Appendix: definitions, constants, reproduction

A. Constants

B. State coordinates

x_K window. \(x_K = K_\tau - (\text{power} - h)\), bucketed as exhaustion_0_31 \([0,32)\), deep_32_63 \([32,64)\), tail_64_95 \([64,96)\); words outside \([0,96)\) are dropped.

bridge cluster. From the reparameterized log-magnitude path, \(z_{25} = x(0.25) - 0.25\,x(1)\). With cuts \(q_{\text{low}}=-1.5\), \(q_{\text{high}}=-0.25\): \(z_{25} \le q_{\text{low}} \Rightarrow\) late_growth; \(z_{25} \ge q_{\text{high}} \Rightarrow\) early_growth; otherwise balanced. The companion features \(z_{50}, z_{75}\) and the final drift enter the regression baseline.

parity. even if power is even, else odd.

C. Block score

Train-split smoothed log-ratio per \((L,\text{state},u\text{-bin},\text{block})\), summed over a test word's sliding windows. Position bins \(u\) are deciles of \((i+1)/\tau\). A state is scored at length \(L\) only if it is stable at that length.

score = 0
for i in range(tau - L + 1):
    block = ",".join(k_cat(k) for k in word[i:i+L])
    score += lookup[(L, state, u_bin(i, tau), block)]   # log2(actual_p / iid_p)

D. File manifest

E. Reproduction

Python 3.10+ with numpy. The scripts require an upstream collatz_escape_word_deficit.py and binary status caches odd_only_status_p{24..28}.bin; edit SRC and CACHE_DIRS at the top of collatz_block_anomaly_score.py to local paths, then run the three scripts in order. Randomness is seeded and the train/test split is by deterministic sample-index parity; reruns reproduce up to floating-point summation order.