Spacetime Contactability in Galactic Civilization Networks: Contact Counts and Percolation

Galactic Civilization Network Theory has described interstellar contact with a single length, the distance light covers during a communicative lifetime L, and has assumed that contact requires contemporaries. This paper removes both assumptions and follows the consequences, from contact counts to large-scale connectivity. Separating the detection range dmax from the light horizon cL gives the reach ratio χ = dmax/cL, and treating a civilization as an event of finite duration splits contact into four networks: listening, audibility, mutual listening and relay. The density parameter is the contactability η = πnd2 max, the mean number of civilizations within detection range of a given one, for a surface density n. A civilization hears 2η others at every χ and for any lifetime distribution, twice the coeval value; audibility and mutual listening sum exactly to 4η; the fraction of contacts that can be returned depends on geometry alone, and is also the probability that a detected civilization is still transmitting; and relay chains settle to a branching factor κη, κ = 1.707. The counts do not determine connectivity. Monte Carlo measurement across five decades in χ finds that, as χ → 0, the non-relay networks approach a critical contactability ηc = 1.32 ± 0.01, a factor 3.4 below the coeval threshold 4.51; that relaying costs about 7% on average; and that there is no critical length of history. In a disc whose contactability falls outward the frontier moves out 3.2 kpc, while the width of the front stays near 3.7 kpc for all four networks. A non-detection bounds the single-epoch count ⟨k⟩obs = η in both frameworks, so the reference bound η(R0) ≤ 2.75 stays where it is. The connectivity threshold, however, falls below that bound, opening a consistency window between non-detection and Galactic-scale connectivity. That bound still allows 5.5 civilizations to be heard over a full communicative lifetime, but fewer than one of those contacts returns a reply once χ ≳ 1.