The migrate-versus-stay crossover: how patient must a computation-maximizing lineage be before relocating to an IMBH platform beats staying home and densifying? Adjust the destination gain, transit survival probability, and transit time; watch the threshold, its half-life, and the hybrid seed-and-stay dominance band update.
Three strategies span the decision space: S (stay and densify to the local Matrioshka ceiling), M (migrate to an IMBH platform after a transit-plus-construction delay τ, discounted by survival probability p_s and destination gain G), and H (dispatch a cheap self-replicating seed while continuing S at home). Comparing the discounted value of M against S gives the crossover condition
ρ + λ < ln(G p_s) / τ ≡ ρ*
where ρ is the discount rate and λ a goal-stability hazard rate that enters additively. Because the threshold depends on the enormous destination-gain multiplier only through its logarithm, order-of-magnitude uncertainty in G moves ρ* by less than a factor of 2.5, the structural result the model leans on.
Strategy H dominates pure stay-and-densify whenever the seed cost fraction f < G p_s e^(-(ρ+λ)τ). At a physically plausible seed cost f = 10⁻⁶ (gram-to-kilogram payloads against a Dyson-scale economy), the dominance band extends to roughly 1.7× the migration threshold at the fiducial parameters; the exact ratio shown above scales with the chosen G·p_s. Outside the band the calculus reverts to pure stay-and-densify; inside it, and everywhere migration itself clears, the hybrid strategy is strictly best.
Because τ ∝ d for a fixed cruise speed, the threshold scales as 1/d: closer destinations relax the patience requirement proportionally. The three named points (ω Cen, 47 Tuc, M54) assume each cluster is the nearest adequate destination for the deciding lineage, at the beamed-sail speed v = 0.2c, using distances from Baumgardt & Vasiliev (2021).
G = 10⁹ (power ratio only, crediting Eddington-luminosity capacity but not fuel import; Paper D §3.2), p_s = 0.5 (Hoang et al. 2017 transit-attrition order-of-magnitude), τ = 10⁵ yr (bare cruise at ≈0.18c to ω Cen, rounded to include ≲10³ yr of construction time). At these fiducials ρ* ≈ 2.0×10⁻⁴ yr⁻¹, t½ ≈ 3,500 yr.