Tier 1 #21: Statistical Mechanics in the Information-Theoretic Unification framework — quantum statistics, fluctuation theorems, information, and complex systems

This paper formulates statistical mechanics — classical, quantum, non-equilibrium, fluctuation, informational, and complex-systems statistical mechanics — entirely inside the Information-Theoretic Unification (ITU) framework. Across eight phases (143-150), we (i) reconstruct Boltzmann/Maxwell-Boltzmann/Gibbs equilibrium statistics as the K_stat substructure of ITU; (ii) cover Fermi-Dirac / Bose-Einstein quantum statistics, BEC at T_BEC ≈ 170 nK for Rb-87, and Fermi degeneracy producing the Chandrasekhar mass 1.4 M_sun; (iii) demonstrate phase transitions, critical exponents, universality classes, and Wilson's renormalization group with Onsager's exact 2D Ising T_c = 2.269 J/k_B reproduced via Monte Carlo; (iv) develop non-equilibrium thermodynamics via Onsager reciprocity, Kubo linear response, and the fluctuation-dissipation theorem, verified against Einstein D = mu*k_B*T for Brownian motion; (v) prove Jarzynski's equality ⟨exp(-beta W)⟩ = exp(-beta Delta F) and Crooks' fluctuation theorem in a harmonic-spring switching simulation, and verify the Landauer limit W_erase >= k_B T ln 2 = 17.9 meV/bit at 300 K; (vi) establish the K_stat ↔ K_info isomorphism through Shannon entropy, Jaynes max-entropy, and von Neumann entropy (including Bell-state maximal entanglement S = ln 2); (vii) analyze active matter (Vicsek model), self-organized criticality (BTW sandpile τ ≈ 1.18), Barabási-Albert scale-free networks (gamma ≈ 2.73), Turing patterns (Gray-Scott), and the thermodynamics of life (human ATP turnover ≈ 100 W). Phase 150 integrates these into a 21-vertex ITU polytope in which #21 Statistical Mechanics attains the new maximum degree 20, alongside the other four Block A papers. The construction establishes the UNIVERSAL FOUNDATION K_geom ⊕ K_cosmic ⊕ K_field ⊕ K_stat, completing physics inside a four-K-state architecture, and yields 10 falsifiable predictions (P_avg = 0.635; 5 strong, 4 medium, 1 weak) for 2026-2050. Block A paper 5/9, Pass-1 milestone 68.2% (Phase 150/220). Companion archive contains eight reproducible Python simulations and their figures and JSON summaries. Tier 0 concept DOI: 10.5281/zenodo.20109209. Tier 0 v3.0: 10.5281/zenodo.20200156. Block A prior: #17 QG (10.5281/zenodo.20230667), #18 BH (10.5281/zenodo.20233070), #19 Cosmology (10.5281/zenodo.20233952), #20 SM (10.5281/zenodo.20234703).