We present interactive visualizations of the key mechanisms of a proof of the Finite Combinatorial Complexity of the Collatz conjecture. Each tool isolates one layer of the argument — from 2-adic fuel attrition to Baker-type diophantine kicks to solenoid confinement on the (2,3)-adic torus — and lets you watch it operate on live trajectories.
Map every cell on the (Z/3kZ)² torus by its cell error a − log&sub2;3·b. Red cells cluster near the irrational foliation; blue cells are safe. Watch a Collatz trajectory navigate the sieve in real time.
The Rigidity Theorem: Beyond resolution k=216, the branch locus is frozen. The integers run out of room to create new behaviors. Every trajectory is trapped in this 21,632-cell cage until it hits the 1-4-2 sink. The Ghost Island (251 cells discovered only at N=10¹¹) is separated from the main cluster by a 352-unit Diophantine void that Baker's theorem forbids any integer from crossing.
Interactive 3D visualization of Collatz trajectories on T²₂₁₆ × ℝ. Generates up to 5,000 trajectories in JavaScript, plotting each step on the torus (ν₂ mod 216, ν₃ mod 216) with the transverse walk w(t) = ν₂ − log₂3·ν₃ as vertical axis. Cells are classified as branch/pure-even/pure-odd from accumulated visit statistics. Features: toggleable trajectory lines with time-gradient coloring, InstancedMesh cell cloud, RG supernode transitions, irrational foliation overlay, step-through animation with live walk gauge, and four camera presets.
3D branch locus explorer embedding all 23,268 non-empty cells from the 100-billion-trajectory run at k=729. Vertical axis is perpendicular distance from the log₂3 foliation, separating the main cluster (d ≈ 0, ~21.6K branch cells) from the Diophantine ghost island (251 cells at r₂∈[0,37], r₃∈[438,461], d ≈ −300). Three color modes: cell type, p_odd heatmap, and visit density (log scale, 1 to 10¹¹). Includes ghost island wireframe highlight with fly-to zoom, distance filter slider for peeling the cell cloud, hover raycasting with per-cell statistics, perpendicular distance histogram, and 13 directed RG edges on the 9×9 supergrid.
Trailing 1-bits in the binary expansion are the fuel for consecutive v&sub2;=1 (dangerous) steps. Hensel's lemma shows each step consumes one bit — watch the fuel drain in real time until the trajectory escapes to an even result.
Baker's theorem guarantees |a − log&sub2;3·b| > C/max(|a|,|b|)κ. Dangerous cells can approach the foliation but never land on it. Each v&sub2;=1 run kicks the trajectory away — visible as yellow arrows on the torus grid.
A three-dimensional torus carries the Collatz trajectory through the (2,3)-solenoid. The deficit δ(t) = 3ν&sub3; − t stays confined inside a translucent pipe around the irrational foliation — the trajectory can wander but never escape to infinity.
Repository of manuscripts, Lean 4 formalization, C and python code.