The Sing Constant S — A Dimensionless Bridge Between Quantum and Cosmic Scales

We propose a new dimensionless invariant, �� = c⁵ / (ħ G HΛ²), which unites the constants of quantum mechanics, gravitation, and cosmology into a single, scale-free ratio. Numerically, �� = 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, �� is unitless and invariant across all reference frames. The logarithmic measure log₂ �� ≈ 406 corresponds to the total information capacity (in bits) of the cosmic system — providing a finite informational foundation for the large-number coincidences of physics and pointing toward a unified informational description of the cosmos.