What question did this study set out to answer?

The aim is to introduce Theory Ω, a hierarchical structure within the real numbers for understanding time and space.

May 22, 2026Open Access

Theory Ω: Hierarchical Structure of Numbers, Space and Time

Key Points

The aim is to introduce Theory Ω, a hierarchical structure within the real numbers for understanding time and space.
Introduced a discrete hierarchical structure using dyadic levels ω_k = 1 − 2^{−k}.
Defined fundamental objects as interintegers n_Ω^k = n + ω_k.
Discussed analogies with multiscale synchronization and wavelet-like encoding.
Established a unique finite binary representation and a well-defined ultrametric.
The framework provides a resolution of the grandfather paradox via a single-identity principle.
Consistent with p-adic metrics, Cantor sets, and multiresolution analysis.

Abstract

We introduce Theory Ω, a discrete hierarchical structure embedded in the real numbers based on dyadic levels ωₖ = 1 − 2^−k within integer intervals. The fundamental objects are the interintegers n_Ωᵏ = n + ωₖ. The system yields a unique finite binary representation, a well-defined ultrametric, and a normalized hierarchical measure. The algebraic extension Ω* is isomorphic to the subring of dyadic numbers ℤ1/2. The theory is interpretative and does not claim direct experimental predictions without additional dynamics. We discuss analogies with multiscale synchronization, hierarchical echoes, and wavelet-like encoding. The framework is consistent with standard mathematics (p-adic metrics, Cantor set, multiresolution analysis) and offers a parsimonious resolution of the grandfather paradox through a single-identity principle. Keywords: ultrametric, dyadic numbers, hierarchical time, structural time travel, p-adic

Theory Ω: Hierarchical Structure of Numbers, Space and Time

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Abstract

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