A within-layer first-pass signature in the remaining_K 64–95 layer

2.2 Local descriptors

• transition_k — the step kt at the transition; in exact tests transition_k=1 means kt=1 exactly.
• pre_k_window_3 — (kt−2, kt−1, kt); the value 1,1,1 is the all-1 local face we abbreviate pre111.
• local_window_4 — the rolling length-4 valuation window near the transition.
• entry_route — how the word occupies the 64–95 layer at the transition.

2.3 Signature hierarchy

We keep the original A signature unchanged and call it A_start. We name a near-neighbour and an umbrella, but the umbrella is descriptive only.

A_all1_pass        64-95 -> 32-63,  transition_k=1,  pre_k_window_3 = 1,1,1
 ├─ A_start        + entry_route = START_IN_LAYER        (= the original A signature)
 └─ A_inflow       + entry_route = INFLOW_FROM_96-127    (near-A; different route)

Other_start        START_IN_LAYER first pass that is NOT A_start

Where the text says "A" without qualification, it means A_start. A_inflow is treated as a separate near-A pass face, not as a relabelling of A_start.

3. The boundary into 64–95

Of the 550 trajectories in the universe, twelve never enter the 64–95 layer at all (group G0); the remaining 538 do enter it (G1). Every trajectory in G1 eventually performs a 64–95 → 32–63 pass.

The twelve G0 trajectories — 7, 9, 18, 19, 25, 91, 121, 242, 243, 484, 485, 486 — are sometimes tempting to read as "near-A failures." The data does not support that reading. None reaches 64–95, so each scores 0/4 on the A conditions for the trivial reason that the source band is never occupied. Under the requested signature clustering they are heterogeneous (every full profile is a singleton); they collapse to eight odd cores. We therefore describe them as boundary non-entrants / low-K cases, set aside from the A neighbourhood rather than placed at its edge.

4. First-pass structure inside 64–95

A does not appear at the first 64–95 event. The first event inside the layer is almost always a 64–95 → 64–95 stay, and no trajectory's first entry event is A-like. Instead, among the 365 trajectories that satisfy A anywhere, A coincides with the first 64–95 → 32–63 pass in all 365 cases (first_A_index = first_pass_index, with none before and none after). In this scanner, A is therefore a within-layer first-pass signature, not an entry signature.

Waiting time inside the layer (events from first entry to first pass) is informative but non-monotone. The A-hit rate rises into a peak and then falls:

Waiting time alone does not describe A cleanly: the longest-wait stratum has the highest pre111 share yet a lower A-hit rate than the mid-wait stratum. Local features carry additional separation. Two thresholds are nearly absorbing in this sample: every A-hit row has a k=1 run of length ≥ 3 and has pre111 before the pass (outside-threshold A-hit rate 0.000 in both cases), and first-pass START_IN_LAYER alone lifts the A-hit rate to 0.847. We read A as a within-layer first-pass face that depends on the wait together with the local all-1 context, not on the wait by itself.

5. A_start and A_inflow

Classifying each G1 trajectory by its first-pass face gives four groups that partition the 538:

A_start and A_inflow share the same local face and differ only by entry route. After the first pass their coarse downstream paths are nearly identical: in both classes the top next-3 and next-5 transition sequences are entirely 32–63 → 32–63 stays and cover the whole class. The median time to the next boundary 32–63 → 16–31 is 23 events for both, and the median time below 16 is 30 for both. The next boundary even wears the same face in both classes:

32-63 -> 16-31    INFLOW_FROM_64-95 / transition_k=3 / pre_k_window_3 = 1,1,3+

The two are not identical on the integer side: A_inflow skews toward larger and longer trajectories (higher mean/median log2(n), word length, and total_K). The descriptive summary is that A_inflow is the same all-1 pass face reached by a different upstream route, not a separate downstream destiny. We keep it under the A_all1_pass umbrella as a near-A face, distinct from A_start by route and provenance.

6. Other_start

Restricting to START_IN_LAYER first passes (431 rows = 365 A_start + 66 Other_start) isolates the cleanest comparison: same entry route, A versus not-A. The 66 Other_start rows are not random residue. The dominant deformation keeps the route and k=1 but breaks the all-1 pre-window:

All 66 fail the pass-time all-1 context (none has pre111, the length-5 suffix 1,1,1, or 1,1,1,1 in local_window_4 at the pass); 20 also fail transition_k=1. Yet downstream they behave much like A_start: their top next-5 sequence is again all 32–63 → 32–63 stays, and the median time to 32–63 → 16–31 is 23, identical to A_start. Other_start is therefore best treated as a separate START_IN_LAYER pass-face family — a dominant near-A deformation plus a low-count heterogeneous tail — that shares A's downstream corridor but not its pass-time face. The separator lives at the pass, in the local all-1 context.

7. Formation and maintenance of the all-1 context

Section 6 locates the separator at the pass; this section follows the all-1 context across the interval from first 64–95 entry to first pass, tracking the first appearance of 1, 11, 111, and 1111 and whether each survives to the pass.

The distinction is not the formation of the all-1 context but its maintenance: A-realizing trajectories maintain 111 to the pass without exception, while every Other_start trajectory either never develops 111 or develops it and then loses it. The maintained interval is short: the median distance from 111 formation to the pass is one event, so the observation is better phrased as "the all-1 context is intact at the pass" than "the all-1 context is held for a long time."

8. Lost111 micro-cases

The eight Other_start trajectories that form 111 and then lose it are the only group with an explicit "made it, then lost it" story, so we read them at the event level and pair each with a matched A_start trajectory (matched on odd core, log2(n), total_K, first-pass wait, and word length; mean differences are small — e.g. |log2(n)| 0.013, |total_K| 0.000).

Every break is the same kind of event: the final 1 is replaced by a larger k (a 2 or a 3+), interrupting the k=1 run. But the break is not at the doorstep: the median distance from break to pass is 6 events, so these are not last-instant slips. Structurally, only 1/8 would have satisfied A_start had 111 been maintained; the other seven also miss the pass-time transition_k=1. We therefore decline to call all eight "near-A failures" and instead describe them as a 111-maintenance-failure micro-subset of Other_start. The representative trajectory cards (e.g. odd cores 639, 2273, 1515) are in lost111_step7_trajectory_cards.md.

9. Connection to the finite-block diagnostics study

This paper is the trajectory-time end of a four-step program that began with a negative result. It is worth stating the link explicitly, because the present finding gives a concrete face to the object that earlier study left open.

9.1 What the diagnostics study closed

The finite-block diagnostics study compared the observed finite-integer escape-word measure against an iid 2-adic reference and asked whether finite-block statistics could generate the difference. Its verdicts were consistent and one-directional: the two diagnostic tests reached class B — the block score separates actual from iid and the separation grows with block length (the +score AUC rises from 0.5363 at L=3 to 0.5643 at L=6) — while the two generative tests reached class C: finite-block reweighting overcorrects and a regularized maximum-entropy block projection does no better than the damped baseline. No test reached A. What survived every test was a reproducible state-distribution residual indexed by bridge shape and parity (bridge coefficient ≈ 1.10–1.13, parity ≈ 0.37–0.44), most visible in the deep-tail focus state late_growth | tail_64_95 | even. The closed claim was a boundary statement: along the iid route, finite-block corrections detect the discrepancy but do not reconstruct it, and the leftover structure is organized by a path-ordered feature.

9.2 Shared coordinates

The two studies are not loosely analogous; they sit on the same axis. The diagnostics focus cell is the deep tail tail_64_95 of the valuation-depth coordinate x_K; Paper 1 showed the closely related remaining_K boundary-distance coordinate binds the same actual − iid difference most sharply, with the deficit running through the 32–63 / 64–95 bands. The present paper lives inside that very region: the 64–95 layer studied here is the deep-tail band where the diagnostics residual was largest. (The coordinates are related remaining-mass quantities, both binned at 64–95, but are not defined identically; see §10.)

9.3 A path-ordered carrier for a path-ordered residual

The diagnostics residual was indexed by bridge shape — a path-ordered feature — and the discussion there singled out "exchangeability breaking with order dependence" as the family the data pointed to: order matters, and finite-block constructions that are largely order-light could not generate the measure. The present finding is a concrete instance of exactly that kind of object. Inside the 64–95 band, the carrier of the conditional excess is not a position-free block statistic but a first-pass face anchored at the boundary, and the cleanest separator of A from its same-route siblings is the maintenance of an all-1 run up to the pass (§7), not its formation. Maintenance is intrinsically order-dependent: it is a statement about a run surviving along the path to the boundary. A finite-block, order-light generator is the wrong instrument to synthesize it — which is consistent with why finite blocks diagnosed but did not generate the discrepancy.

9.4 What this does and does not settle

This is a point of contact, not an identification. Showing that A is a boundary-anchored, maintenance-defined face does not generate the iid-reference discrepancy, recover the bridge/parity residual numerically, or close any test that the diagnostics study left at B or C. The route split here (START_IN_LAYER vs INFLOW_FROM_96-127, with A_inflow skewing to larger log2(n)) and the A signature's integer-side parity/prefix shadow reported in Paper 2 are suggestive neighbours of the diagnostics parity coordinate, but we do not assert they are the same coordinate. The honest statement is that the present first-pass face is one realization of the continuations the diagnostics study named — stopping boundaries and order-dependent structure — and that it is consistent with, but does not resolve, that study's open residual.

10. Interpretation boundaries

• The universe is a sampled, final-state-conditioned ensemble of long words, not an exhaustive enumeration of integers. All shares and rates are within-sample.
• All statements are about a scanner-defined coordinate system (remaining_K, original_n_strict mode). A different mode (odd_core, odd_only) shifts counts; the appendix reports the dependence explicitly.
• The cross-study link of §9 is a correspondence, not an identity. The diagnostics study's x_K valuation-depth coordinate and the remaining_K boundary coordinate used here are related remaining-mass quantities binned at 64–95, but they are not defined identically, and the two studies use different reference measures (iid 2-adic word vs resampled valuation word) and universes.
• The findings are concentration and correspondence statements. No generative mechanism for the A signature, and no causal role for the all-1 context, is claimed or implied.
• Low-count cells (e.g. Other_inflow = 3, the long tail of Table 4, the eight Lost111 cases) are candidate structure, not load-bearing claims.
• "Maintenance" is a descriptive label for "the all-1 context is present at the pass after being present earlier in the same layer occupancy"; it is not a dynamical assertion.

11. Conclusion

The picture that the observations support, stated descriptively:

• A_start is a within-layer first-pass face: the START_IN_LAYER all-1 first pass out of 64–95, coinciding with the first pass whenever it occurs.
• A_inflow is an inflow analogue of the same all-1 pass face, sharing the downstream corridor but reached by a different upstream route.
• Other_start is a separate START_IN_LAYER pass-face family; it shares A's downstream corridor but not its pass-time all-1 face.
• The twelve no-transition trajectories are boundary non-entrants, set aside from the A neighbourhood.
• The cleanest observational separator of A_start from its same-route siblings is the maintenance of the all-1 context to the pass, not its first formation.
• Read against the prior finite-block diagnostics study, this maintenance-defined, boundary-anchored face is a concrete path-ordered carrier in the same deep-tail band where that study's bridge/parity residual was largest — a point of contact with its open residual, not a resolution of it (§9).

Appendix A — Rozier–Terracol overlay

As an external benchmark only, we overlay the Rozier–Terracol Appendix C acyclic paradoxical starting integers (arXiv:2502.00948) onto the remaining_K coordinate and the A signature. Their notion of "paradoxical" — overshoot of the initial value past the stopping time — is defined differently from the mass-deficit / conditional-excess configuration studied here, so this is a correspondence check, not a claim that the two notions coincide.

From 593 Appendix C occurrences (550 distinct starting integers), the full A signature is hit by 365/550 = 0.664 under original_n_strict and 377/550 = 0.685 under odd_core mapping; the broader 64–95 → 32–63 corridor is hit by ~0.97. A robustness recheck (extraction audit, mapping-mode comparison, strict-event definitions, baselines, negative controls) returns the verdict partially robust: the strict A-event share stays high under both mappings, while random same-range negative controls land far lower (full-A shares around 0.17–0.21).

Two caveats are recorded rather than hidden. First, the overlap depends on mapping Rozier–Terracol integers through their odd cores into the odd-accelerated coordinate, while ordinary stopping coordinates are computed on the original integer. Second, the most strongly enriched matched baselines (e.g. residue/prefix matching) are the supported ones; log2-matched baselines have no actual support because the benchmark integers sit below the sampled actual universe. A non-hit taxonomy (which benchmark integers miss strict A, and why) is given in rozier_nonhit_classification.md; most non-hits still traverse the broad corridor but miss strict A by route or local k-context. This material supports an appendix-level benchmark note only.

Appendix B — Output file index

Primary reports underlying each section. Per-trajectory CSVs are referenced inline above.

Observational follow-up paper. No mechanism, causality, or proof is claimed.