Published February 1, 2026 | Version v1
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Emergence of the Planck Length from Wave Support Limits in the Sobolev-Ozok Lattice Framework

Authors/Creators

  • 1. Independent Researcher

Description

This paper derives the Planck length as an emergent minimal spatial scale from internal consistency requirements of wave support in the Sobolev-Ozok Lattice (SOL) framework. Rather than assuming Planck-scale discreteness from dimensional analysis, black-hole arguments, or spacetime curvature, the derivation is based on a single physical requirement: that a propagating wave solution must be supportable by the finite update capacity of the underlying lattice.

Starting from a classical wave excitation of the lattice, the analysis shows that increasing spatial resolution necessarily amplifies gradient energy and excitation cost. Beyond a critical scale, the energy required to sustain a distinguishable wave mode exceeds the maximum energy the lattice can accommodate without violating its local update rule. This leads to a natural halt of resolution, defining a fixed-point length scale determined by the balance between wave-support energy and lattice coherence stiffness.

Within the SOL framework, this emergent scale is identified with the Planck length once the coherence tension modulus governing k = 1 gradient energies is independently fixed through weak-field (Newtonian) gravitational matching. Importantly, the minimal length is derived prior to and independently of gravitational assumptions; its agreement with the Planck length represents a nontrivial consistency closure rather than a definitional input.

The result provides a geometric and dynamical origin for the Planck-scale lattice spacing used in subsequent SOL derivations of gravitation, cosmology, and coherence dynamics. More broadly, the work demonstrates how minimal length can arise from wave-support limits in discrete frameworks, without invoking quantum-gravitational collapse or curvature-based cutoffs. Alternative microscopic update rules may, in principle, lead to different minimal scales.

Declaration of Tools Used:

This manuscript was prepared and typeset using LaTeX via Overleaf. Language refinement and stylistic polishing were assisted by the Overleaf AI Editor. All scientific content, mathematical derivations, conceptual development, and conclusions are original and authored by the undersigned.

This paper is part of the Sobolev–Ozok Lattice (SOL) research program.

Project webpage (papers, figures, updates):

https://ozokozcasol.github.io/Sobolev-Ozok-Lattice/

 

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Cites
Preprint: 10.5281/zenodo.18447688 (DOI)

References

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