A Geometric Vocabulary for Gravitational and Dynamical Physics: Circumferences, Geometric Means, and Spatial Dimensions Across 55 Equations

The empirical formula a₀ = cH₀(Ω_Λ/Ω_m)^(1/6)/(2π), tested against 3,389 SPARC rotation curve measurements, decomposes the MOND acceleration constant into three identifiable components: a Hubble circumference scale, a geometric mean between two cosmic epochs, and a spatial-dimensionality factor. This paper asks: for every equation in gravitational, cosmological, and dynamical physics containing the same dimensionless factors, can we identify what boundary the circumference belongs to and what two regimes the geometric mean balances between? We catalog 55 equations across gravity, cosmology, horizon thermodynamics, oscillatory dynamics, and wave physics. For each, we state the standard form, rewrite it using geometric notation — C(X) for circumference, GM(A,B) for geometric mean, S(X) for surface area, D for spatial dimensions — and identify the physical boundary and regime pair. We classify each identification as already standard in the literature, consistent with the a₀ result but not new, genuinely new, or speculative. The geometric reading applies cleanly to every equation in the catalog. The principal new result is the nested geometric mean form of the deep-MOND limit: g_obs = GM(GM(c²/C_now, c²/C_eq), g_bar), which to our knowledge has not previously appeared in the literature. We identify two structural templates — boundary-set quantities and oscillation quantities — and show that the a₀ formula uniquely bridges both. We outline a program of follow-up investigations targeting specific equations and equation families.