What question did this study set out to answer?

This research aims to demonstrate that ordinary matter, dark matter, and dark energy are phases of a single scalar field governed by the sine-Gordon equation.

April 14, 2026Open Access

Metric Phase Transition Theory: Three Phases of the Sine-Gordon Metric Field as the Unified Origin of Matter, Dark Matter, and Dark Energy

Key Points

This research aims to demonstrate that ordinary matter, dark matter, and dark energy are phases of a single scalar field governed by the sine-Gordon equation.
Developed Metric Phase Transition Theory (MPT) to link three universe components.
Applied the sine-Gordon equation to derive physical constants directly from one model.
Utilized symbolic mathematics (SymPy) for verification of results and error analysis.
Each component of the universe is linked to a specific solution of the sine-Gordon equation.
Derived 45 physical quantities with high accuracy from a single postulate.
Predicted specific characteristics for dark matter and dark energy that are testable.

Abstract

Metric Phase Transition Theory (MPT) proposes that the three components of the observable universe -- ordinary matter (5%), dark matter (27%), and dark energy (68%) -- are not separate substances but three phases of a single scalar metric field governed by the sine-Gordon equation. Their ratio follows from the kink spectrum without free parameters. The three fundamental solution classes of this equation correspond directly to the three observed components: topological kinks (baryons/ordinary matter), breathers (dark matter), and radiation (dark energy). No new particles or fields are introduced. From the single stability condition that the kink supports exactly one bound state, the golden ratio phi = (1+sqrt (5) ) /2 emerges as the unique eigenvalue. From phi, through an exact algebraic cascade verified by symbolic mathematics (SymPy), MPT derives: The Standard Model gauge group SU (3) x SU (2) x U (1) exact The effective SM degrees of freedom g* (SM) = 106. 75 exact The fine structure constant 1/alpha = 137. 037 (error +0. 001%) All three lepton masses via the Koide formula (errors < 0. 10%) The proton mass (error 0. 000%) The CMB temperature (error +0. 020%, within FIRAS 1-sigma) The Hubble constant (error -0. 48%, within Planck 1-sigma) An algebraically exact dark energy density formula: rhoLambda = mₚi⁴ * (1/kappa₂3) ᵏappa₂3 The cosmic energy budget from the kink spectrum In total, 45 physical quantities are derived from one postulate (every particle is a metric kink) and one external measurement (the baryon fraction Omegab = 0. 0493). All exact results are verified using SymPy. The cosmological constant problem is resolved dynamically: dark energy is exponentially suppressed by the kink spectrum, not by fine-tuning. Three falsifiable predictions are stated: A dark matter particle at 93 MeV/c² with self-interaction cross-section sigma/m = 0. 21 cm²/g H₀ ~ 67 km/s/Mpc (Planck-consistent, Hubble tension prediction) w = -1 exactly for dark energy The paper includes complete step-by-step derivations with units, SymPy verification code for all exact results, full error analysis tracing every discrepancy to a single identified source, and a complete reference section.

Metric Phase Transition Theory: Three Phases of the Sine-Gordon Metric Field as the Unified Origin of Matter, Dark Matter, and Dark Energy

Key Points

Abstract

Cite This Study