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 Aₙ, Bₙ, Cₙ, Dₙ + 5 exceptional E₆/E₇/E₈/F₄/G₂, with E₈ 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 Kgeom ⊕ Kcosmic ⊕ Kfield ⊕ Kₛtat ⊕ Kₛolid ⊕ Kflow ⊕ Kₘath, expressing physics + mathematics in seven fundamental K-states, and yields 10 falsifiable predictions (Pₐvg = 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).