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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 H0=(50–100) 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 H0=H80 (from the inverse of the square root of the 80th triangle–square number) falls within 1σ of the SH0ES team measurement of H0=73.04±1.04 km s−1 Mpc−1 (Riess et al., 2022), and more recent estimate of H0=73.29±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 Λ may be the only kind of matter permeating the universe.