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June 7, 2026Open Access

A Scale-Invariant Spectral Substrate: Emergent Three-Dimensionality and the Dark-Energy Scale from a Single Principle

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Authors

KFKarol Frank

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Overview

Theoretical framework reveals dark-energy dynamics in a three-dimensional universe, suggesting new insights into cosmological constants.

Key Points

  • This work explores whether the number of spatial dimensions and the vacuum density can emerge from a unified spectral framework.
  • Derivation of spatial dimensions from conserved labels (frequency and U(1) charge) within a complex spectral field.
  • Application of heat-kernel arguments to demonstrate three-dimensional behavior at ultraviolet scales.
  • Use of maximum-entropy principles to establish scale invariance and its implications for dark energy.
  • Emergence of two spatial axes from conserved labels, with a third requiring a spectral kinetic term.
  • Unique selection of vacuum-energy functional that addresses the cosmological constant problem, avoiding the Planck catastrophe.
  • Demonstration that the observable universe is fundamentally three-dimensional across the Planck-to-Hubble scale.

Cite This Study

Karol Frank (2026) studied this question.

synapsesocial.com/papers/6a250be87def13d035e1be00https://doi.org/10.5281/zenodo.20552989
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  5. 5The Theory of the Scale Dimension: A Five-Dimensional Framework for the Hierarchy, Dark Matter, and the Cosmological Constant2026