Resonance reformation and orbital expansion under optimized conditions with mass model M1 in Table 2. Panel a shows the evolution of planets’ semi-major axis (black), pericenter (blue), and apocenter (red) in the optimized collision phase, in which planets d₁ and d₂ are adiabatically replaced by planet d. Panel b represents the expansion of period ratio from integer ratio in Phase III, with Δ defined as Δ_bc = P_c/P_b−2 and Δ_ce = P_e/P_c−3. In panel c and d, the blue dots show the 3BR angle between b, c, and e: φ_3BR = 2 λ_b−5 λ_c+ 3λ_e. The figure shows that the b, c, and e resonance reforms after the simplified collision phase (panels a and c), whereafter the period ratio expands with tidal damping (panels b) before the resonance finally breaks at a larger planet period ratio, ceasing the expansion.