First-Pass Face Patterns at the 64-95 → 32-63 Boundary

R1Observed band ladder

Main observation. First-pass coverage forms a band ladder from the upstream feeder through the 64-95 → 32-63 intersection row and then into downstream reconvergence.

Across the sampled universe, first-pass coverage organizes into an observed band ladder running from upstream to the terminal bins:

96-127 → 64-95 → 32-63 → 16-31 → 8-15 → 4-7 → 0-1

Table 1 gives the per-boundary role, dominant compact face, and dominant share. Roles are observational labels for the patterns found: feeder describes a diffuse upstream surface; intersection row describes the boundary where multiple axes co-occur (R3–R4); merged downstream face / terminal drop describe rows that converge onto one dominant compact face.

Three observations follow directly from Table 1, stated descriptively:

• The feeder 96-127 → 64-95 has the highest face diversity (entropy 4.069) and the lowest dominant-face share (0.178) of any row — it is diffuse, not concentrated on one face.
• The intersection row 64-95 → 32-63 has the lowest face diversity of the upper boundaries (1.643) together with the highest class separation in the table (15.212) — i.e. relatively few faces, but those faces line up closely with class identity. R3-R4 develop this co-occurrence reading.
• All three downstream merged rows and the terminal row have dominant shares ≥ 0.987 and class separation 0.000: previously separated classes are observed to converge onto a shared compact face after the pass (R2).

R2The 64-95 → 32-63 first-pass face and its classes

Main observation. The 64-95 → 32-63 first pass partitions the 538 entering trajectories into four classes, with the A face defined by route plus local all-1 pass context.

R2.1 Entry partition. Of the 550-trajectory universe, 12 trajectories never enter the 64-95 layer (G0); the remaining 538 do enter it (G1), and every G1 trajectory eventually performs a 64-95 → 32-63 pass.

R2.2 First pass vs. first entry. The first event inside the 64-95 layer is never an A face: it is a 64-95 → 64-95 stay, not a pass; no trajectory's first-entry event is an A face (R2.3). Among trajectories that ever satisfy the A face, it coincides with the first 64-95 → 32-63 pass in every observed case. The A face is therefore described as a within-layer first-pass face: it is defined at, and observed to coincide with, the first pass — never with first entry.

R2.3 Four-class partition of G1. Classifying each G1 trajectory by its first-pass descriptor gives four classes that partition the 538 trajectories (Table 2). The term A face (not "A signature") is used for the joint condition from = 64-95, to = 32-63, transition_k = 1, pre_k_window_3 = 1,1,1 (i.e. pass-face all-1, D8), observed under one of two entry routes.

Checks: 365 + 104 + 66 + 3 = 538; 538 + 12 = 550.

A_start and A_inflow share the same local pass face (k=1, pre3=1,1,1) and differ only by entry route. Other_start and Other_inflow are the observed non-A complement at the same boundary, by route. We do not describe any of Other_start/Other_inflow collectively as "near-A failures"; within Other_start, the leading deformations keep START_IN_LAYER and often k=1 but break the all-1 pre-window (cf. Figure 2, "Other_start breakdown" panel).

R2.4 Shared downstream corridor. After the first pass, A_start and A_inflow are observed to follow a common coarse downstream path: their top next-3 and next-5 transition sequences are predominantly 32-63 → 32-63 stays, and the next boundary's dominant face is the same for both, INFLOW_FROM_64-95 | k=3 | pre3=1,1,3+ (cf. Table 1, row 3). These trajectories share a downstream corridor: an observed coarse-path similarity after the first pass.

R3Boundary differential comparison

Main observation. The intersection row is distinguished by a combination of route split, all-1 pass-face context, and class separation, not by maximal neighbor distance alone.

R3.1 Not a maximal-neighbor-distance claim. The intersection row (64-95 → 32-63) is not singled out by raw neighbor distance. The summed descriptive distance around it is 4.676, which is below the downstream-neighbor average of 5.996 (Table 3, where it is listed under its internal analysis label B_sort). Downstream boundaries show large raw label changes for an uninteresting reason: transition_k, pre3, and entry_route are boundary-specific labels, so neighboring boundaries differ in their labels almost by construction.

R3.2 What does single it out: a combination. What distinguishes the intersection row (64-95 → 32-63) is the combination of moderate face diversity (1.643, Table 1) with high class separation specifically among the four first-pass classes. Table 4 decomposes the class-distance total (15.212) at this boundary by class pair: A_start vs A_inflow contributes 4.000 (driven by compact_face_l1 and entry_route_l1, since the two classes differ only by route); A_start vs Other_start contributes 4.606; A_inflow vs Other_start contributes 6.606 (the largest single pair, combining a route difference with the all-1/non-all-1 split). At the upstream feeder (96-127 → 64-95), the same class-pair decomposition sums to only 8.000 overall, and is driven almost entirely by the A_inflow vs. others contrast (A_start vs. Other_start is 0.000 there); downstream of 64-95 → 32-63, all class-pair distances are 0.000 (Table 1).

R3.3 Feature-association screen. A descriptive feature screen over the 64-95 → 32-63 occurrence table (internally, the "sorting-power" screen over boundary B_sort) finds that the feature most strongly associated with class identity is full_face, with a mutual-information-like association score of 1.251 and purity 1.000. We read this as: the full joint first-pass descriptor aligns with class membership almost exactly, which is close to a restatement of how the classes were defined (R2.3) rather than a new discovery. This reflects the class definitions rather than an additional observation.

R3.4 Other boundary-level extremes. The most diffuse boundary by face-diversity is the feeder 96-127 → 64-95 (entropy 4.069, 24 compact faces); the cleanest merged boundary is 16-31 → 8-15 (dominant share 0.998). The upstream feeder supplies the route label (START_IN_LAYER vs. INFLOW_FROM_96-127) used at 64-95 → 32-63, but it does not by itself uniquely identify the later A_inflow class; the A/non-A separation is carried at the 64-95 → 32-63 face itself (cf. Table 4, where the upstream feeder's A_start-vs-Other_start distance is 0.000).

R4Boundary intersection map

Main observation. The 64-95 → 32-63 row is the only boundary where all six displayed observable axes are active in this scan.

R4.1 Thresholds. The intersection map evaluates six observable axes (route split, k split, pre3 split, all-1 context, class separation, face diversity) per boundary on a 0/1/2 scale: distribution axes use 0 if the dominant value's share is ≥ 0.98, 1 if only one non-dominant value has support ≥ 10, 2 if multiple values have support ≥ 10; class separation is 2 for the 64-95 row or total class-distance ≥ 8, 1 for nonzero but weaker separation; reconvergence is 2 on downstream rows where major classes share the dominant compact face; face diversity is 2 when compact-face entropy ≥ 2 or ≥ 20 compact faces are present.

R4.2 Result. 64-95 → 32-63 has the strongest intersection of active axes in the table: active-axis count 6 and total score 12 — the only row that scores on all of route split, k split, pre3 split, all-1 context, class separation, and face diversity simultaneously (Table 5; Figure 4). Downstream rows score mainly, and in several cases solely, on reconvergence. The upstream feeder 96-127 → 64-95 is diffuse (high face diversity) but scores only 3/8 overall, and is explicitly weak as a standalone separator of the later A/non-A classes (cf. R3.4, Table 4).

R5All-1 formation, pass-face all-1, and continuous maintenance

Main observation. Pass-face all-1, continuous pre111 maintenance, and 1111 present at pass are distinct quantities and have different observed values.

R5.1 Pass-face all-1 is true by definition. For A_start and A_inflow, pass-face all-1 (D8) is 1.000 because it is how the classes are defined: the first pass is k=1, pre3=1,1,1. This is a pass-event quantity (D8) and is distinct from Table 6's "pre111 ever appears before pass" column, which instead reports whether pre111 is reached anywhere in the entry-to-pass window, not necessarily at the pass itself (Table 6 below; source field share_first_pre111_present).

R5.2 Continuous pre111 maintenance is strictly lower. Once pre111 first appears within the entry-to-pass window, whether it is held at every subsequent event through the pass is a different and strictly harder condition. The observed shares are A_start = 0.729, A_inflow = 0.548, Other_start = 0.000, Other_inflow = 0.000 (Table 6). This does not contradict R5.1. It is a different, stricter, temporal quantity, and the two are not reported as the same number.

R5.3 Break statistics. Where continuous maintenance fails for A_start/A_inflow, the interrupting break is one of two kinds: pre3 goes to 1,1,2 or to 1,1,3+. For A_start: 1,1,2 in 59 cases (0.596 of breaks), 1,1,3+ in 40 (0.404). For A_inflow: 1,1,2 in 29 (0.617), 1,1,3+ in 18 (0.383). In both classes, the same final step is the one perturbed; the run interrupts rather than collapses.

R5.4 Formation timing. Across the all-1 formation analysis (all1_formation_class_summary.csv), the median distance from the first appearance of 111 to the pass is 1 event for both A_start and A_inflow — that is, the first 111 often appears close to the pass, rather than early in the entry-to-pass window.

R5.5 1111 present at pass is a separate, third quantity. The class-summary column share_maintaining_1111_until_pass reports A_start = 0.564, A_inflow = 0.548. Despite its column name, this is a pass-presence judgment — whether the length-4 run 1111 is present at the pass — directly analogous to R5.1 but one digit longer, and is not a continuous/strict maintenance measure (D10). It is therefore reported as "1111 present at pass" or "pass-event 1111," not as "strict" or "continuous" maintenance, and it is kept visually and verbally distinct from R5.2's 0.729/0.548.

R5.6 Where the separator lives. Among START_IN_LAYER rows, Other_start mostly never reaches 111 before the pass at all (share with pre111 ever appearing before pass = 0.121, Table 6) — only a small subset (8 of 66) reach it and then lose it (the lost-111 micro-cases; cf. Figure 2's "Other_start breakdown" panel). The observational reading is therefore: the separator between A_start and same-route Other_start lives at the pass, in the local all-1 context, not in some earlier formation event.

R6Miss-event local type vs. pre-exit waiting behavior (key-safe join audit)

Main observation. A key-safe join links miss-event local types to pre-exit waiting behavior for all 228 miss rows at exact event-position equality, where the waiting behavior is uniformly no_exit_layer_observed. A separate drift_down overlap of 69 rows is a weak trajectory-band candidate, not a direct event-level join, and the two are not mixed.

This result adds a key-safe join audit between miss-event local types and pre-exit waiting behavior. It is an event-position-key audit, kept structurally separate from the first-pass face results above. As elsewhere in this note, it reports where keys match and where descriptors co-locate. Two link types are reported and are never mixed: a strong event-level join (exact event-position equality) and a weak trajectory-band candidate overlap (trajectory/band coexistence only).

R6.1 No native event-position key in the miss file. miss_local_type_final_detail.csv carries miss values and band_jump, but on inspection it has no event-position column of its own. A direct event-level join therefore could not be performed from that file as-is.

R6.2 Event index recovered for all rows. miss_event_index was recovered for 228 of 228 miss rows from miss_event_detail.csv. In the source script this recovered index is first_pass_index-derived (miss_event_index = int(row['first_pass_index']) in miss_event_local_state_audit.py), so it is an event-row position rather than a separately named odd-step counter.

R6.3 Strong event-level join reaches the waiting layer for 228/228. Using the composite key sample_id + trajectory_id + miss_before_band/band_label + miss_event_index/first_pass_index, the miss rows join to the waiting side at exact event-position equality for 228 of 228 rows. Under this strong join, the observed waiting behavior is no_exit_layer_observed for all 228 rows.

R6.4 The drift_down overlap is a weak candidate, not a direct join. Separately, miss_after_band → drift_down shows 69 overlap rows. These are a weak trajectory-band candidate overlap, selected on sample_id + trajectory_id + target_band_from_band_jump (the right-hand side of band_jump), and they are not a direct event-level join. They can show that a miss-event trajectory and a waiting-hall band record coexist under compatible trajectory/band keys; they cannot show that the miss event occurred at the same event position as a waiting-hall phase boundary. The 69 candidate rows and the 228 no_exit_layer_observed event-join rows are kept separate and are not merged.

Check: 13 + 28 + 0 + 0 + 6 + 22 = 69. These are candidate-overlap counts only; C1 and C2 contribute no rows to the 69.

R6.5 C3 is fully inside the candidate overlap, but n is small. All 6 of the 6 C3 rows fall in the drift_down candidate overlap (6/6, a within-type candidate rate of 100.00%). This is visually striking next to C1 and C2 (both 0), but C3 is an n = 6 cell, so it is treated here as candidate structure, not a load-bearing claim; a within-type rate near 100% on six rows should not be read as more than that small sample supports. The six C3 rows share a common descriptive profile (Table 8). The consistency of that profile across all six rows is consistent with their joint appearance in the candidate overlap, but at n = 6 we do not claim it generalizes.

R6.6 70 → 63 is a remaining_K transition, not a Collatz one-step. The 70 → 63 seen in the C3 rows is not an ordinary Collatz step on an integer value. It is also not an actual trajectory value, a band representative value, a normalized value, or an odd-only integer value. It is a remaining_K_before → remaining_K_after transition: the event consumes transition_k = 7, so 70 − 7 = 63 in remaining-K coordinates.
As a plain Collatz check, 70 is even, so one even step gives 70/2 = 35, not 63; the 70 → 63 step does not arise from a single odd/even rule application. The actual integer-side accelerated step at these events is a separate quantity — for example (3·6869 + 1) / 2⁷ = 161 on one C3 row, and (3·171989 + 1) / 2⁷ = 4031 on others — and must not be conflated with the 70 → 63 remaining-K coordinates.