Inertial Ricci Flow on Discrete Compact Manifolds via Adaptive Hash Grids Convergence, Lattice Curvature Scaling, and Geometric Momentum.

We address the problem of numerically solving the Ricci flow equation∂g₍μν₎/∂t = −2R₍μν₎on ten-dimensional compact discrete manifolds, where naive grid discretization becomes computationally intractable due to the curse of dimensionality. The central contribution is a physical analogy: we treat the metric tensor g₍μν₎ as a dynamical field carrying inertia, in direct correspondence with the momentum density of the electromagnetic field in classical field theory. This observation leads to a Heavy Ball integrator for the Ricci flow PDE, here termed inertial geometric flow, which introduces a velocity field v₍μν₎ evolved according to vⁿ⁺¹₍μν₎ = βvⁿ₍μν₎ − Δt·Rⁿ₍μν₎ with momentum coefficient β = 0.90. This integrator is combined with an adaptive hash-grid architecture that instantiates only geometrically active nodes, thermodynamic pruning, periodic boundary conditions encoding compact topology, and native multi-threading. We report convergence of a massively deformed initial metric (R₍μν₎ᵐᵃˣ = 44) to a Ricci-flat configuration on discrete tori Tᴸ⁴ for L ∈ {8,16}, achieving R₍μν₎ᵐᵃˣ < 10⁻⁶ in 520 steps (L = 8, 4,096 nodes) and 1,253 steps (L = 16, 65,536 nodes). The isotropy of the converged metric reaches 8.44×10⁻¹⁵, at the level of double-precision machine epsilon, and the residual momentum velocity satisfies ‖v₍μν₎‖ₘₐₓ < 3×10⁻⁸ A uniform residual lattice curvature κ(L) is identified, arising from the intrinsic curvature of the periodic hypercubic lattice, and shown to scale as κ(L) ∝ L⁻² with constant C ≈ 1.14×10⁻², confirming O(Δx²) convergence toward the continuous flat torus. The characteristic overshoots produced by the momentum integrator in the high-curvature regime are interpreted as the geometric analogue of ringing in an underdamped electromagnetic circuit.