Coordinate Audit for the Paradoxical Sequence Benchmark

1.Why this audit is needed

Three different constructions are used in discussions of the Paradoxical Sequence, and they are easy to conflate:

1. the Rozier–Terracol (RT) definition itself, computed on the Appendix C occurrences (n, j, q);
2. our sampled-universe baseline, where the k-word is W = trace_escape(n, 1 << power) and remaining_K is defined inside that word;
3. the RT550 overlay, where the RT integers are re-walked with the odd-accelerated map all the way to 1 and then read in a remaining_K coordinate.

These are not the same coordinate system. A statement like "RT550 is A-like" changes truth value depending on which construction and which counting mode (integer-level any-event vs first-hit strict vs occurrence-level) is meant. This README pins each published number to its construction so that no result is quoted outside its own coordinate system.

2.Coordinate systems at a glance

One-line rule: Experiment 1 reproduces RT; Experiment 2-A is our own result; Experiment 3 only overlays RT550 onto our coordinate. They are sibling experiments, not a pipeline.

Definition recomputed (none of our coordinate machinery is used). Input is one Appendix C occurrence (n, j, q). The map is T(n) = (3n+1)/2 for odd n, T(n) = n/2 for even n. The finite interval n, T(n), …, T^j(n) is recomputed; q is the number of odd terms among T^0(n), …, T^(j-1)(n). The RT paradoxical condition is 3^q / 2^j < 1 and T^j(n) ≥ n. No remaining_K, no trace_escape, no A-face coordinate appears anywhere in this experiment.

Global checks (all pass):

• occurrences: 593; distinct starting integers: 550; distinct duplicated integers: 40
• recomputed q matches Appendix C q for all rows: True
• coefficient 3^q / 2^j < 1 for all rows: True
• RT paradoxical condition holds for all rows: True
• cyclic rows: 0; acyclic rows: 593; near-cycle rows with T^j(n) − n = 1: 20
• rows containing 11: 5; rows containing 103: 588
• rows starting and ending at odd integers: 148

By group (j, q):

Status. This is the clean RT baseline. Any later remaining_K, A-face, or 64-95 -> 32-63 statement about these integers must be introduced only as a separate projection from these reproduced occurrences into another coordinate (that projection is Experiment 3, never Experiment 1).

This is the main coordinate system of the Paradoxical Sequence chapter, and it does not use RT550. The universe is our own sampled long-word / final-state-defined universe. The word is W = trace_escape(n, 1 << power), and inside that word remaining_K_before = sum(W) − sum(W[:pos]), remaining_K_after = remaining_K_before − transition_k. Band names like 64-95 refer to remaining_K bands inside W, not to integer values.

Inverse-sign observation across the three target transitions (from-band mass deficit together with positive conditional pass excess):

Deep dive on 64-95 -> 32-63 (the cleanest paradoxical transition in this sample): the conditional excess is concentrated in the START_IN_LAYER route (actual pass 0.136904 vs iid 0.132609, conditional delta 0.00429505), not in INFLOW_FROM_96-127 (conditional delta -0.000589567).
Within START_IN_LAYER, the strongest readable positive component is transition_k = 1 with short all-1 local context — pre_k_window_3 = 1,1,1 and local_rolling_k_window_4 = 1,1,1,1. The strongest joint cell is transition_k=1 & pre_k_window_3=1,1,1: actual pass 0.122009 vs iid pass 0.0924706, conditional delta 0.0295384. Counter-patterns (transition_k=2, 1,1,2, 1,1,3+) read as iid-enriched pass or actual-stay-associated patterns.

Status. Because this reproduces on our own universe, the chapter's claims about 64-95 -> 32-63, START_IN_LAYER, transition_k=1, pre111, and the A face do not depend on RT550. Low-support rows are flagged in the CSVs and are hypothesis material only.

This is an overlay, not the RT definition and not the trace_escape window. Each RT550 integer is walked with the odd-accelerated map to 1 (odd_core strips the initial power of two;
original_n_strict prepends the initial v2(n)), and remaining_K is computed on that generated k-word. The RT (j,q) finite interval is not used as the window, and the RT550 rows carry no power box, so trace_escape is not applicable. Recomputation against the existing overlay CSV shows 0 mismatching rows.

Counting modes must be separated. The same overlay produces very different numbers depending on how hits are counted:

original_n_strict gives the same picture with slightly different values (e.g. hits_full_A_signature 365/550 = 0.663636; first_hit_strict_64_to_32 5/550 = 0.009091; A_events_over_focus_events 365/19137 = 0.019073).

The narrow statement this supports. RT550 shows a high integer-level any-event overlap with the A-like 64-95 -> 32-63 signature: most integers have some A-like 64-95 -> 32-63 event somewhere along the odd-accelerated-to-1 path (377/550 in odd_core).
It does not support the stronger statements that the first 64-95 event is usually A-like (first-hit strict is only 7/550), that A-like events are common among all focus events (occurrence-level is 377/18882 ≈ 2.0%), or that RT's own (j,q) interval coincides with this remaining_K window.
Also notable: among the 545 rows entering 32-63, the first 32-63 -> 16-31 exit occurs at R = 32 in 545/545 — in this overlay the 32-63 interior does not differentiate RT550 rows; the structure visible in this coordinate sits at the 64-95 -> 32-63 entrance.

6.What can and cannot be compared

Can be said (each pinned to its coordinate):

• (Exp 1) The Appendix C occurrences reproduce exactly under the RT definition: 593/593 satisfy 3^q/2^j < 1 and T^j(n) ≥ n; 0 cyclic.
• (Exp 2-A) In our sampled universe, 64-95 -> 32-63 shows the inverse-sign pattern, and the conditional excess is carried by START_IN_LAYER × transition_k=1 × pre111 — a result that does not depend on RT550.
• (Exp 3) RT550, overlaid onto the odd-accelerated-to-1 remaining_K coordinate, shows a high integer-level any-event overlap with A-like 64-95 -> 32-63 events (0.685 in odd_core; 0.664 in original_n_strict).

Cannot be said (without further work):

• that the three constructions are "the same remaining_K coordinate" — they are not; only Exp 2-A uses the trace_escape window, and only Exp 3 goes to 1;
• that "RT550 is A-like" without naming the counting mode — first-hit strict is 7/550 and occurrence-level is ≈ 2.0%, so the unqualified sentence is misleading;
• that Exp 2-A validates RT or that RT validates Exp 2-A — they are computed on different universes and only correspond via the Exp 3 overlay;
• that any of this connects to the Waiting Hall miss/control structure — that material uses yet another finite window, and no chapter-level connection is claimed in this README;
• anything causal. Enrichment in a finite table is not a cause.

7.Safe wording for the manuscript

Recommended sentences (verbatim or close):

Experiment 1 is a pure reproduction of the Rozier–Terracol Appendix C occurrences under their own definition; no coordinate of ours is used.

Experiment 2-A is an independent coordinate analysis on our sampled universe, with W = trace_escape(n, 1 << power) and remaining_K_before = sum(W) − sum(W[:pos]); the Paradoxical Sequence chapter's claims rest on this experiment and do not use RT550.

Experiment 3 overlays the reproduced RT550 set onto an odd-accelerated remaining_K-to-1 coordinate. This is an overlay, not the RT (j,q) interval and not the trace_escape window.

At the integer-level any-event count, RT550 shows a high overlap with the A-like 64-95 -> 32-63 signature in this overlay (377/550 in odd_core); the first-hit strict count is small (7/550), and the occurrence-level share is ≈ 2.0%. Claims about RT550 and the A face should always name the counting mode.

These analyses use different finite windows and should not be identified without a coordinate audit.

Phrases to avoid: "the same remaining_K coordinate" (without qualification), "RT550 is A-like" (without counting mode), and any causal verbs ("causes", "drives", "explains"). Prefer suggests, is enriched in, overlays onto, corresponds under this projection.

8.Source reports and files