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

This repository contains a descriptive finite-sample audit of prefix-local descriptors in odd-accelerated Collatz trajectories.

The analysis asks whether quantities computed only from the observed prefix n_0 ... n_t are associated with targets computed over a fixed finite-future window n_{t+1} ... n_{t+H}.

Among the three tested prefix-local descriptors, cumulative_log_drift was the strongest descriptor across all five finite-future targets. Reference- coupling checks were then used to test whether the observed association was merely produced by the descriptor and targets sharing the current reference value n_t.

A bridge-matched comparison with an existing retrospective waiting-hall classification found a limited result labelled partial_prefix_shadow: within the 532-event overlap subset, the drift_down group was lower in both cumulative_log_drift and future_net_log_change. The corresponding drawdown comparison remained unresolved.

This is a finite-sample descriptive audit. It is not a convergence predictor, a causal mechanism, a new invariant, or a proof concerning the Collatz conjecture.

This report combines three light, reuse-oriented audits.

1. Light prefix audit. Among three prefix-local descriptors (record_age, above_start_dwell, cumulative_log_drift), cumulative_log_drift was the strongest prefix-side descriptor across all five finite-future targets examined.
2. Reference-coupling reuse audit. The association was tested against the possibility that it is merely an artifact of shared reference definitions (both the descriptor and the targets refer to the current value n_t). For future_net_log_change and future_max_log_drawdown, the association survived future-only redefinition, current-state matching, residualization, and a within-current-state permutation null, and was labelled persists_beyond_shared_reference.
3. Waiting-hall bridge alignment audit. The prefix-side descriptor was then compared against a retrospective waiting-hall classification through a bridge key. Within the bridge-matched overlap with the light-audit sample, the result was partial_prefix_shadow.

Key conclusion. Within the bridge-matched overlap between the waiting-hall archive and the light prefix audit, the drift_down group shows a partial prefix-side shadow: it is lower in cumulative_log_drift and lower in future_net_log_change. This supports a limited connection between the retrospective waiting-hall classification and a prefix-side finite-future signal, but it does not extend cleanly to future_max_log_drawdown and should not be generalized to the full waiting-hall archive.

This is a descriptive finite-sample audit. It is not a proof, not a convergence predictor, not a causal mechanism, and not a new theory.

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 comparison asks a narrow question: does a classification defined after the fact leave any trace on the entrance side, i.e. in quantities computable from the prefix alone?

Bridge matching confirmed the waiting-hall table linkage (74,116 event keys). The actual prefix-side comparison, however, is restricted to the 532-key overlap with the existing light-audit sample, expanded to 3,382 rows. All alignment statements below are statements about that overlap subset only.

No new sampling was performed at any stage; the same seed, split, sample and H list were reused throughout.

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

Sign direction is consistent: higher cumulative_log_drift goes with lower future_net_log_change (negative share 0.99) and larger future_max_log_drawdown (positive share 0.99). future_drop_below_start is weak and is deferred.

Redefining the targets so they depend only on the interior of the future window (future_internal_net_log_change, future_internal_max_log_drawdown) left the cumulative_log_drift trends essentially intact (absolute trend ratio ≈ 1 at almost every H). The shared reference point is therefore not a sufficient explanation on its own. record_age and above_start_dwell stay unresolved in this family.

Group means over the 3,382-row overlap subset, by the retrospective trajectory_behavior_class label:

Contrast of drift_down against all other groups:

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

The safest reading, in order of confidence:

• cumulative_log_drift is a useful prefix-local axis — a ruler that touches no terminal information — and it is consistently associated with several finite-future targets in this sample.
• That association is not fully explained by the shared reference value n_t, since 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: it is 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 retrospective grouping is not purely hindsight — some component of it is visible from the entrance side.
• The word for this is partial prefix-side shadow, not "explained". One group, two aligned quantities, one unresolved quantity, one small overlap subset.

• Small overlap. The alignment rests on 532 bridge-validated event keys (3,382 H-expanded rows). The drift_down group dominates the overlap; direct_to_exit has 2 unique events and no_exit_layer_observed has 10. Group contrasts involving those cells are effectively anecdotal.
• Light 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 live concern. future_net_log_change and future_max_log_drawdown are referenced to n_t, which also appears inside cumulative_log_drift. The audit reduces this concern; it does not eliminate it.
• Row-level regeneration. Public light CSVs are aggregate-only; row-level values were regenerated from the same seed/sample for bridge-validated keys. Recomputation matched the existing seed-1 trends 28/28, but this is a reuse pipeline, not an independent replication.
• record_age and above_start_dwell remain unresolved in the coupling audit and should not be promoted to the main story.

This work does not claim:

• a proof, disproof, or partial proof of the Collatz conjecture;
• a predictor of final convergence to 1 — no target here is a terminal outcome;
• a causal mechanism, a "descent force", "mean reversion", or a "restoring law";
• a new theory, a new invariant, or a newly discovered quantity — cumulative_log_drift is close to standard drift / log-height descriptors, and the only unusual element is its use as a prefix-local baseline axis for comparing fixed finite-future targets under shared splits;
• that the prefix side can reproduce the waiting-hall classification;
• that the partial_prefix_shadow finding extends to the full waiting-hall archive, to groups other than drift_down, or to future_max_log_drawdown.

Permitted vocabulary: prefix-side descriptor, prefix-local axis, finite-future target, finite-horizon trend, resampling stability, partial prefix-side shadow, unresolved.

No standard-run, no new targets, no thickened bootstrap, and no change to the waiting-hall labels were performed in producing this report.