Cosmological Unification via Scaling–Duality: From Holographic Equipartition to a Master ODE for H(a)

We propose a symmetry-driven cosmological framework in which
ultraviolet (UV) and infrared (IR) corrections are organized by a sin
gle Scaling–Duality symmetry, H → ˜H ≡ H2
∗/H, together with a
horizon-thermodynamic expansion of the correction sector. Writing
the modified constraint as H2 = (8πG/3)ρm + f(H) with ρm ∝ a−3,
we derive a single autonomous “Master ODE” for the expansion his
tory, H′ = −3(H2 −f(H))/(2H −f′(H)) with N ≡ lna. We adopt
a minimal publishable truncation—a cosmological-constant-like piece
(n = 2), a dual pair (n = 1,3) with coefficient pairing c3 = κc1 (κ =
±1), and a logarithmic running term γ—and present a referee-robust
phase-space analysis. In particular, we prove a universal attractor re
sult: every nondegenerate de Sitter fixed point in the E ≡ H/H∗ flow is
automatically attractive with eigenvalue λ = −3 in N time. We isolate
the only sharp background pathology (branch points 2E −F′(E) = 0)
and provide a simple admissibility test for singularity-free evolution.
A reproducible scan protocol (with explicit numerical routines) and a
minimal Python implementation are provided.