The Collatz Conjecture as Gradient Flow: Empirical Evidence for Switched Dissipation

We present an empirical framework demonstrating that the Collatz iteration exhibits gradient flow dynamics with forced dissipation sampling. Through computational analysis of trajectories for n ≤ 1000, we establish: (1) a well-defined potential function V (n) (stopping time) with gradient ∇V = −1 in 99.85% of cases; (2) perfect laminar channel structure at powers of 2 with zero turbulence; (3) a 2.03:1 ratio of dissipative to forcing steps, yielding expected strain ⟨∇S⟩ ≈ −0.288; (4) correlation of 0.9994 between turbulence and stopping time. The critical mechanism is the “+1” switch in 3n + 1, which forces parity flip and enables mandatory sampling of the dissipative (even) channel. These results suggest Collatz dynamics are not stochastic but arise from deterministic gradient descent on an arithmetic manifold with intrinsic 2-adic density structure.