Line-Like Concentration in Collatz Outlier-Gap Clouds

This exploratory computational study investigates a geometric object derived from the shortened Collatz map. For a fixed threshold M, all distinct visited values z>M arising from trajectories with seeds n≤M are collected, sorted, and converted into successive gaps Δz. Plotting (z,Δz) on logarithmic axes produces an outlier-gap cloud with a broad wedge and visually salient line-like concentrations. The study combines reproducible visualization with permutation-based controls. A toy construction shows that rightward thinning can amplify the visibility of sparse diagonal structure. For M=100,000, a 45-bin local-permutation null model yields an exploratory excess of thin-strip concentration, with a Monte Carlo tail fraction of approximately 0.002 under the primary specification and approximately 0.006 for a phase-independent sliding-window statistic. The effect is not robust to finer local conditioning, however, with substantially weaker results for 60 and 90 bins. Additional analyses include reproducible halo visualizations for M=100,000, 1,000,000, and 50,000,000, a point-normalized cross-scale control, an exploratory test of comparatively detached peripheral line-like groups, and a confluence-preserving Random Tree A control without Collatz arithmetic. The random-tree control reproduces much of the coarse wedge geometry and some right-tail concentration, indicating that these features are not uniquely Collatz-specific. The paper therefore does not claim the discovery of a null-model-robust arithmetic filament structure. Its main result is the combination of a reproducible line-like visual phenomenon with an exploratory statistical deviation whose interpretation remains sensitive to the choice of local null model. A complete standalone Python script for reproducing the M=100,000 outlier-gap construction and halo plot is included in the appendix.