# Paradoxical Sequence Analysis

## Purpose

This is an observational sequence-level decomposition of the remaining_K inverse-sign pattern: mass deficit in a band, but conditional excess for the downstream transition. It does not claim a mechanism.

Target transitions:

- `96-127 -> 64-95`
- `64-95 -> 32-63`
- `32-63 -> 16-31`

## Overall Results

- `32-63 -> 16-31`: from-mass delta `-0.180211`, actual pass rate `0.106222`, iid pass rate `0.101604`, conditional delta `0.00461753`, support `4.09928`.
- `64-95 -> 32-63`: from-mass delta `-0.0752273`, actual pass rate `0.116249`, iid pass rate `0.108388`, conditional delta `0.00786124`, support `0.608096`.
- `96-127 -> 64-95`: from-mass delta `-0.0130593`, actual pass rate `0.176505`, iid pass rate `0.167431`, conditional delta `0.00907383`, support `0.0503482`.

The inverse-sign structure is present in all three target transitions in this sample. The strongest conditional excess is upstream at `96-127 -> 64-95`, while the largest from-band mass deficit remains lower, especially around `32-63` and `64-95`.

## Transition Notes

### 96-127 -> 64-95
- route `START_IN_LAYER`: actual from `0.0186444`, iid from `0.0317038`, actual pass rate `0.176505`, iid pass rate `0.167431`, conditional delta `0.00907383` (ok support).

Top pass-side pattern concentrations:
- `pre_k_window_3` `1,2,2`: pass share delta `0.0215746`, support `0.0013634` (low).
- `pre_k_window_3` `2,1,2`: pass share delta `0.0208569`, support `0.00167386` (low).
- `pre_k_window_5` `1,2,1,2,2`: pass share delta `0.0195416`, support `0.000175214` (low).
- `k_prefix_3` `2,1,3+`: pass share delta `0.0175019`, support `0.0014928` (low).
- `pre_k_window_5` `3+`: pass share delta `0.0150536`, support `0.00140668` (low).
- `pre_k_window_3` `3+`: pass share delta `0.0150536`, support `0.00140668` (low).
- `pre_k_window_3` `2,1,1`: pass share delta `0.0143111`, support `0.00287898` (ok).
- `pre_k_window_5` `2,1,2,1,2`: pass share delta `0.0139795`, support `0.000263472` (low).

### 64-95 -> 32-63
- route `INFLOW_FROM_96-127`: actual from `0.0642345`, iid from `0.102433`, actual pass rate `0.0512316`, iid pass rate `0.0518211`, conditional delta `-0.000589567` (ok support).
- route `START_IN_LAYER`: actual from `0.2022`, iid from `0.239229`, actual pass rate `0.136904`, iid pass rate `0.132609`, conditional delta `0.00429505` (ok support).

Top pass-side pattern concentrations:
- `transition_k` `1`: pass share delta `0.0495034`, support `0.353739` (ok).
- `pre_k_window_3` `1,1,1`: pass share delta `0.0339716`, support `0.09395` (ok).
- `pre_k_window_3` `1,2,3+`: pass share delta `0.0162109`, support `0.0120311` (ok).
- `pre_k_window_5` `2,1,1,1,1`: pass share delta `0.0146934`, support `0.0125406` (ok).
- `pre_k_window_5` `1,2,1,2,3+`: pass share delta `0.0119648`, support `0.00179419` (low).
- `pre_k_window_3` `3+,1,1`: pass share delta `0.0119032`, support `0.0278929` (ok).
- `pre_k_window_5` `3+,1,1,1,1`: pass share delta `0.01061`, support `0.00752601` (ok).
- `pre_k_window_5` `2,1,1,2,1`: pass share delta `0.0084737`, support `0.00568132` (ok).

### 32-63 -> 16-31
- route `INFLOW_FROM_64-95`: actual from `0.607691`, iid from `0.743697`, actual pass rate `0.0509682`, iid pass rate `0.0497946`, conditional delta `0.00117357` (ok support).
- route `START_IN_LAYER`: actual from `1.35184`, iid from `1.39605`, actual pass rate `0.13106`, iid pass rate `0.129204`, conditional delta `0.00185568` (ok support).

Top pass-side pattern concentrations:
- `pre_k_window_5` `1,1,1,1,2`: pass share delta `0.0114368`, support `0.0949464` (ok).
- `pre_k_window_3` `1,1,2`: pass share delta `0.00985669`, support `0.334897` (ok).
- `transition_k` `2`: pass share delta `0.00878829`, support `1.03331` (ok).
- `pre_k_window_5` `3+,2,1,1,2`: pass share delta `0.00848144`, support `0.0128147` (ok).
- `pre_k_window_5` `1,3+,2,1,1`: pass share delta `0.00801259`, support `0.0306705` (ok).
- `pre_k_window_3` `1,3+,3+`: pass share delta `0.00415706`, support `0.0400729` (ok).
- `pre_k_window_5` `3+,2,1,1,1`: pass share delta `0.0039522`, support `0.0259383` (ok).
- `k_prefix_3` `1,1,2`: pass share delta `0.00368308`, support `0.390549` (ok).

## Focus State

The requested focus state is `late_growth|deep_32_63|even`. Rows below use all entry routes combined.

- `32-63 -> 16-31`: actual from `0.258959`, iid from `0.315629`, mass delta `-0.0566699`, actual pass rate `0.0551497`, iid pass rate `0.0546024`, conditional delta `0.000547325`.
- `64-95 -> 32-63`: actual from `0.0418732`, iid from `0.0541741`, mass delta `-0.0123009`, actual pass rate `0.177095`, iid pass rate `0.17506`, conditional delta `0.00203578`.

## Observational Limits

- This uses the same sampled long-word/final-state-defined universe as the remaining_K chain check, not an exhaustive all-word enumeration.
- Feature rows are weighted occurrence summaries. Rolling windows can count multiple windows per occurrence, so they should be read as concentration diagnostics rather than exclusive decompositions.
- Low-support rows are marked in the CSV files. They are useful for generating hypotheses, but should not be used as stable claims.
- The analysis identifies sequence groups that carry the inverse-sign observation; it does not prove that those groups cause the discrepancy.

## Next Candidates

1. Re-run only the strongest feature families with a larger iid sample to test stability.
2. Split `START_IN_LAYER` vs `INFLOW_FROM_*` inside `late_growth|deep_32_63|even` for the `64-95 -> 32-63` and `32-63 -> 16-31` pair.
3. Add prefix-cylinder labels to the representative examples if a paper-facing sequence table is needed.