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We prove that the cosmological bounce in the K-functional framework is a stable attractor. Two independent arguments establish that the same physical constants emerge every cycle. First, the K_rec gradient at the monopole fixed point is determined by the unique rank-4 compact Lie algebra satisfying three structural constraints (S² boundary, complex fundamental, three K-sectors), forcing identical coupling ratios at every bounce. Cross-terms between K-sectors vanish because the generators are traceless on separate tensor factors. Second, the self-consistency loop through nuclear physics (QCD confinement → nucleon mass → TOV limit → critical radius → K-sector boundaries → gauge couplings → QCD confinement) has gain |dF/dg| ≈ 0.23 < 1, verified numerically: a 2× perturbation converges in 7 iterations. The QCD confinement scale acts as a thermostat with a 20:1 buffer between the binding energy and the nucleon mass. The result is robust: even for σ_πN = 80 MeV (upper bound of the pion-nucleon sigma term), the gain remains below 0.41. The cosmological cycle has the structure of a thermodynamic heat engine with efficiency η_cycle = N_SM/S_horizon ~ 10⁻¹²⁰, cycle-invariant. Combined with the critical density identity ρ_crit = N_SM M_P⁴/(12π² S_horizon) (Paper 13), the heat engine efficiency determines the total energy content of the universe to 0.37% accuracy. No anthropic selection is needed: the constants are the unique fixed point of a contracting map.