Spectral Stability, Fatou Basin Contraction, and Machine-Checked Multipliers in Holomorphic Extensions of Generalized Collatz Dynamics

The Collatz (3n+1) conjecture asserts that iterated application of the compressed Syracuse operator T(n) = n/2 for even n and T(n) = (3n+1)/2 for odd n maps every positive integer into the periodic cycle {1, 2}. While computational sweeps have verified convergence up to 2^71 ≈ 2.36 × 10^21 and logarithmic density theorems establish that almost all orbits attain almost bounded values (Tao, 2022), the discrete parity selector (-1)^n obstructs classical differential stability analysis. In this paper, we embed the discrete Syracuse operator T(n) and the uncompressed Collatz operator C(n) into the complex plane C via entire functions of exponential type π:f(z) = [1 + 4z - (1 + 2z)cos(πz)] / 4 and F(z) = [2 + 7z - (2 + 5z)cos(πz)] / 4. We establish five principal results:1. Multiplier Cohomology Invariance and Universal Cycle Attractiveness: We prove via an exact telescoping product identity that any positive integer periodic orbit C ⊂ N+ of T is unconditionally a strictly attracting Fatou cycle of f with rational multiplier μ_f(C) = 3^(N_odd) / 2^N = Πn ∈ C odd 3n / (3n + 1) ∈ (0, 1), and prove that 2-adic Syracuse reduction preserves the exact multiplier between the 2-cycle {1, 2} and the uncompressed 3-cycle {1, 4, 2}, yielding μ_f({1, 2}) = μ_F({1, 4, 2}) = 3/4 = 0.75 < 1.2. Exact Koenigs Linearization Curvature Jets: By the Koenigs Linearization Theorem, the second iterate g = f∘2 conjugates holomorphically to Φ_j(g(z)) = (3/4)Φ_j(z) on the immediate Fatou basin B({1, 2}), with closed-form quadratic Taylor coefficients c_2(1) = -(13/2)π^2 and d_2(2) = -(9/2)π^2.3. Universal Negative Schwarzian Derivative and Fatou–Julia Partition: We prove that the Schwarzian derivative S(f)(n) is strictly negative at every integer n ∈ Z (S(f)(n) ≤ 3π^2 - (3/8)π^4 ≈ -6.919596 < 0), and map the complete critical and periodic point partition on [0, 3], verifying that B({1, 2}) captures the real critical point c_2 ≈ 1.9583122157 ∈ (1.5, 2) while the repelling Julia 2-cycle O_rep ≈ {1.0236861198, 2.0334209268} ⊂ J(f) (μ ≈ 1.214780 > 1) isolates the integer basin from Chamberland's irrational 2-cycle O* ≈ {1.1925319070, 2.1386563355}.4. The 2-Adic Haar–Fatou Resonance Identity: We prove via Birkhoff's Ergodic Theorem on (Z_2, μ_Haar) that the asymptotic 2-adic Lyapunov exponent λ_Haar = (1/2)ln(1/2) + (1/2)ln(3/2) = (1/2)ln(3/4) ≈ -0.14384104 coincides identically with the per-step logarithmic Fatou multiplier of {1, 2}, supported by multi-decade empirical scaling across 500,000 trajectories (λ̄ = -0.175737 ± 0.0777 < 0).5. Machine-Checked Formalization: We certify 20 algebraic, spectral, and Diophantine theorems in the Lean 4 interactive theorem prover (CollatzAttractor.lean) with zero unproven assumptions (0 sorry). Mathematics Subject Classification (2020): Primary 37F10, 11B83; Secondary 30D05, 68V15, 37P35.