A Coulomb Invariant of the Hydrogen Atom and an Effective Entropic Scale Proportional to πα

The hydrogen atom is conventionally introduced through its discrete energy spectrum, where the principal quantum number n determines the binding energy. This work presents an effective informational reading of the hydrogen atom by combining the Bohr energy and radial scalings with the standard quantum-mechanical degeneracy of the hydrogen level, gₙ = 2n². The central observation is that the product of the Bohr radial scale Rₙ = n² a₀ and the binding energy |Eₙ| is independent of n, yielding the Coulomb invariant Rₙ|Eₙ| = alpha hbar c / 2, where alpha is the fine-structure constant. When this invariant is inserted into the dimensionless structure appearing in a Bekenstein-type entropy bound, one obtains 2 pi Rₙ|Eₙ| / (hbar c) = pi alpha. This quantity is not interpreted as the total entropy of the atom, nor as a saturation of an entropy bound, but as an effective entropy-like dimensionless scale associated specifically with the electromagnetic binding energy. Separately, the degeneracy gₙ = 2n² defines a coarse-grained entropy Sₙ^eff = kB ln (2n²), which quantifies the multiplicity of unresolved quantum labels. The contribution is conceptual and structural: the same quantum number n organizes spatial scale, binding energy, degeneracy, and coarse-grained informational capacity, while the Coulomb product Rₙ|Eₙ| isolates an n-independent scale controlled by the electromagnetic coupling.