicenseThis work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).

How to CiteMarnov, M. (2025). Singularity as Foundation: Fundamental Laws of the Observed Universe. Zenodo. `https://doi.org/10.5281/zenodo.17010049