A Single Global Scale Linking Galactic Acceleration and a Homogeneous Dark-Energy Density Scale

This paper studies a conditional relation between a global length scale, the low-acceleration gravitational regime, and the homogeneous dark-energy density scale.The construction assumes a single global length L and a geometric weak-field variable whose static limit is proportional to L g / c^2. The deep regime is described by an AQUAL-type response function. Scale transformations induce a renormalization map on this response, whose positive fixed points are linear in the weak-field variable. If the physical response approaches such a fixed point in the infrared, then mu(X) = X / xi + o(X), where xi is a positive dimensionless coefficient.Defining a0 = xi c^2 / L gives the usual deep-field relations g = sqrt(g_N a0) and v_flat^4 = G M_b a0 in the isolated spherical limit, up to asymptotically vanishing corrections. These relations are standard MOND/AQUAL results and are not claimed as new phenomenology.The distinctive hypothesis considered here is that the same global length also controls a homogeneous dark-energy density scale, Lambda_rho = F_Lambda / L^2. This gives a0 / (c^2 sqrt(Lambda_rho)) = xi / sqrt(F_Lambda). If the dimensionless ratio K = xi / sqrt(F_Lambda) is constant over cosmological evolution, then a0 evolves as the square root of the homogeneous dark-energy density scale.For a separately conserved dark-energy sector, the paper derives the corresponding CPL redshift law and compares it quantitatively with an alternative scaling a0 proportional to H. The result is presented as a conditional and testable co-evolution hypothesis. The microscopic origin of the common scale, the slicing dynamics, the value of K, the crossover function, and the full relativistic completion remain open.