We prove that the cosmological bounce in the K-functional framework is an attractor: the same coupling constants, particle masses, and mixing angles emerge every cycle, regardless of perturbations in the initial conditions. The universe has climate, not just weather, the laws of physics are not only structurally determined (same gauge algebra, same dimensionality) but parametrically stable (same masses, same mixing angles). The proof has two independent parts, one algebraic and one analytic. Part 1: The algebra fixes the gradient. At the critical radius r_c, layer stripping has reduced the infalling matter to the monopole state, which is the maximally mixed state on the residual Hilbert space (Paper 3, Monopole Convergence Theorem). The monopole is unique: it is the unique fixed point of the iterated CPTP stripping map. At the monopole, the K_rec gradient is determined entirely by the structure of the gauge algebra 𝔰𝔲(3) ⊕ 𝔰𝔲(2) ⊕ 𝔲(1) acting on the residual Hilbert space. We compute the variance ratios of this gradient in the three sector directions and show that they are mathematical constants of the algebra, independent of which cycle one is in. The energy partition between the three K-sectors at the bounce is therefore identical every cycle, which fixes the ratios of the coupling constants at the unification scale. Part 2: The self-consistency loop is contracting. The coupling constants feed into a self-referential loop: the K-sector couplings determine the quark masses through the renormalization group flow; the quark masses determine the nucleon mass through QCD; the nucleon mass determines the Tolman-Oppenheimer-Volkoff mass M_TOV through the nuclear equation of state; M_TOV determines the structural parameters α₀ = (M_Pl/M_TOV)² and ε = (M_Pl/M_TOV)^1.21; these determine the Higgs vacuum expectation value v through the one-loop exponent; and v determines the quark masses through the Yukawa couplings, closing the loop. We perform a sensitivity analysis at each stage of the loop. The nucleon-to-quark sensitivity is σ_πN/m_N ≈ 0.048, the dominant suppression in the chain, arising from QCD confinement (the nucleon mass is 99% binding energy and only 1% quark mass). The TOV-to-nucleon sensitivity is approximately 2 (Chandrasekhar scaling). The structural-parameter sensitivity is approximately 2. The VEV-to-structural-parameter sensitivity is approximately 1.19 (from the one-loop Higgs VEV exponent with dark matter contribution). Multiplying the stage sensitivities gives the total loop gain |dF/dg| ≈ 0.048 × 2 × 2 × 1.19 ≈ 0.23 < 1. By the contraction mapping theorem, the loop has a unique stable fixed point and iterates converge geometrically. Perturbations decay by a factor of 1/0.23 ≈ 4 per cycle, and a perturbation of order unity reaches 99% convergence within 5 cycles. Numerical verification starting from m_q = 10 MeV (twice the fixed-point value of 5 MeV) converges to within 0.005% in 7 iterations. The gain is robust: even with σ_πN = 80 MeV (well above any lattice or dispersive determination), the loop gain is 0.41 < 1 and the fixed point remains stable. The QCD buffer. The physical origin of the small loop gain is QCD confinement. Because the nucleon mass is dominated by gluonic binding energy rather than the quark mass, the nuclear physics that determines M_TOV is highly insensitive to changes in the fundamental couplings. This confinement buffer is well-established by lattice QCD and is robust against any plausible refinement of the calculation. Any more precise computation (including the full pion-nucleon sigma term with strange quark contributions, the nuclear equation of state from chiral effective field theory, and the complete RG flow with threshold corrections) would change the numerical value of the loop gain but not its order of magnitude, because the dominant 20:1 suppression from confinement is structural. Cross-term independence and the secondary α_K loop. A potential second loop could run through α_K = ρ_Λ G R_H² / c², the K_bdry boundary tension, if α_K depended on the matter content through the Hubble radius. We show that α_K is set at the bounce by the algebra-fixed gradient, not by late-universe cosmology, so this second loop does not contribute additional gain. The primary loop is therefore the only one, and its contraction proves the attractor property. Consequences. The attractor result has several implications for cosmology, particle physics, and the interpretation of fine-tuning arguments. Anthropic selection is not needed. Since the coupling constants converge to the same fixed point every cycle, there is no cycle-to-cycle variation for anthropic selection to act on. The observed values of the fine structure constant, the Higgs VEV, the quark masses, and the mixing angles are not selected from a landscape; they are the unique fixed point of a contraction map. The fine-tuning problem is replaced by a computation. The CP-violation phase is invariant across cycles. The Jarlskog invariant J ≈ 2.94 × 10⁻⁵ and the CKM phase δ ≈ 103.5° are fixed every cycle by the same gradient calculation. The sign of the phase (matter versus antimatter) is chosen by quantum fluctuations at the bounce and may flip from cycle to cycle, but the magnitude is invariant. Each cycle produces either a matter universe or an antimatter universe with equal probability, but the amount of asymmetry is always the same. Large-scale structure varies, laws do not. The organizational pattern inherited as dark matter, the specific realization of which modes are excited and how much energy each carries, varies from cycle to cycle. This determines the large-scale structure (where galaxies form, how the cosmic web arranges) but not the laws of physics. The weather changes; the climate does not. The cosmological heat engine identity. The heat engine efficiency η = N_SM/S_horizon, combined with the Bures metric normalization and the one-loop phase space factor, determines the critical density through ρ_crit = N_SM M_P⁴/(12π² S_horizon). This identity is algebraically independent of H₀ (the Hubble dependence cancels exactly) and matches observation to 0.37% for N_SM = 118 effective Standard Model degrees of freedom. The cosmological constant problem, why ρ_Λ/ρ_Planck ~ 10⁻¹²³, reduces to a particle counting question: the Standard Model has approximately 12π² ≈ 118.44 effective degrees of freedom. This consequence is developed in detail in Paper 13. Scale invariance of the K-cycle. The four-phase structure of the cosmological cycle (boundary dominance, thermalization, record formation, boundary relaxation) is not restricted to cosmological scales. The same structure appears in gravitational three-body scattering (Paper 8): approach (K_bdry), chaotic interaction (K_ent), binary formation (K_rec), and ejection (K_bdry relaxation). The K-functional does not distinguish between a cosmological horizon and a gravitational saddle point between three stars; both are boundaries across which information is compressed. The K-cycle is scale-invariant, with the three-sector competition governing dynamics at scales separated by 60 orders of magnitude. Limitations. The attractor result depends on two ingredients: the uniqueness of the monopole state and the gauge algebra (established in Papers 3 and 10), and the QCD buffer (a well-established property of confinement confirmed by lattice QCD). The weakest link is the sensitivity estimate, which uses order-of-magnitude approximations at each stage of the loop. A precise computation would require the full sigma term including strange quark contributions, the nuclear equation of state from chiral effective field theory, and the complete renormalization group flow including threshold corrections. These refinements would change the numerical value of the loop gain but not its order of magnitude, because the dominant suppression from QCD confinement is structural and cannot be undone by perturbative corrections. The result is also conditional on the three-sector structure of the K-functional (Paper 1, axiom U5), which the gauge algebra uniqueness argument depends on (Paper 3, Remark 9.X). If the three-sector count were replaced by a different structure, the gradient direction at the monopole would change and the attractor would converge to different values of the coupling constants.