Algorithmic Non-commutative Class Field Theory, Volume III: Higher-Rank Buildings, Non-Equilibrium Topology and Higher-Order Games
Abstract
This monograph is the third volume of "Algorithmic Non-commutative Class Field Theory". It studies spectral, dynamical and arithmetic structures on finite quotients of Ã₂ (and, where possible, Ã_d) buildings, in particular on the complexes X₃, Y₂₁, …, Y₁₆₈ of Volumes I–II and on the canonical 2-tower of Cartwright–Steger type. Every numbered statement is either proved or explicitly marked as numerical, and every numerical statement is reproduced by the accompanying Python code. Chapter 1 (Higher-rank buildings). The Z/3 type grading forces Tr(C^(3))^k = 0 for 3 ∤ k. The cubic trace identity Tr(C^(3))^3 = n(q−1)²(q+1) holds under the flag condition. It extends to every rank: Tr(C^(d+1))^(d+1) = n ∏_{i=1}^{d} (1 − q^i), whose sign is (−1)^d. Garland vanishing H¹(Y; E) = 0 is proved in local form for every unitary local system. Also proved: the Hecke pencil identity det(I − uC^(3)) = det P(u) and the derivative identity linking it to |Jac^H(Y)|. Chapter 2 (Simplicial sandpiles). A sign obstruction excludes abelian higher sandpiles on thick complexes. Also proved: the Hodge factorization of critical groups, and the exact mean avalanche size ⟨s⟩ = n² L⁺_ss/(n−1). This gives Θ(n) on expanding families against Θ(n log n) on flat tori. Chapter 3 (Phase oscillators on buildings). The gauge sector of chiral Kuramoto dynamics is a gradient flow, with frustrated twisted ground states. The Jacobian at cluster equilibria splits into blocks M(λ). On Y₁₆₈ transverse Hopf bifurcations occur, with multiplicity 21 forced by the deck group. In every case examined the loss of stability is hard and ends in a frequency-locked state. Chapter 4 (The geodesic edge operator). The unitary remainder of L_E is established unconditionally, and the two-circle Riemann hypothesis is equivalent to the Ramanujan property. For the crossover from arithmetic degeneracy to GUE statistics: the ratio statistic is pinned at a first-order plateau ⟨r⟩₁ ≈ 0.508 for ε ≤ 1/dim W, and reaches 0.600 only at the first-order splitting scale. Chapter 5 (Chamber games). Linear chamber public-goods games reduce to pairwise games, and nonlinear chamber games are exact potential games. The chapter also treats chiral games, interfaces and entrainment by clocks; there is no topological catalysis by thickness. Chapter 6 (Hecke algebras, Iwasawa modules and the 2-adic tower). The Iwahori–Hecke algebra acts on chambers, and Garland rigidity rules out any Z_p-direction. The Iwasawa module of Hecke Jacobians satisfies descent and an analytic class number formula along the canonical non-analytic pro-2 tower. The 2-part v₂|Jac^H(Y_m)| is computed exactly up to n = 1,572,864 vertices: 1742, 11595, 36937, 254662 at levels 5–8. It satisfies liminf v₂/n ≥ c_e, with c_e ≤ 0.0645. Unconditional finite-level characteristic elements in K₁ are constructed. Open problems are stated explicitly. They include: whether Z₂[[P]] is an Ore domain; whether c_e > 0; and whether some branch of the equivariant transverse Hopf bifurcation is supercritical. Contents of the record: the book (PDF, 104 pages), its LaTeX sources, the verification and computation scripts (Python: numpy, scipy, sympy, mpmath, matplotlib), the numerical data and run logs, and five vector figures. Source repository: https://github.com/Ruqing1963/anccft-volume3