We prove a trichotomy theorem for integer convergence in the polynomial family xⁿ = x + 1 at the Pisot boundary. For n = 2 (golden ratio), convergence to integers is monotonic. For n = 3 (plastic constant), convergence is oscillatory — the unique dimension where damped ringing and integer approach coexist. For n ≥ 4, conjugate roots escape the unit disk and convergence fails. The decay is governed by a survival partition expressing the Perrin recurrence as a probability constraint, with a universal half-life of exactly one bit in entropy units. Vieta's formulas prove that the convergence margin λ₃ = 1 − |σ|² is exact, identifying the decay fraction with the self-coupling coefficient of the companion Lagrangian (Pisot Dimensional Theory). A recursion budget mechanism explains the asymmetry: the cubic reabsorbs its surplus because conjugate roots lie inside the unit disk; the quartic cannot, because its conjugates escape. The Barbero–Immirzi parameter of loop quantum gravity is identified as γ = λ₄ρ — the four-dimensional self-coupling measured on the three-dimensional convergent scale — matching the LQG value to 0.9%. A previously unnoticed near-invariant of the polynomial family is established: the product nπ²λₙ converges to approximately 7 cycles per Hubble time across all dimensions n ≥ 2, with the physical spacetime dimensions n = 3 and n = 4 bracketing the value. This universal resonance matches the damped cosmological oscillation observed independently by Ringermacher and Mead (AJ 2015, MNRAS 2020) in Type Ia supernova data at 7.15 ± 1.0 HHz, to 0.16% accuracy with zero free parameters. The composite damping rate — Hubble friction times the Pisot conjugate modulus — matches the observed decay to 4.2%. The dark matter density parameter λ₃(1 + λ₄/2) = 0.267 matches Planck to 0.9%. These results establish the Pisot boundary as a phase transition between algebraic self-sufficiency and algebraic dependence, with the first independent observational confirmation from published astronomical data.