A common eigenmode postulate and a Λ-only model predict an exact Hubble constant that prefers SH0ES result: H0,80th=73.62694±0.00083 km s−1Mpc−1

The cosmological constant is probably the most fundamental aspect of nature’s laws and a deeper understanding of it may be the most direct route to the theory of everything. This research adopts the framework of a basic triangle number harmonic sum of energy-eigenmodes with an important postulate: that cosmologies contain a common energy-eigenmode. This ensures that cosmologies relate to the triangle–square numbers (TSn) and guarantees that the Hubble horizon is an integer multiple of the Planck length. Importantly, this allows a single value to be found in the range of $H_0=\left(\text{50–100}\right)$ km s⁻¹ Mpc⁻¹ for the Hubble constant leading to the anticipated resolution of the tension and a solution to the vacuum catastrophe. Remarkably, the predicted expansion rate $H_0=H_{80}$ (from the inverse of the square root of the 80th triangle–square number) falls within $1\sigma$ of the SH0ES team measurement of $H_0=\text{73.04}\pm \text{1.04}$ km s⁻¹ Mpc⁻¹ (Riess et al., 2022), and more recent estimate of $H_0=\text{73.29}\pm \text{0.90}$ km s⁻¹ Mpc⁻¹ (Murakami et al., 2023). Furthermore, the combinatorics of the prime factors of the square root of the 80th TSn are observed to give a scale hierarchy of the masses of the elementary particles of the standard model to a first-order approximation. Hence, if the postulate is correct, then the cosmological constant $\Lambda$ may be the only kind of matter permeating the universe.