4.2 remaining_K bins

The bins are half-open intervals; a transition A → B occurs at position \(t\) when \(R_t\) lies in bin A and \(R_{t+1}\) lies in bin B.

0-1      : 0   <= R < 2
2-3      : 2   <= R < 4
4-7      : 4   <= R < 8
8-15     : 8   <= R < 16
16-31    : 16  <= R < 32
32-63    : 32  <= R < 64
64-95    : 64  <= R < 96
96-127   : 96  <= R < 128
128-191  : 128 <= R < 192
192+     : 192 <= R

4.3 transition_k

For a transition occurrence at position \(t\), transition_k is the valuation step \(k_t\). In capped pattern summaries, values greater than \(2\) are reported as 3+; in exact event tests such as the A signature, transition_k=1 means \(k_t = 1\) exactly.

4.4 pre_k_window_3 and local windows

For a transition occurrence at position \(t\), \( \texttt{pre\_k\_window\_3} = (k_{t-2}, k_{t-1}, k_t) \), truncated at the start of the word when \(t<2\). pre_k_window_5 is the length-5 analogue. Pattern tables cap entries greater than \(2\) as 3+. The local_rolling_k_window_4 is the set of length-4 valuation windows whose index range contains the occurrence \(t\), subject to the word boundary; local_rolling_k_window_3 is the length-3 analogue.

4.5 entry route

For each remaining_K bin, let \(s\) be the first position whose pre-step remaining_K value lies in that bin. An occurrence at position \(t\) in that bin has entry route START_IN_LAYER if \(s=0\), and otherwise INFLOW_FROM_<previous bin>, where the previous bin is the bin at position \(s-1\).

4.6 pass and stay

For a target transition from bin \(A\) to bin \(B\), an occurrence starting in bin \(A\) is pass if its post-step bin is \(B\), and stay if its post-step bin is still \(A\). Other exits are excluded from the pass/stay conditional comparison.

4.7 A signature

The canonical A signature is an occurrence-level condition; all conditions must hold on the same transition occurrence.

from_bin       = 64-95
to_bin         = 32-63
entry_route    = START_IN_LAYER
transition_k   = 1
pre_k_window_3 = 1,1,1

4.8 focus_state

focus_state is a composite state label fixed by the pipeline; the specific state attended to here is late_growth|deep_32_63|even.

4.9 actual and iid word samples

The actual sample consists of accelerated Collatz valuation words generated from sampled odd integers in the specified power/layer ranges of the pipeline. The iid control sample consists of independent valuation words generated by drawing valuation steps from the tilted valuation distribution used in the pipeline and stopping when the accumulated log-position crosses the layer escape threshold. Each iid word carries the Monte Carlo weight assigned by the sampling procedure.

4.10 Weighting and derived statistics

Sequence-level tables are occurrence-weighted unless otherwise stated. Integer-side projection tables report the associated integer support separately, because a single integer may contribute more than one qualifying occurrence. from mass is the weighted mass placed in a from-bin; pass rate is the conditional probability of passing given occupancy; conditional delta is actual pass rate − iid pass rate; mass delta is actual from mass − iid from mass; pass share is the share of total pass mass attributable to a sub-population.

4.11 Paradoxical configuration (observational label)

A from-bin exhibits the paradoxical configuration when mass delta < 0 (mass deficit) and conditional delta > 0 (conditional excess) hold simultaneously. The term is used as an observational label only.

5. The remaining_K chain: mass deficit with conditional excess

We examine three adjacent downward transitions and report, for each, the from-bin mass in both ensembles, the conditional pass rates, and their differences.

In all three transitions the from-bin is thinner in the actual ensemble than in the surrogate, while the conditional pass rate is higher in the actual ensemble. Mass placement and local transition strength carry opposite signs; this is the paradoxical configuration we decompose.

The three transitions are not equally read. The conditional excess is numerically largest at 96-127 → 64-95, but that transition has by far the smallest support, and its leading sub-patterns are dominated by low-support cells. The mass deficit grows monotonically down the chain and is most stably measured at 32-63 → 16-31. The transition 64-95 → 32-63 sits in the middle: a clearly negative mass delta, a positive conditional delta, and adequate support. We therefore treat 64-95 → 32-63 as the primary object of decomposition, 32-63 → 16-31 as secondary, and hold 96-127 → 64-95 as provisional.

6. Paradoxical sequences: sequence-level carriers

Having established the reversal, we ask whether the conditional excess is spread evenly across trajectories or concentrated in particular sequence-level signatures. We compare four groups within each from-bin: actual pass iid pass actual stay, iid stay.

6.1 Entry-route decomposition of 64-95 → 32-63

Splitting the cleanest transition by entry route is the single most informative cut.

The conditional excess is concentrated in START_IN_LAYER. The inflow route carries a comparable from-mass deficit but a slightly negative conditional delta; it does not carry the excess. We therefore read the configuration not as "the actual ensemble falls downstream more readily" in general, but more specifically as "trajectories that begin in the 64-95 bin fall downstream more readily in the actual ensemble." This is one increment of information beyond the raw conditional delta.

6.2 Pattern concentration within START_IN_LAYER

Within START_IN_LAYER, the strongest positive pass-side concentrations are short and all-1: transition_k=1 (pass-share delta +0.0434), pre_k_window_3=1,1,1 (+0.0238), local_rolling_k_window_3=1,1,1 (+0.0130), local_rolling_k_window_4=1,1,1,1 (+0.0119), all at adequate support. The joint cells sharpen this further.

The counter-patterns — transition_k=2, pre_k_window_3=1,1,2, pre_k_window_3=1,1,3+, local_rolling_k_window_4=2,1,1,1 — are depleted on the actual pass side, all at adequate support, and read mostly as iid-enriched pass patterns or actual-stay-enriched patterns rather than support artifacts. They are essentially the complement of the positive component, which we take as evidence that the concentration is structural rather than noise.

6.3 Focus-state check

We attended to the composite state late_growth|deep_32_63|even. For 64-95 → 32-63 this state shows a positive conditional delta (+0.00204, ok support) concentrated in START_IN_LAYER, consistent with the overall reading. For 32-63 → 16-31 the same state shows a smaller, route-split signal. The focus state does not introduce a qualitatively different structure; it is consistent with, but not sharper than, the route- and pattern-level decomposition.

7. Band Signature Map

We next ask whether the A signature is specific to 64-95 → 32-63 or whether the other bands present the same face. For each transition and route we rank positive pass-side concentrations under a fixed template.

The cross-band reading is twofold. First, the bands do not share one identical signature; the common element is the type of configuration — concentrated conditional excess under mass deficit — rather than a single carrier. Second, the 64-95 → 32-63 all-1 face is the cleanest, 32-63 → 16-31 is a distinct transition_k=2 face, and 96-127 → 64-95 is too low-support to classify with confidence. We therefore describe the bands as exhibiting band-specific faces of a common paradoxical configuration, rather than a single universal signature.

A measurement caution is carried in the map itself: narrow joint cells can report large conditional deltas simply because they are narrow. The map distinguishes readable carriers (adequate support, stable across the template) from top joint cells (largest conditional delta, possibly narrow), and we read the former rather than the latter.

8. Integer-side projection

We project the 64-95 → 32-63 A signature back onto integer coordinates and ask whether actual integers carrying the signature are biased in integer space. This is a bias check on a correspondence, not a property claim about integers. We form four occurrence-weighted groups of actual integers, all within START_IN_LAYER at the 64-95 bin: A (signature pass), B (non-signature pass), C (signature stay), D (non-signature stay).

8.1 Scale

The signature-pass group sits at slightly lower log2(n): the A-vs-B mean shift is −0.894 (median −0.766), and the A-vs-C mean shift is −0.498 (median −0.333). Within the same bin and route, integers carrying the signature are observed at somewhat smaller scale.

8.2 Residue and prefix structure

The A group is enriched, relative to the other groups, in the maximal-residue classes: mod 8 residue 7 leads (≈0.382), mod 16 residue 15 (≈0.261), mod 32 residue 31 (≈0.179), mod 64 residue 63 (≈0.097). Correspondingly, the leading parity prefixes are all-1 (1111, 111111) and the leading valuation prefix is 1,1,1,1. These are the integer-coordinate images one would expect of a transition_k=1 / all-1 sequence signature.

9. Matched control

A shadow alone does not distinguish "the sequence signature predicts pass beyond its integer image" from "the sequence signature is merely the image of an integer-side coordinate." We match A against control groups sharing its integer-side coordinates and ask whether the signature still separates pass from stay after matching.

The reading is a qualified yes. No single integer-side coordinate — log2(n), any single residue modulus, or any single parity prefix — accounts for the signature; matched on any one of these, the matched delta is negative. The signature does survive exact matching on the full length-4 valuation prefix, and it survives the supported combined controls that hold log2(n), a residue modulus, and a parity prefix simultaneously. We conclude that, in this sampled matched comparison, the integer-side shadow does not fully explain the A signature, but the surviving signal is joint-coordinate dependent rather than a one-dimensional residue or scale effect. We make no claim beyond predictiveness after the listed controls.

10. Discussion

The measurements assemble into a single descriptive chain. A difference between observed trajectories and a structureless surrogate, previously localized to a remaining_K chain, takes the form of a sign reversal: mass deficit with conditional excess. The excess is not diffuse; it is concentrated in a small number of sequence-level signatures. The clearest carrier, at 64-95 → 32-63, is a compact joint signature — START_IN_LAYER, transition_k=1, all-1 local context — which we have called the A signature. Across bands the configuration is shared but the signature is not: each band has its own face. Projected onto integer coordinates, the A signature leaves a weak but reproducible shadow, and that shadow is multivariate: it resists explanation by any single integer-side coordinate yet persists under joint control.

The external benchmark (Appendix A) is consistent with the A signature marking a broad corridor rather than an isolated cell: a substantial fraction of an independently defined set of paradoxical starting integers maps onto it, and the non-mapping integers are mostly one condition away from it. We read this as the A signature occupying a canonical face of the corridor — a representative, not a unique, carrier — and we do not infer sameness of phenomenon from overlap.

11. Limitations

• The analysis universe is a sampled long-word, final-state-conditioned ensemble, not an exhaustive enumeration. All shares and supports are ensemble quantities.
• Low support is flagged throughout at the weighted threshold 0.002; low-support cells are candidates, not claims. In particular the 96-127 → 64-95 band is provisional.
• Rolling windows are local to the transition neighborhood and do not represent full-word window frequencies.
• The matched-control survival is joint-coordinate dependent; matching removes only the listed integer-side coordinates, not all possible integer structure.
• The surrogate ("iid word") ensemble removes sequential structure while preserving the tilted valuation marginal of the pipeline; its construction and weighting are given in §4.9 and are pipeline choices.
• The word paradoxical is used only as an observational label for the mass-deficit / conditional-excess configuration and is not the Rozier–Terracol notion.
• No causal, generative, or mechanistic claim is made anywhere in this paper.

12. Conclusion

We have reported, in observational terms, that observed accelerated Collatz trajectories and a structureless surrogate differ on a remaining_K chain in a sign-reversing way — mass deficit together with conditional excess — and that the conditional excess is concentrated in a small number of band-specific sequence signatures. The cleanest carrier is the 64-95 → 32-63 A signature. The signature leaves a weak, multivariate, reproducible integer-side shadow that no single integer coordinate explains. An external benchmark overlaps substantially with the signature without being identical to it. The result is a map of where the reverse-sign phenomenon is carried and how it corresponds to integer coordinates; it is not an explanation, and is offered as a measurement.

Appendix A. Rozier–Terracol external benchmark overlay

A.1 Purpose and benchmark set

We overlay the Rozier–Terracol Appendix C acyclic paradoxical starting integers (arXiv:2502.00948) on the remaining_K coordinate system and the 64-95 → 32-63 signature, as an external benchmark only. Extraction reproduces 593 Appendix C occurrences and 550 distinct starting integers, of which 40 appear in multiple (j,q) groups; the odd/even split is 294 / 256.

A.2 Overlap with the A signature

The benchmark overlaps strongly with the A carrier and does not align at all with the checked 32-63 → 16-31 face.

A.3 Robustness recheck

Because the overlap is strong, we audited extraction, mapping mode, event strictness, counting mode, baselines, and negative controls.

First-occurrence vs. trajectory. Under the first-64-95-occurrence-only definition, the A overlap is 0. The overlap is a trajectory-level event: a substantial subset of benchmark integers encounters a strict A signature somewhere along the trajectory, not at first entry into the 64-95 bin. This is consistent with the §4.5 definition, under which START_IN_LAYER marks where the word begins, not the first entry event.

Same-occurrence validity. A separate audit confirms that transition_k=1 and pre_k_window_3=1,1,1 are satisfied within the same occurrence (validity 1.0 in all modes; one strict A occurrence per integer), ruling out a cross-occurrence pooling artifact.

Negative controls. Full A overlap for control sets is markedly lower: random same-size range 0.205, consecutive from minimum 0.204, odd random 0.175, and shifted sets (n±1, 2n+1) 0.33–0.35. The benchmark overlap substantially exceeds these.

A.4 Classification of non-overlapping integers

The integers that do not hit the full A signature are not a random remainder. For the original-n strict mode the 185 non-hits classify as follows; 164 are near-A (three of four A conditions), the most common missing condition being START_IN_LAYER. The odd-core mode gives the same picture (154/173 near-A).

A.5 Reading and limits

We read the benchmark overlay as consistent with the A signature occupying a canonical face of a broad corridor through which many of the benchmark integers pass. We do not claim that the Rozier–Terracol paradoxical sequences and the configuration studied here are the same phenomenon; the two notions are defined differently. Odd-core dependence is reported explicitly. Baselines lacking actual support (notably the log2-matched baselines, since the benchmark integers are much smaller than the sampled actual universe) are not used as claims. This is an external benchmark overlay only.

Suggested figures and tables

Editorial only; drawn entirely from existing outputs. None requires additional computation.

1. Figure 1 — the reverse-sign chain. Twin bars per transition: mass Δ (negative) beside conditional Δ (positive) for §5, support encoded by bar width or opacity.
2. Figure 2 — entry-route decomposition. START_IN_LAYER vs. INFLOW_FROM_96-127, conditional Δ and pass-share side by side (§6.1).
3. Figure 3 — A-signature concentration. Pass-share delta by pattern within START_IN_LAYER, sorted, with the all-1 carriers and their counter-pattern complements (§6.2).
4. Figure 4 — Band Signature Map. Small multiples, one tile per band, with the 96-127 → 64-95 tile marked provisional (§7).
5. Figure 5 — integer-side shadow. Overlaid log2(n) distributions for A/B/C/D, plus a residue-enrichment strip for mod 8/16/32/64 (§8).
6. Figure 6 — matched-control ladder. Enrichment ratio per control with the 1.0 line marked, separating single-coordinate (below 1) from joint controls (above 1) (§9).
7. Figure 7 (Appendix) — Rozier overlap tree. The 550 → A-hit / non-hit split with the four non-hit classes, annotated with near-A count (A.4).
8. Tables 1–5, A1–A3. As printed above.