# Why the 47 controls sit at max_avoidance_depth = 2

Finite-sample descriptive exploration only. No mechanism, proof, causality, or global Collatz claim is made.

## Inputs

- Waiting-hall table: `C:\Users\yauki\Documents\Codex\2026-06-30\task-waiting-hall-interior-map-audit\outputs\waiting_hall_interior_detail.csv`
- Exit-avoidance chains: `C:\Users\yauki\Documents\Codex\2026-06-30\task-exit-avoidance-path-space-audit\outputs\exit_avoidance_chains.csv`
- Exit-avoidance events: `C:\Users\yauki\Documents\Codex\2026-06-30\task-exit-avoidance-path-space-audit\outputs\exit_avoidance_path_events.csv`

## Counts

- controls: 47
- miss depth 2 rows: 57
- miss depth 3 rows: 120
- controls whose selected event band is inside the max-depth avoidance run: 0 / 47
- miss depth 2 rows whose selected event band is inside the max-depth avoidance run: 1 / 57
- miss depth 3 rows whose selected event band is inside the max-depth avoidance run: 119 / 120

## Confirmed similarities

- Controls and miss rows still share the same local shape support by construction; every control has a miss-observed residue-pair plus transition-k pair.
- Controls and miss-depth-2 rows both have max-depth chain length 2, so depth alone cannot distinguish them.
- Transition-k remains close across the depth-controlled comparison: M_miss_depth3:5:43; M_miss_depth3:4:39; M_miss_depth3:6:22; S_minus_M_control_depth2:5:18; M_miss_depth2:5:18; S_minus_M_control_depth2:6:14; M_miss_depth2:7:14; M_miss_depth2:4:12; S_minus_M_control_depth2:4:10; M_miss_depth2:6:10; M_miss_depth3:8:7; M_miss_depth3:7:4.

## Confirmed differences

- The 47 controls are not simply depth-2 miss rows shifted upward in one coordinate; they differ in behavior class and chain placement.
- Controls by trajectory behavior: `drift_down:30; mid_band_wait_then_drop:17`.
- Miss depth 2 by trajectory behavior: `no_exit_layer_observed:57`.
- Controls selected row behavior: `drift:47`; miss depth 2 selected row behavior: `miss:57`.
- Controls selected event band is inside the max-depth avoidance run only 0/47 times; for miss-depth-2 it is 1/57.
- For the controls, the selected `S \ M` event usually occurs earlier/upstream of the two-band avoidance run; the depth-2 run is typically downstream.

## What depth 3 adds

- Miss-depth-3 rows add one more consecutive avoided band in the max-depth chain pattern.
- Miss-depth-3 max-depth run bands: `32-63>16-31>8-15:120`.
- Control max-depth run bands: `16-31>8-15:47`.
- The common control run is `16-31>8-15` followed by capture at `4-7`; common miss-depth-3 runs include `32-63>16-31>8-15` before capture at `4-7`.

## Unexpected regularities

- All 47 controls have `max_avoidance_depth = 2`, but the selected control event is generally not the chain's depth-run event. This separates the shape-control event from the later exit-avoidance run.
- The control chain pattern is highly concentrated. See `s_minus_m_47_chain_pattern_counts.csv`.
- The controls are constant on several coordinates; see the constants table below.

## Failed hypotheses

- `The selected control event itself is the depth-2 avoidance run`: mostly false. It is inside the max-depth run for only the count shown above.
- `The controls are just miss-depth-2 rows shifted upward`: not supported as stated, because the selected-row behavior is `drift`, trajectory behavior is split between `drift_down` and `mid_band_wait_then_drop`, and the max-depth run is usually downstream of the selected event.
- `Depth=2 alone explains the 47 controls`: false; miss-depth-2 rows exist but have miss labels and different selected-row behavior/position.

## Interpretation stress test

Candidate interpretation to falsify: `The 47 controls are displaced miss-shapes whose position sends them into drift rather than miss.`

What survived descriptively:

- The controls are genuinely displaced in `exit_distance` and `remaining_K_before` while retaining miss-like shape+k cells.
- Their selected rows are all drift-labeled, not miss-labeled.
- They do not enter the miss-front support after adding position coordinates.

What weakens or complicates it:

- The depth-2 chain fact is trajectory-level and downstream; it should not be read as the immediate consequence of the selected control event.
- Some controls belong to `mid_band_wait_then_drop`, not only `drift_down`, so `drift` is not a single fine type.
- This remains a finite-sample description of observed rows, not a rule.

## Numeric comparison

| comparison_group | feature | n | min | median | mean | max | unique |
| --- | --- | --- | --- | --- | --- | --- | --- |
| M_miss_depth2 | remaining_K_before | 57 | 36.0 | 68.0 | 83.017544 | 133.0 | 10 |
| M_miss_depth2 | remaining_K_after | 57 | 29.0 | 63.0 | 77.368421 | 127.0 | 8 |
| M_miss_depth2 | exit_distance | 57 | 3.0 | 3.0 | 3.859649 | 8.0 | 6 |
| M_miss_depth2 | transition_k | 57 | 4.0 | 5.0 | 5.649123 | 9.0 | 6 |
| M_miss_depth2 | wait_length | 57 | 0.0 | 6.0 | 9.719298 | 46.0 | 26 |
| M_miss_depth2 | R_drop | 57 | 4.0 | 5.0 | 5.649123 | 9.0 | 6 |
| M_miss_depth2 | max_depth_run_length | 57 | 2.0 | 2.0 | 2.0 | 2.0 | 1 |
| M_miss_depth3 | remaining_K_before | 120 | 35.0 | 35.0 | 35.933333 | 67.0 | 6 |
| M_miss_depth3 | remaining_K_after | 120 | 27.0 | 31.0 | 30.641667 | 63.0 | 6 |
| M_miss_depth3 | exit_distance | 120 | 3.0 | 3.0 | 3.666667 | 8.0 | 5 |
| M_miss_depth3 | transition_k | 120 | 4.0 | 5.0 | 5.291667 | 11.0 | 8 |
| M_miss_depth3 | wait_length | 120 | 0.0 | 8.0 | 9.225 | 35.0 | 20 |
| M_miss_depth3 | R_drop | 120 | 4.0 | 5.0 | 5.291667 | 11.0 | 8 |
| M_miss_depth3 | max_depth_run_length | 120 | 3.0 | 3.0 | 3.0 | 3.0 | 1 |
| S_minus_M_control_depth2 | remaining_K_before | 47 | 99.0 | 99.0 | 103.680851 | 165.0 | 5 |
| S_minus_M_control_depth2 | remaining_K_after | 47 | 92.0 | 94.0 | 98.319149 | 159.0 | 5 |
| S_minus_M_control_depth2 | exit_distance | 47 | 35.0 | 35.0 | 35.595745 | 37.0 | 3 |
| S_minus_M_control_depth2 | transition_k | 47 | 4.0 | 5.0 | 5.361702 | 9.0 | 6 |
| S_minus_M_control_depth2 | wait_length | 47 | 15.0 | 27.0 | 27.765957 | 77.0 | 23 |
| S_minus_M_control_depth2 | R_drop | 47 | 4.0 | 5.0 | 5.361702 | 9.0 | 6 |
| S_minus_M_control_depth2 | max_depth_run_length | 47 | 2.0 | 2.0 | 2.0 | 2.0 | 1 |

## Constant coordinates across the 47 controls

| feature | unique_values | top_value | top_count | exceptions | distribution |
| --- | --- | --- | --- | --- | --- |
| group | 1 | S_minus_M_control | 47 | 0 | S_minus_M_control:47 |
| max_avoidance_depth | 1 | 2 | 47 | 0 | 2:47 |
| max_depth_eventual_capture_band | 1 | 4-7 | 47 | 0 | 4-7:47 |
| max_depth_exit_layer_id | 1 | lower_5pct | 47 | 0 | lower_5pct:47 |
| max_depth_run_length | 1 | 2 | 47 | 0 | 2:47 |
| miss_event | 1 | 0 | 47 | 0 | 0:47 |
| near_behavior | 1 | drift | 47 | 0 | drift:47 |
| waiting_class | 1 | long_wait | 47 | 0 | long_wait:47 |

## Near-constant coordinates with one or two exceptions

| feature | unique_values | top_value | top_count | exceptions | distribution |
| --- | --- | --- | --- | --- | --- |
| max_depth_final_status | 2 | eventually_reentered_exit_layer:caught | 45 | 2 | eventually_reentered_exit_layer:caught:45; eventually_reentered_exit_layer:avoid_then_caught:2 |

## Depth-transition table preview

| group | max_avoidance_depth | exit_distance | near_behavior | band | count |
| --- | --- | --- | --- | --- | --- |
| M_miss | 1 | 3 | miss | 32-63 | 23 |
| M_miss | 1 | 4 | miss | 32-63 | 15 |
| M_miss | 1 | 4 | miss | 64-127 | 8 |
| M_miss | 1 | 3 | miss | 128-255 | 2 |
| M_miss | 1 | 3 | miss | 64-127 | 1 |
| M_miss | 1 | 5 | miss | 64-127 | 1 |
| M_miss | 1 | 8 | miss | 32-63 | 1 |
| M_miss | 2 | 3 | miss | 64-127 | 23 |
| M_miss | 2 | 3 | miss | 128-255 | 9 |
| M_miss | 2 | 4 | miss | 64-127 | 9 |
| M_miss | 2 | 6 | miss | 64-127 | 6 |
| M_miss | 2 | 5 | miss | 128-255 | 3 |
| M_miss | 2 | 4 | miss | 128-255 | 2 |
| M_miss | 2 | 5 | miss | 64-127 | 2 |
| M_miss | 2 | 4 | miss | 32-63 | 1 |
| M_miss | 2 | 7 | miss | 64-127 | 1 |
| M_miss | 2 | 8 | miss | 64-127 | 1 |
| M_miss | 3 | 3 | miss | 32-63 | 63 |
| M_miss | 3 | 4 | miss | 32-63 | 40 |
| M_miss | 3 | 5 | miss | 32-63 | 13 |
| M_miss | 3 | 8 | miss | 32-63 | 2 |
| M_miss | 3 | 3 | miss | 64-127 | 1 |
| M_miss | 3 | 7 | miss | 32-63 | 1 |
| S_minus_M_control | 2 | 35 | drift | 64-127 | 27 |
| S_minus_M_control | 2 | 36 | drift | 64-127 | 10 |
| S_minus_M_control | 2 | 37 | drift | 64-127 | 7 |
| S_minus_M_control | 2 | 36 | drift | 128-255 | 2 |
| S_minus_M_control | 2 | 37 | drift | 128-255 | 1 |

## Suggested next exploration

- Compare the downstream run `16-31>8-15 -> 4-7 capture` in controls against miss-depth-3 rows where the run starts one band earlier at `32-63`.
- Build paired cards for controls and nearest same-shape miss rows, then compare the entire downstream chain pattern rather than only the selected event.
- Split the controls into `drift_down` vs `mid_band_wait_then_drop` before making any stronger statement about the drift label.

## Output files

- `s_minus_m_47_chain_cards.md`
- `s_minus_m_47_chain_cards.csv`
- `control_miss_depth2_depth3_event_chain_detail.csv`
- `depth_transition_table.csv`
- `depth2_depth3_numeric_summary.csv`
- `depth2_depth3_distributions.csv`
- `s_minus_m_47_constant_coordinates.csv`
- `s_minus_m_47_near_constant_coordinates.csv`
- `s_minus_m_47_chain_pattern_counts.csv`