This analysis provides a straightforward explanation for the origin of the universe and its contents. It accounts for key PDG 2025 data [1] without introducing mathematical singularities or infinities. If simpler explanations are developed in the future, this analysis will be superseded.

The analysis has three important implications:

1) Continuum mathematics, with its associated infinities and singularities, only approximates an underlying finite and discrete mathematics describing the universe;

2) Classical general relativity is not the best way to account for origin of the universe; and

3) The Standard Model of particle physics can be extended by abandoning the point particle approximation.

Strong evidence shows the universe began 13.8 billion years ago as a small, closed space at high temperature that expanded and cooled since then. At a fundamental level, information specifies the distribution of matter within the universe, and a tremendous amount of information is required to describe structures, ranging from atoms to clusters of galaxies, in our three spatial dimensions.

Holographic analyses [2] (based on quantum mechanics, general relativity, black hole thermodynamics, and Shannon information theory) show that only a finite number of bits of information on the event horizon will ever be available to describe the universe and its contents. That implies the universe is a closed system described by discrete mathematics.

The event horizon at distance R H = 3 Λ = 1.661 × 10 28   cm from any observer’s location is the farthest distance observers can ever see out into our vacuum-dominated universe with cosmological constant Λ = 1.088 × 10 − 56   cm − 2 . Using PDG 2025 data, mass of the observable universe inside the event horizon

with Hubble constant H 0 = 67.4   km ⋅ sec − 1 ⋅ Mpc − 1 , critical energy density ρ c r i t = 3 H 0 2 8 π G g / cm 3 = 8.533 × 10 − 30 g / cm 3 , gravitational constant G = 6.67430 × 10 − 8   cm 3 ⋅ g − 1 ⋅ sec − 2 , and vacuum energy fraction Ω Λ = 0.685 . Holographic analysis indicates the bits of information describing systems with definite mass m within the universe are available on spherical surfaces surrounding the system with radius r = m M H R H , so holographic radii of objects with definite mass m are r = m 0.187   g / cm 2 .

With Planck length l P = ℏ G c 3 = 1.61625 × 10 − 33   cm , holographic analysis finds only N = ( π ln ( 2 ) ) ( R H l P ) 2 = 4.741 × 10 122 bits of information encoded on areas of l P 2 on the event horizon will ever be available to describe the observable universe within the event horizon. Half the bits on the horizon provide information associated with bits of matter within the horizon and the other half provide information associated with bits of anti-matter within the horizon. Mass of bits of matter within the universe is m b i t = M H 2.371 × 10 122 = 2.17 × 10 − 67   g .

ΛCDM cosmology, the current standard model of cosmology, is the simplest general relativistic theory providing a good account of the universe. Radius R of our closed, homogeneous, and isotropic three-dimensional space is specified by Friedmann’s equation

with today’s radiation energy density ρ r 0 and matter energy density ρ m 0 , including cold (slow-moving) dark matter. Vacuum energy (dark energy) density ρ V constitutes 68.5% of total energy density in our three dimensions. Negative pressure associated with vacuum energy density accelerates the observed expansion of our three dimensions. Invisible cold dark matter, constituting 26.5% of total energy density and 98.1% of matter density in our three dimensions, interacts only by gravitation and binds visible matter in stars into galaxies and galaxy clusters.

General relativity is not reliable at distances less than l P , where quantum mechanical effects become important, so general relativity cannot explain our universe originating as a state with radius R < l P or a “singularity” with radius R = 0 . In contrast, this quantum cosmology explains the universe as originating with R > l P .

Friedmann’s equation multiplied by 1/2

describes a unit mass particle at coordinate R moving in one dimension in potential

with zero total energy. Hamiltonian quantization of Friedmann’s equation [3] results in the Schrodinger equation

for wavefunction ψ ( R ) specifying radius R of our closed three-dimensional space, with total energy E .

The universe can be described as a product space composed of our three closed space dimensions with radius R and a compact space with radius r . The equation

describes a unit mass particle at coordinate r moving in one dimension in potential

with total energy E . The corresponding Schrodinger equation

determines wavefunction ψ ( r ) specifying radius r of the compact closed space. Describing the universe as a product space with wavefunction ψ ( R ) ψ ( r ) allows origin of the universe by a quantum fluctuation from nothing, with expectation value 〈 R 〉 〈 r 〉 .

The universe, a product space with wavefunction ψ ( R ) ψ ( r ) , probably originated by quantum fluctuation from nothing. Nothing (the absence of any thing, or any information) is a homogenous, isotropic state of indefinite size with zero energy and zero angular momentum. The quantum fluctuation producing the universe created an unstable initial state, resulting in the inflationary three-dimensional space we inhabit today and all of its contents.

In order of decreasing strength, forces acting today in the universe are the strong force, the electromagnetic force, the weak force, and gravity. Long-range forces are gravity, important at macroscopic scales, and electromagnetic force important at atomic scales. Planck mass M P = ℏ c G = 2.176 × 10 − 5   g , characteristic of gravity, has quantum mechanical Compton wavelength l P = ℏ M P c = ℏ G c 3 . Electromagnetic force is characterized by electron mass m e = 9.109 × 10 − 28   g , the lowest stable mass in the universe, and electron Compton wavelength is l e = ℏ m e c . If those forces were unified when the universe originated, Planck mass equaled electron mass, and quantum wavelength of the universe’s initial state was

Origin of the universe by quantum fluctuation from nothing 13.8 billion years ago might seem unlikely, but it only had to occur once. If other universes originated in the same way at other times, they are profoundly elsewhere. We could never have contact with such universes, so they are scientifically irrelevant.

This description of the universe identifies time with quantum mechanical time, classical time, and cosmic time in a Friedmann universe. It allows for quantum fluctuations extending indefinitely into the past with no beginning of time and no “problem of time”. In contrast, general relativity describes the relative duration of events observed in different systems moving relative to each other within the universe.

In the initial state of the universe, ψ ( R ) specified radius R of a closed three-dimensional space filled with electromagnetic radiation, with ρ r = ρ r 0 ( R H 4 l e 2 ) . Since ψ ( R ) is zero at R = 0 [4], the universe could not begin as a point singularity with zero radius. The quantum mechanical equation

with E = 0 allows origin of the universe by a quantum fluctuation from nothing into a product space with wavefunction ψ ( R ) ψ ( r ) , radius l e , energy E = 0 and d R d t = d r d t = 0 .

Immediately after the universe originated as an unstable product space, radius of the compact space collapsed from radius l e to l P , inflating our three space dimensions by a factor of 2.389 × 10 22 from radius l e to R I = 9.223 × 10 11   cm . When the universe originated, energy of our three dimensions was − ( 4 π G 3 c 2 ) ρ r 0 ( R H 4 l e 2 ) and energy of the compact space was ( 4 π G 3 c 2 ) ρ r 0 ( R H 4 l e 2 ) . Inflation duration T I is determined by ( d r d t ) 2 + K r ( r − l P ) 2 + c 2 = E . At universe origin, ( d r d t ) = 0 , r = l e = 3.861 × 10 − 11   cm , E = ( 4 π G 3 c 2 ) ρ r 0 ( R H 4 l e 2 ) , K r = [ ( 4 π G 3 c 2 ) ρ r 0 ( R H 4 l e 2 ) − c 2 ] / ( l e − l P ) 2 , so

T I is much less than Planck time 5.39 × 10 − 44 sec identified by Heisenberg’s uncertainty principle as the shortest measurable time interval, so inflation happened almost instantly after the universe began.

Energy in the compact space after inflation, maintaining compact space radius at l P , is now the constant dark energy accelerating today’s expansion of our three space dimensions. Collapse of the compact space injected energy into our three dimensions that became matter. If bits of matter have slightly lower energy than bits of anti-matter, many more bits of matter than anti-matter were produced when energy was injected into our three space dimensions by compact space collapse. When almost all of the tremendous number of matter and anti-matter bits annihilated to two photons, only matter was left.

Rigid spheres of dark matter with mass M and radius r have self-interaction cross-section σ = 4 π r 2 . Observations of eight galaxy clusters [5] find σ M = 0.082 cm 2 g , so r = 0.082 M 4 π and dark matter particle density ρ = M 4 3 π r 3 = 45 M g / cm 3 . Rigid spheres of dark matter with density equaling matter density 1.4 × 10 19 g / cm 3 in our three space dimensions immediately after inflation have mass M = 6.7 × 10 − 26   g = 74 m e = 38   MeV / c 2 .

Formation of rigid impenetrable spheres of dark matter involves Friedmann’s equation

defining radius R of closed systems bound by effective gravity with gravitational constant s G , and its Hamiltonian quantization to Schrodinger’s equation

for wavefunction ψ ( R ) with total energy E . Schrodinger’s equation with gravitational constant s G , has the mathematical form of Schrodinger’s equation for hydrogen atoms and solving it results in 〈 R 〉 = s G M π c 2 , ρ = M / [ 4 π 3 ( s G M π c 2 ) 3 ] , and s = c 2 G [ 3 ρ ( π M ) 2 ] 1 3 = 1.8 × 10 12 .

Schwarzschild black holes with mass M have radius r S = 2 G M c 2 , density ρ S = 3 c 6 32 π G 3 M 2 , and Hawking temperature T = ℏ c 3 8 π G M k , so rigid impenetrable spheres of dark matter result in a minimum black hole mass in today’s universe. An aggregate of close-packed impenetrable spheres, each with density 1.4 × 10¹⁹ g/cm³, has density 0.74 ( 1.4 × 10 19 g / cm 3 ) = 1.0 × 10 19 g / cm 3 . In today’s universe, the minimum mass of black holes composed only of rigid impenetrable spheres with mass M = 6.7 × 10 − 26   g is 0.15 M ⊙ , about 15% of the mass of V723 Mon, the smallest black hole found to date. Hawking temperature of minimum mass black holes is 4.0 × 10 − 7   K , far below today’s 2.7255 K CMB temperature.

Energy of bits of matter ( 1 − f ) m b i t c 2 and energy of bits of anti-matter ( 1 + f ) m b i t c 2 with f = 5.079 × 10 − 10 resulted in more matter than anti-matter in the universe. Radiation temperature after inflation was T I = 2.7255   K R I / R H = 4.908 × 10 16   K , with today’s cosmic microwave background (CMB) temperature 2.7255 K. Ratio of nucleons to anti-nucleons, resulting from energy difference between matter and anti-matter bits, is the ratio of Boltzmann factors

so nucleon excess is 6.04 × 10⁻¹⁰. Nucleons are baryon constituents of all atomic nuclei, so nucleon excess 6.04 × 10⁻¹⁰ results in PDG 2025 baryon to photon ratio 6.04 × 10⁻¹⁰.

Standard Model particle physics treats elementary particles as point particles with angular momentum ± ℏ / 2 , but point particles have zero volume, infinite energy density, and cannot rotate or have angular momentum. The Standard Model involves nine particles with charge q e (electrons, muons and tauons with q = − 1 , three quarks with q = 2 / 3 , and three quarks with q = − 1 / 3 ), but does not explain why only three particles are in each charge state.

In contrast, charged elementary particles treated as spheres rotating around an axis through their center have volume components, surface components, and axial components. Cubic equations for particle holographic radii in each charge state are

Rewritten as A r 3 + B r 2 + C r = 0 with A = 4 3 π ( ρ − ρ v ) , B = 4 π ρ s , and C = 3 2 π ρ a , with average component mass densities ρ v g/cm³, ρ s g/cm², and ρ a g/cm, the equations have positive discriminant and three (and only three) positive solutions corresponding to holographic radii of three particles in a charge state. Angular momentum ± ℏ / 2 is determined by clockwise or anti-clockwise rotation about the central axis and charge on the central axis precludes energy loss from accelerated charge.

Electron mass, the lowest mass particle persisting indefinitely, is specified by holographic analysis involving electron holographic radius r e . Electrostatic potential energy of electron charge e and positron charge − e separated by 2 r e , a precursor to electron-positron pair production, is V = − e 2 2 r e = − α ℏ c 2 r e with fine structure constant α . Two adjacent spheres with radii r e have total energy E = 2 m e c 2 − α ℏ c 2 r e = 0 when r e = α ℏ c 4 m e c 2 . Two equations for r e result in

PDG 2025 parameters predict electron mass 0.5% higher than actual to three significant figures, but setting Λ = 1.08800 × 10 − 56   cm − 2 and increasing Ω Λ by 0.5% to 0.6883855 (within PDG 2025 error bars) specifies electron mass to six significant figures. Since gravitational constant G is known to six significant figures, the calculation cannot be extended to greater precision until G is measured more precisely.

Specifying up quark and down quark holographic radii as r u = 2 r e for up quarks and r d = 3 r e for down quarks results in up quark mass m u = 4 m e and down quark mass m d = 9 m e (within estimated ranges in PDG 2025 Quark Particle Listings).

Mass of electron neutrinos, described as spherical excitations of vacuum energy density ρ V = 5.840 × 10 − 30 g / cm 3 with radius one quarter their Compton wavelength, is

Neutrino oscillation data [6] predict

resulting in neutrino mass sum 0.0609 eV, below the lowest upper limit 0.07 eV from terrestrial and astronomical observations.

This analysis accounts for key PDG 2025 data regarding our universe and its contents, using simple mathematics with no singularities or infinities.