The Sing Constant S — A Dimensionless Invariant Linking Quantum Mechanics, Gravitation, and Cosmology

We propose a new dimensionless invariant, S = c⁵ ⁄ (ħ G HΛ²), which unites the constants of quantum mechanics, gravitation, and cosmology into a single, scale-free ratio. Numerically, S = 1.037 352 0301 × 10¹²². This invariant expresses the total relational capacity of the universe — the ratio between its quantum lower bound (Planck domain) and its cosmological upper bound (de Sitter horizon). Unlike the cosmological constant Λ, which has dimensional units, S is unitless and invariant across all reference frames. The logarithmic measure log₂S ≈ 406 corresponds to the total information capacity (in bits) of the cosmic system — providing a finite informational basis for the large-number observed in physics and pointing toward a unified informational description of the cosmos.