Boundary-front geometry and finite trajectory atlas

1.What this note is — and its relation to collatz-waiting-hall

This note is a derivative of the Waiting Hall work. It is treated as a separate note.

• collatz-waiting-hall asked which coordinates isolate the observed miss rows in the waiting hall: rows that should have been caught at a location but slipped past at that moment. Combining the position condition and the shape condition led to the finite-sample selector identity P_pos ∩ S_shape = M.
• This note starts from the 47 remaining non-miss rows, S_shape \ M, and organizes how their trajectories pass through the bands. The focus is no longer a single selector condition, but the route those trajectories take: boundary-front geometry, trajectory routes, observed flow, and the shape of the descending paths themselves.

The earlier work was about separating a signal; this note is about building a coordinate map of where the observed rows travel. It moves from classification to a finite trajectory atlas.

2.Main results at the point of this write-up

Stated as observations. Interpretation is held to §4.

The previous chapter left off with P_pos ∩ S_shape = M, where S_shape (rows whose residue_pair_mod32 sits in the miss-event residue support) contains all 228 miss events and 47 non-miss rows (S_shape \ M). The core results of this note concern those 47 controls and, for contrast, the 120 depth-3 miss rows.

2.3 At 32-63, three descriptors must be kept apart — only two separate the groups

Local event classification is identical for both, so it does not separate them. Chain status and boundary-front position do.

2.6 Boundary-front structure appears across multiple bands

Rather than looking only at miss rows, the audit re-organizes all bands by how far each row sits from the boundary. The lower-edge, near-exit, and deeper front classes appear in every band with enough rows, including bands where no miss event is observed.

Rows with the same shapes as miss rows appear only in 32-63 (159), 64-127 (97), and 128-255 (19). And when asking which coordinate groups row status more cleanly, boundary-front position beats local event classification only in the 32-63 band.

In short: once the view is widened from miss rows to all bands, the band where the miss-related contrast is especially clean is 32-63.

Failed hypotheses

Tested and not supported in this finite table: shape alone is miss-only (false; the 47 controls are the leakage); shape + transition_k is miss-only (false; the same 47 remain); transition_k alone separates controls from miss (false); local event classification explains the 32-63 split (false; both avoid_then_caught); controls contain a 32-63 stage, classified differently; the two groups reconverge immediately after 16-31 (not supported); controls are depth-2 miss rows shifted upward (not supported); 32-63 is unique (not supported — the contrast is clearest there, not exclusive to it).

3.Subsequent progress: from classification to a finite trajectory atlas

The follow-up audits stopped asking "is this row a miss?" and started drawing the trajectories as routes and numeric paths across the same finite table. This is a change of viewpoint, not a stronger claim.

3.1 Trajectory grammar — routes as a finite dictionary

Treating each trajectory as an ordered route over states band|boundary_front|chain_status (consecutive repeats compressed) gives 2,750 trajectories and 144 distinct compressed routes. The most frequent route (64-127 … -> 4-7|…|caught) covers 1,081 trajectories (prob 0.393). Route families summarize the shape: capture_sink_4_7 (105 routes, 2,581 trajectories), branch_at_32_63 (114 routes, 2,489 trajectories), avoid_channel (85, 2,360), near_front_avoid_route (107, 2,206), upstream_mixed_64_127 (76, 1,754). High-entropy prefixes sit at 32-63, i.e. 32-63 reads more like a branching intersection than a single special point.

3.2 Empirical flow map — adjacent transitions

Converting adjacent stage rows into state_t -> state_{t+1} gives 9,806 observed adjacent transitions across 21 source states and 17 destination states, with the state coding band × boundary_front × chain_status. The largest edges form a compact downward band map (e.g. 16-31|near_exit_front|avoid -> 8-15|lower_edge_front|avoid at prob 0.974; 32-63|lower_edge_front|avoid_then_caught -> 16-31|near_exit_front|avoid at 0.760). Branching concentrates at a few 32-63 states; 4-7 is the main capture-side receiving layer.

3.3 Numeric trajectory anatomy — path shape instead of labels

Reducing the semantic labels and keeping the numeric path (R_before, R_after, R_drop, transition_k, exit_distance, remaining_K_before) gives 2,750 trajectory traces from 12,556 numeric stage rows. Terminal-aligned medians let one ask where miss / control traces separate; a compact gradient signature shows miss traces with a different terminal shape (median exit_distance_slope = -0.800, k_variation = 7.0) from control/other traces (0.100, 5.0). Here exit_distance can be viewed as a graded coordinate along the path rather than a front label, and 32-63 appears less like a standalone object and more like a region where the remaining-K trace is large enough for several nearby numeric shapes to be visible.

3.4 Paired numeric difference — local deformation of a shared road

Subtracting each matched control from its miss partner at terminal-aligned indices gives 210 miss/control pairs and 1,173 aligned pair-step rows. The differences collapse toward zero at the last indices (-2, -1, 0 all median 0) while the visible separation sits at earlier steps (-4, -3). This can be viewed as a local deformation of a shared road rather than a distinct route class.

3.5 Alignment coordinate comparison — the view depends on the ruler

Keeping the same numeric rows but changing the alignment (terminal step, start step, exit_distance, R_before) changes what is visible. Terminal alignment makes the last steps collapse; start alignment asks where traces separate from the entry side; exit_distance and R_before alignment turn trajectory-time into coordinate slices (the largest single-slice difference, 28.00, appears under exit_distance alignment at coordinate 0). The point is descriptive: the apparent miss/control gap is partly a function of which coordinate is used as the ruler.

4.Interpretation

5.Current summary

We construct a finite-sample coordinate atlas of observed accelerated trajectories.

The progression this note records is a change of viewpoint: from a miss-vs-background classification to a finite trajectory atlas made of routes, flow, numeric paths, and coordinate slices.

6.Limitations

7.Files

Primary reports