An independent finite-sample observational chapter
exit_distance. The findings are descriptive and coordinate-based; the coordinates describe where events occur, not causes.
This chapter gives a finite-sample description of the waiting hall as a small state space organized by exit_distance.
The principal observation is that the waiting hall is internally structured rather than uniform. Along exit_distance, three regions emerge: an exit layer (0–2) with no observed miss events, a miss-front (3–8) containing all 228 observed miss events, and a k-structure corridor (12–30) where the valuation distribution changes but no miss events occur. A selector audit further shows that miss events are not isolated by any single coordinate; in this sample, miss-only cells first appear when a positional coordinate is paired with a local-shape coordinate.
These findings are descriptive and coordinate-based. Their scope is limited to the finite sample analyzed here.
exit_distance 3–8 (the miss-front), concentrated around 3–4.exit_distance 0–2 (the exit layer).12–30 corridor holds 34,565 events and 0 misses; a k-structure region, not a miss region.| Coordinate | Kind | Meaning |
|---|---|---|
exit_distance | positional | Distance from the lower-edge exit layer of the band; 0 is the exit, larger is deeper inside. Computed as remaining_K_before − band_lower_edge. |
remaining_K_before | positional | Remaining valuation mass before the step. |
transition_k | local-shape | The valuation k of the step. |
residue_pair (e.g. mod32) | local-shape | The residue-pair coordinate of the event. |
| miss local type | label | Existing A / B / C1 / C2 / C3 / C_unassigned labels, reused as-is. |
"Near behavior" labels (drift, wait, miss, exit) are observational labels derived from position plus existing behavior/miss joins, not new types. No new miss type is introduced.
exit_distance, descending toward the first exit. A conceptual map, not a boundary map.An initial partition divided the hall by within-band position into four coarse regions: upper_hall, mid_hall, lower_hall, and the exit_layer (distance 0–2). This was an initial, ratio-based partition only.
| Zone | Events | Mean wait | Median | k up / down / flat | Miss |
|---|---|---|---|---|---|
| upper_hall | 16,322 | 23.467 | 22.0 | 4,702 / 4,367 / 4,859 | 0 |
| mid_hall | 22,555 | 22.183 | 22.0 | 6,732 / 6,346 / 8,450 | 0 |
| lower_hall | 27,456 | 18.876 | 19.0 | 8,957 / 9,023 / 8,531 | 228 |
| exit_layer | 10,197 | 17.489 | 20.0 | 2,346 / 1,372 / 6,363 | 0 |
Coarse miss-type mix in lower_hall: B:85, A:64, C_unassigned:50, C1:13, C2:10, C3:6.
The four regions are not interchangeable: event counts, wait times, the k up/down/flat balance, and miss location all differ. The exit layer is overwhelmingly flat (6,363 of 10,197).
The four boxes are a coarse partition; the fact that every miss lands in one box is suggestive, not conclusive. R2 replaces the boxes with a continuous coordinate to test the shape of the region.
Replacing the boxes with the continuous coordinate exit_distance, three regions emerge without being predefined: an exit layer (0–2), a miss-front (3–8), and a k-structure region (12–30).
| Zone | Distance | Events | Miss | Miss rate | Dominant k |
|---|---|---|---|---|---|
| miss_front_3_8 | 3–8 | 14,743 | 228 | 0.015465 | 1 |
| k_structure_corridor_12_30 | 12–30 | 34,565 | 0 | 0.000000 | 1 |
The two regions share dominant k=1 and are close in overall k distribution (zone-level L1 difference 0.16214), but differ decisively in miss profile: 228 misses vs 0.
Because the 12–30 region has zero miss events in this sample, it is best described as a k-structure region, not a miss region — even though it is adjacent in exit_distance. Some individual distances look k-similar across the two regions, but zone-level miss and behaviour profiles separate them.
Four landmark distances were checked for support and shape.
| Distance | Events | Dominant k | Top-2 share | k3 share | Entropy | Reading |
|---|---|---|---|---|---|---|
| 15 | 2,456 | 3 | 0.8388 | 0.5908 | 1.6309 | k3 spike |
| 18 | 2,222 | 1 | 0.9154 | 0.0351 | 1.0182 | k1-heavy reset, not a k3 point |
| 23 | 2,228 | 1 | 0.8999 | 0.0471 | 1.2150 | k1-heavy side of a flip |
| 27 | 2,035 | 1 | 0.9253 | 0.0403 | 1.0209 | k1-heavy side of an outer flip |
| Zone candidate | Distance | Events | Signature |
|---|---|---|---|
| approach_to_k3_spike | 12–14 | 5,136 | 12:1/2 → 13:1/2 → 14:2/1 |
| k3_spike | 15 | 2,456 | 15:3/1 |
| k2_to_k1_reset | 16–18 | 5,950 | 16:1/2 → 17:2/1 → 18:1/2 |
| k1_plateau_with_k4_tail | 19–21 | 5,699 | 19:1/2 → 20:1/2 → 21:1/2 |
| k2_k1_flip | 22–23 | 4,456 | 22:2/1 → 23:1/2 |
| outer_k1_k2_flip | 24–27 | 7,398 | 24:1/2 → 25:1/2 → 26:2/1 → 27:1/2 |
| outer_mixed_tail | 28–30 | 3,470 | 28:1/2 → 29:2/1 → 30:1/2 |
The corridor is not a smooth gradient. Distance 15 is the single clearest local spike (k=3 dominant, share 0.5908). Distances 22→23 and 26→27 both flip from a k2-heavy point to a k1-heavy point; distance 18 is a reset-like k1-heavy point after the 16–17 transition, not a spike.
The zone split is supported by the audited counts and should not be interpreted as a mechanism. The corridor is best described as a short series of alternating local k-regimes with no miss events in this sample.
exit_distance 3–8, never in the exit layer (0–2).3→15:64, 4→15:47, 3→14:38, 4→14:18, …), thinning toward 5–8.k mix is dominated by small k (k1:9,142; k2:3,358; k3:1,433; …).32-63→32-63:9,704; 64-127→64-127:4,143; 128-255→128-255:668).Within the miss-front the labels appear as B:85, A:64, C_unassigned:50, C1:13, C2:10, C3:6. In expanded parity, the named/short types sit on short words (OEEEEEO:85 for the B-like group, OEEEEO:64 for the A-like group), while the remaining mass reaches longer words (OEEEEEEO:44, OEEEEEEEO:19, OEEEEEEEEO:8, … OEEEEEEEEEEEO:1).
In this finite sample, A and B behave as compact, short-window types, while C_unassigned appears as a more diffuse residual that spreads further across the longer-window coordinates rather than collapsing onto a single cell. This is an observation about where the labels concentrate, not a claim that any of them is necessary, sufficient, or mechanistic.
A selector audit over the full event table (76,530 rows; 228 miss; 76,302 non-miss background) used four candidate coordinates — two positional and two local-shape. A cell is "miss-only" when it contains all 228 miss events and 0 non-miss events.
| Coordinate | Matched | Miss | Non-miss | Miss rate | Miss-only? |
|---|---|---|---|---|---|
| exit_distance | 14,743 | 228 | 14,515 | 0.015465 | no |
| remaining_K_before | 13,802 | 228 | 13,574 | 0.016519 | no |
| residue_pair_mod32 | 275 | 228 | 47 | 0.829091 | no |
| transition_k | 4,946 | 228 | 4,718 | 0.046098 | no |
| Pair | Matched | Miss | Non-miss | Miss-only? | Pairing |
|---|---|---|---|---|---|
| exit_distance + residue_pair_mod32 | 228 | 228 | 0 | yes | position + shape |
| exit_distance + transition_k | 228 | 228 | 0 | yes | position + shape |
| remaining_K_before + residue_pair_mod32 | 228 | 228 | 0 | yes | position + shape |
| remaining_K_before + transition_k | 228 | 228 | 0 | yes | position + shape |
| residue_pair_mod32 + transition_k | 275 | 228 | 47 | no | shape + shape |
| exit_distance + remaining_K_before | 13,802 | 228 | 13,574 | no | position + position |
Every miss-only pair combines one positional coordinate with one local-shape coordinate. The two pairs that fail are exactly the two that stay on one side: position+position leaks 13,574 non-miss; shape+shape leaks 47. Three- and four-coordinate selectors remain miss-only but add nothing once a position–shape pair already separates the misses; the four-coordinate tuple is redundant in this scan.
In this dataset, "miss-only" means that the cell contains misses and no non-miss background. The key structural pattern is that a miss is pinned down not by position alone nor shape alone, but by a position coordinate and a shape coordinate together.
The selector audit above can be written compactly as a single statement about two selectors over the audited event set. This is a compact way to state the observation within the audited sample.
Let Ω_N be the audited set of band-internal event rows, and let M ⊆ Ω_N be the observed miss-event set:
Define the miss-front position selector and the miss-supported local-shape selector (the latter built from the residue-pair values that the miss events themselves occupy):
S_shape is defined on residue_pair_mod32 alone. transition_k is not added to the main definition: adding it does not remove the non-miss rows that S_shape admits (the leakage stays at 47), so the shape side carries no extra separating information here — the separation is supplied by the position side.
Here S_shape is a miss-supported selector: it is defined from the residue-pair values observed in M, then applied back to Ω_N. Thus the identity is a finite-sample support check, not an independent prediction rule.
In this finite sample each selector contains all of M, but neither alone isolates it:
Their conjunction removes the excess rows exactly:
We call this last line a finite-sample selector identity: an empirical equality that holds inside Ω_N. Within the audited rows, the conjunction of these two selectors picks out exactly the observed miss-event set.
This identity is descriptive and limited to the audited sample.
The rows in S_shape \ M are not diffuse noise. In this sample all 47 lie outside the miss-front, at exit_distance 35, 36, 37, and all carry near_behavior = drift:
So the same residue-pair shape that appears in the miss-front also appears at these upper positions, but there it is realized as drift rather than miss. Imposing P_pos removes exactly this upper-position drift counterpart, leaving P_pos ∩ S_shape = M. (Adding transition_k to the shape selector does not remove these rows; position does.)
Viewed along the two axes of the state space:
3–8 window contains 14,515 non-miss rows;Ω_N — their conjunction is exactly M.This is the same "where × how" interpretation as the rest of the chapter, written as an equality that holds in this finite sample. It does not imply a mechanism, proof, counterexample, or global Collatz behavior.
k balance, and miss location vary by position.C_unassigned is the most diffuse label (R4).P_pos ∩ S_shape = M inside Ω_N (selector identity).As a state space, the hall has a "where" axis (exit distance / position) and a "how" axis (local shape: k, residue, parity). The miss-front is where those two axes jointly select miss events; the corridor is where the "how" axis varies but no miss events are selected; the exit layer is the lower reference region.
The claims in this chapter are limited to the finite sample studied here. The regions are descriptive groupings, not proven thresholds. The coordinates used here, including exit_distance, residues, and transition_k, are used to describe where events occur; they are not claimed to be causes.
Accordingly, the conclusions of this chapter are limited to the finite sample analyzed here and should not be interpreted as claims about the Collatz problem as a whole. "Miss-only selector" means only that, in this sample, the selected cell contains miss events and no non-miss background.
No new miss type is introduced; the existing A / B / C1 / C2 / C3 / C_unassigned labels are reused as-is.
| Figure / artifact | Role |
|---|---|
waiting_hall_interior_map.png | R1 — coarse four-box hall |
band_position_flow_map.png | R1 — downward flow between hall zones and exits |
k_by_hall_zone_heatmap.png | R1 / R2 — k distribution by coarse zone |
exit_distance_miss_rate_plot.svg | R2 / R4 — miss rate vs continuous exit distance |
k_change_by_exit_distance.png | R2 / R3 — k-change score per distance (landmarks 15 / 18 / 23 / 27) |
k_structure_corridor_heatmap.png | R3 — per-distance k histogram for 12–30 |
| selector diagram (Diagram 2 above) | R5 — position × shape miss-only cells |
| state-space map (Diagram 1 above) | R6 — stacked exit-layer / miss-front / corridor |
glossary_ of_ Terms.png | Glossary — Collatz terminology |
Five finite-sample reports, reorganized into a single narrative (R1 → R6) rather than listed report by report:
| Report | Section | python |
|---|---|---|
waiting_hall_interior_report.md | R1 — coarse four-box hall | waiting_hall_interior_audit.py |
hall_zone_comparison_report.md | R2 — miss-front vs corridor | hall_zone_comparison_audit.py |
zone_coordinate_contrast_report.md | R2 / R4 — k-similar pairs, residue/parity | zone_coordinate_contrast_audit.py |
k_structure_corridor_report.md | R3 — the 12–30 corridor | k_structure_corridor_audit.py |
minimal_miss_selector_report.md | R5 — minimal miss-only selectors | minimal_miss_selector_audit.py |
This chapter is independent of, and does not modify, the Paradoxical-Sequence chapter (first-pass faces at the 64-95 → 32-63 boundary). It studies the space before that boundary.