Classical Appearances of π_g: A Geometric Substitution Note

This expository note records representative classical formulae in which the circle constant π is rewritten using the geometric limiting constant π_g. Under the previously established identity π_g = π, these substitutions preserve the corresponding classical identities exactly. The note distinguishes the completed geometric limit from finite polygonal approximants and includes a reproducible binary64 numerical illustration using N-Series acceleration and Gaussian normalization. The numerical illustration is a consistency check, not an independent proof of the Gaussian integral or of π_g = π. No new classical identities or computational speed advantage are claimed. The accompanying archive includes the LaTeX manuscript source, numerical verification script, recorded output, reproducibility instructions, copyright notice, and SHA-256 checksums.