Descriptive finite-sample audit · odd-accelerated Collatz

Prefix-Local Descriptors and a Partial Prefix Shadow of the Waiting-Hall Classification

Lightweight prefix audit → reference-coupling reuse audit → waiting-hall bridge alignment

01The 3-minute version

We asked a simple question about Collatz trajectories: if we look only at the first part, do we see any differences in what happens next?

What we found

We looked at how much the numbers rose and fell in the first part. The trajectories then showed some differences in how they moved for a while afterward.

This does not mean that the first part predicted the later motion exactly. Some differences were visible; others remained unclear.

This is not a proof of the Collatz conjecture. We did not test every number, and we did not discover why the pattern occurs. It is an observation within the range of data examined here.

That is the short version. The sections below move step by step through the audit summary, comparisons, numerical results, limitations, and source files.

02Audit summary

This report combines three lightweight, reuse-oriented audits. First, among three prefix-local descriptors, cumulative_log_drift was the strongest descriptor across all five finite-future targets examined. Second, a reference-coupling audit tested whether this was merely an artifact of shared reference definitions such as n_t; for future_net_log_change and future_max_log_drawdown, the signal persisted after accounting for the shared reference. Third, the prefix-local axis was compared against a retrospective waiting-hall classification, yielding a partial_prefix_shadow verdict.

This is a descriptive finite-sample audit, not a claim about proof, convergence, or causality.

Safe takeaway

Within the bridge-matched overlap between the waiting-hall archive and the lightweight prefix audit, the drift_down group shows what we call a partial prefix shadow: it has lower values for cumulative_log_drift and future_net_log_change.

This supports a limited connection between the retrospective waiting-hall classification and a prefix-local finite-future signal. It does not extend cleanly to future_max_log_drawdown, and it should not be generalized to the full waiting-hall archive.

03What was compared

Table 1 · The three sides of the comparison
Side	Quantity	Computed from
Prefix-local axis	`cumulative_log_drift`, `above_start_dwell`, `record_age`	only n₀ … n_t — no terminal information
Finite-future targets	`future_net_log_change`, `future_max_log_drawdown`, `future_record_update`, `future_above_start_dwell`, `future_drop_below_start`	only the window n_t+1 … n_t+H
Retrospective classification	waiting-hall labels (`trajectory_behavior_class`, `near_behavior`, `waiting_class`, `position_label`)	pre-existing waiting-hall archive; not relabeled here

Horizons are fixed at H = 4, 8, 12, 16, 20, 24, 28, 32, 40, 48, 64, 80, 96, 128 (14 values). Terminal quantities — remaining_K, steps-to-1, total trajectory length, terminal suffix — are never used on the descriptor side. Feature-side leakage flag count: 0.

The question is deliberately narrow: does a classification defined retrospectively leave any trace in quantities computable from the observed prefix alone?

04Inputs and overlap

Table 2 · Sources, matching, and the subset actually compared
Component	Source or key	Count
Lightweight prefix audit	seeds 1–10; discovery 5,000 / validation 5,000; 50 bootstrap replicates	5 targets × 3 descriptors × 14 H
Reference-coupling reuse audit	seed 1, same sample; no new sampling	0 mismatches across 28 comparisons
Bridge file rows	`exit_neighborhood_per_trajectory.csv`	104,272
Bridge ↔ waiting-hall matched event keys	sample_id + trajectory_id + event_index	74,116
Overlap with lightweight-audit seed/split sample	bridge-validated keys used for the comparison	532
Rows after horizon expansion	532 keys × horizons	3,382

Bridge matching confirmed the waiting-hall table linkage at 74,116 event keys. The prefix-local comparison itself is restricted to the 532-key overlap with the existing lightweight-audit sample, expanded to 3,382 rows across horizons. Every alignment statement below is a statement about that overlap subset only. No new sampling was performed at any stage.

05Main findings

4.1 Lightweight prefix audit

cumulative_log_drift was the best-ranked prefix-local descriptor for every target examined.

Table 3 · Best prefix-local descriptor per finite-future target (validation)
Finite-future target	Best prefix-local descriptor	Stable H	Median validation R²
`future_above_start_dwell`	`cumulative_log_drift`	14 / 14	0.396
`future_record_update`	`cumulative_log_drift`	14 / 14	0.369
`future_max_log_drawdown`	`cumulative_log_drift`	14 / 14	0.205
`future_net_log_change`	`cumulative_log_drift`	14 / 14	0.162
`future_drop_below_start`	`cumulative_log_drift`	11 / 14	0.045

Direction is consistent: a higher prefix-local drift value goes with a lower future_net_log_change (negative share 0.99) and a larger future_max_log_drawdown (positive share 0.99). future_drop_below_start shows only a weak signal and is not pursued further here.

4.2 Reference-coupling reuse audit

Both the descriptor and the targets refer to the current value n_t. Redefining the targets so that they depend only on future-window values, excluding the current state, left the drift trends essentially intact (absolute trend ratio ≈ 1 at almost every H).

Table 4 · Coupling verdict per descriptor × target family (14 horizons)
Descriptor × target family	Future-only attenuation	Persists after state matching	Non-near-zero residual H	Outside null	Final label
`cumulative_log_drift` × `net_log_change`	little 14 / partial 0 / lost 0	8 / 14	13 / 14	10 / 14	persists_beyond_shared_reference
`cumulative_log_drift` × `max_log_drawdown`	little 13 / partial 1 / lost 0	8 / 14	13 / 14	11 / 14	persists_beyond_shared_reference
`above_start_dwell` × net / drawdown	—	1 / 0	12 / 13	9 / 9	unresolved
`record_age` × net / drawdown	—	0 / 0	13 / 13	12 / 11	unresolved

The shared reference point is therefore not a sufficient explanation on its own for the net-change and drawdown families. The other two descriptors remain unresolved and are not included in the primary interpretation.

4.3 Waiting-hall bridge alignment

Table 5 · Group means over the 3,382-row overlap subset, by trajectory_behavior_class
Waiting-hall group	Unique events	Rows (H-exp.)	Mean cumulative_log_drift	Mean future_net_log_change	Mean future_max_log_drawdown
`drift_down`	490	3,029	−2.72	−3.17	4.37
`mid_band_wait_then_drop`	30	268	−2.59	−1.32	4.02
`no_exit_layer_observed`	10	63	0.02	−5.64	6.55
`direct_to_exit`	2	22	0.29	−1.30	2.75

Table 6 · Contrast of drift_down against all other groups
Quantity	drift_down − other	Direction
`cumulative_log_drift`	−1.963	prefix-local axis: lower
`future_net_log_change`	−0.416	same direction
`future_max_log_drawdown`	−0.071	unresolved — not consistently aligned

Verdict: partial_prefix_shadow (fold-wise stable rows for drift_down: 15). The group has a lower mean value on the prefix-local axis and a lower mean finite-future net log change. The drawdown target does not follow consistently, so the verdict is not terminal_consistent_prefix_signal.

06Interpretation

cumulative_log_drift works as a prefix-local axis based exclusively on prefix information and is consistently associated with several finite-future targets in this sample.
That association is not fully explained by the shared reference value n_t: it survives future-only redefinition, current-state matching, residualization, and a within-current-state permutation null for the net-change and drawdown families.
The waiting-hall classification is retrospective — built from what a trajectory eventually did. Finding that its drift_down subgroup is also low on the prefix-local axis, and low in future_net_log_change, indicates that this particular grouping is not solely an artifact of retrospective labeling: some component of it is visible in the observed prefix.
We refer to this pattern as a partial prefix shadow, not as an explanation. It concerns one group, two aligned quantities, one unresolved quantity, and one small overlap subset.

07Limitations

Small overlap. The alignment rests on 532 bridge-validated event keys (3,382 rows after horizon expansion). drift_down dominates the overlap; direct_to_exit has 2 unique events and no_exit_layer_observed has 10, so contrasts involving those cells are too sparse to support stable inference.
Lightweight audit, not standard audit. Seeds 1–10, 50 bootstrap replicates, representative residue/stride checks only. The reference-coupling stage is a single-seed (seed 1) reuse audit.
Sample scope. Odd original_n ≤ 1,000,000, odd-accelerated map, sampled subsets — not all integers.
Definitional coupling remains a relevant concern. The targets are referenced to n_t, which also appears inside the descriptor. The audit reduces this concern; it does not eliminate it.
Reuse pipeline. Public lightweight-audit CSVs are aggregate-only; row-level values were regenerated from the same seed/sample for bridge-validated keys. Recomputation matched all 28 existing seed-1 trends, but this is reuse, not independent replication.
record_age and above_start_dwell stay unresolved in the coupling audit.

08Scope of the claims

The results concern descriptive finite-future associations; they do not provide a proof, disproof, or partial proof of the Collatz conjecture.
Eventual convergence to 1 lies outside the scope of the targets, none of which is a terminal outcome.
The analysis is descriptive rather than causal and does not identify a "descent force", "mean reversion", or restoring law.
cumulative_log_drift is treated as a baseline descriptor close to standard drift / log-height measures, not as a new theory or newly discovered quantity. Its distinctive use here is as a prefix-local baseline axis for comparing fixed finite-future targets under shared splits.
The prefix-local quantities are not claimed to reproduce the waiting-hall classification.
The partial_prefix_shadow finding is limited to the stated overlap, the drift_down group, and the two aligned quantities; it is not extended to the full archive or to future_max_log_drawdown.

09Audits, CSVs, and code

Table 7 · Source artifacts behind this summary
File	Stage	Contents
`future_behavior_target_light_audit_README.md`	Lightweight audit	Setup: seeds, bootstrap count, descriptors, targets
`light_target_summary.csv`	Lightweight audit	Best prefix-local descriptor and R² per target
`light_observation_target_summary.csv`	Lightweight audit	Every descriptor × target pair; stable-H counts, sign shares
`light_full_run_recommendations.csv`	Lightweight audit	Coarse priority order for a possible standard run
`reference_coupling_reuse_audit_README.md`	Coupling audit	Reuse setup; seed 1; 0 result-diff mismatches
`coupling_attenuation_by_H.csv`	Coupling audit	Original vs. future-only trend per H; attenuation labels
`matched_current_state_summary.csv`	Coupling audit	Within-current-state-cell trends after matching
`reference_coupling_verdict.csv`	Coupling audit	Final label per descriptor × target family
`waiting_hall_bridge_prefix_alignment_README.md`	Bridge audit	Bridge key, match counts, verdict
`waiting_hall_bridge_join_manifest.csv`	Bridge audit	Source files, join keys, matched row counts
`waiting_hall_bridge_group_alignment.csv`	Bridge audit	Group means for every waiting-hall label column
`waiting_hall_bridge_alignment_verdict.csv`	Bridge audit	partial_prefix_shadow verdict and group contrasts

This report used no standard run, added no new targets, did not increase the number of bootstrap replicates, and made no changes to the waiting-hall labels.