Tier 1 #24: Mathematical Physics in the Information-Theoretic Unification framework — Lie groups, integrable systems, Yang-Mills, conformal field theory, index theorems, knot theory, random matrix theory, and Clay Millennium connections

This paper formulates mathematical physics — Lie groups and Cartan classification, soliton theory and Lax pairs, Yang-Mills gauge theory and the Clay Millennium mass gap, conformal field theory and the Virasoro algebra, Atiyah-Singer index theorems and ABJ anomalies, knot theory and the Jones polynomial, random matrix theory and the Riemann Hypothesis — entirely inside the Information-Theoretic Unification (ITU) framework. Across eight phases (167-174), we (i) classify Lie groups by Cartan's theorem (4 families A_n, B_n, C_n, D_n + 5 exceptional E_6/E_7/E_8/F_4/G_2, with E_8 dim 248 maximal), verifying SU(2) Pauli commutators and SU(3) fundamental Casimir 4/3; (ii) demonstrate exactly solvable models — KdV soliton, sine-Gordon kink with topological charge Q = ±1, NLS bright soliton, Toda lattice (energy drift 3×10⁻⁶), and the Yang-Baxter equation satisfied to machine precision ||LHS - RHS|| = 2.22×10⁻¹⁶; (iii) develop Yang-Mills theory with QCD asymptotic freedom (Nobel 2004), Cornell linear confinement (1.95 GeV at 1 fm), Wilson loop area law, BPST instanton, and the Clay Millennium Problem #5 mass gap (glueball 0⁺⁺ = 1.71 GeV lattice candidate, $1M unresolved); (iv) construct 2D CFT via Virasoro algebra, Kac minimal models, Ising CFT (c = 1/2) reproducing all 2D Ising critical exponents matching Phase 145 exactly, WZW SU(2)_k, Cardy entropy formula linking CFT to BH (Phase 122), AdS/CFT (Phase 111); (v) establish index theorems — Atiyah-Singer (1963, Abel Prize 2004), Dirac index = Pontryagin n, ABJ chiral anomaly with π⁰ → γγ decay 7.76 eV matching experiment 7.82 eV to 0.8%, SM anomaly cancellation, Witten index for SUSY breaking, Chern number reproducing Phase 155 topology; (vi) develop knot theory — Jones polynomial (1984, Fields 1990) distinguishing chirality, Kauffman bracket → Jones (sympy-verified exactly), Khovanov categorification, Witten Chern-Simons (1989, Fields 1990) from 3D QFT, Fibonacci anyons (d_τ = φ = 1.618) for universal topological QC; (vii) cover random matrix theory — Wigner semicircle, GOE/GUE/GSE classification, Clay Millennium Problem #4 Riemann Hypothesis (30 zeros computed, $1M unresolved), Montgomery-Dyson pair correlation showing ζ-zero ↔ GUE duality, Odlyzko 10⁻⁸ agreement at 10²⁰ scale, Hilbert-Pólya conjecture, BGS quantum chaos universality. Phase 174 integrates into a 24-vertex ITU polytope in which #17-#24 all attain new maximum degree 23 (218 edges, ⟨k⟩ = 18.17). The construction establishes the MATHEMATICAL FOUNDATION BLOCK K_geom ⊕ K_cosmic ⊕ K_field ⊕ K_stat ⊕ K_solid ⊕ K_flow ⊕ K_math, expressing physics + mathematics in seven fundamental K-states, and yields 10 falsifiable predictions (P_avg = 0.525; 3 strong, 5 medium, 2 weak including Clay Millennium #4 RH and #5 Yang-Mills) for 2026-2080. Tier 1 papers #23 (Fluid Dynamics, Clay #6 Navier-Stokes), #24 (Math, Clay #4 RH + #5 Yang-Mills) thus address three of the seven Clay Millennium Problems explicitly. Block A paper 8/9, Pass-1 milestone 79.1% (Phase 174/220). 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), #21 Stat Mech (10.5281/zenodo.20237082), #22 CM (10.5281/zenodo.20249191), #23 Fluid Dynamics (10.5281/zenodo.20249794).