Inertial Ricci Flow on Discrete Compact Manifolds via Adaptive Hash Grids: Convergence, Lattice Curvature Scaling, Geometric Phonons, and the Discrete Ricci Floor

We present a numerical framework for solving the Ricci flow equation ∂ₜ g₍μν₎ = −2 R₍μν₎ on discrete ten-dimensional manifolds using an adaptive sparse hash-grid architecture (Version 3). The metric tensor is represented as a dynamical field with inertia via a Heavy Ball integrator (β = 0.90 adaptive), motivated by an analogy with classical electrodynamics. Compact topology T⁴ₗ is encoded via periodic boundary conditions on four dimensions. Full convergence to Max|R₍μν₎| < 10⁻⁶ is achieved at L = 8 (520 steps, 136 s) and L = 16 (1253 steps, ~600 s) on a mobile ARM processor with 8 threads and no GPU. The residual lattice curvature satisfies κ(L) = C/L² with C = 0.01138, confirmed at L ∈ {8, 16}. At L = 32, two previously unreported phenomena are identified: (1) the discrete Ricci floorRᵐⁱⁿ ≈ N₍compact₎ · C / (L² · dx²), a hard resolution limit of the lattice Ricci operator at fixed dx = 1.0, where the measured offset 4.287×10⁻⁵ = 3.858 · κ(32) ≈ 4 · κ(32) encodes the number of compact dimensions; and (2) geometric phonons — quantized oscillation modes of the metric field with period T(L) = 2.5 · L steps, amplitude ≈ (2/3) · κ(L), and a single dominant FFT frequency, representing the fundamental acoustic mode of the discrete torus. Two algorithmic innovations are introduced: • an adaptive momentum coefficient β(R) that prevents resonance at large L, and• a multigrid prolongation operator P : T⁴₁₆ → T⁴₃₂ that seeds L = 32 from the converged L = 16 metric. The implementation requires only a C++17 compiler, ~128 MB RAM, and no external libraries, running on any device with SIMD-128 support including mobile ARM processors.