### Digit tests (stats_02_digits.py; Monte-Carlo p-values, 100 000 draws, seed 20260914)
| test | B1 | B2 | B3 |
|---|---|---|---|
| last digit vs uniform, all numbers: chi2 (9 df), p | 47.4, p=1e-05 | 78.9, p=1e-05 | 38.1, p=2e-05 |
| same, type-randomised null (reuse kept), p | p=0.035 | p=0.279 | p=0.360 |
| last digit vs uniform, distinct values: chi2, p | 18.2, p=0.032 | 9.4, p=0.400 | 7.7, p=0.569 |
| share even, all numbers (binomial p) | 0.590 (p=4.3e-05) | 0.528 (p=0.14) | 0.531 (p=0.14) |
| share even, distinct values (binomial p) | 0.564 (p=0.032) | 0.525 (p=0.55) | 0.502 (p=1) |
| share ending 0 or 5, distinct values (p vs 0.20) | 0.211 (p=0.61) | 0.265 (p=0.032) | 0.190 (p=0.76) |
| share with repeated final digits (dd), distinct values (p vs 0.10) | 0.062 (p=0.031) | 0.116 (p=0.45) | 0.082 (p=0.4) |
| last two digits uniform, distinct values (MC p) | p=0.41 | p=0.98 | p=0.89 |
| first digit vs Benford, all numbers: chi2, p | 52.1, p=1e-05 | 33.4, p=5e-05 | 60.8, p=1e-05 |
| first digit vs Benford, distinct values: chi2, p | 23.6, p=0.0031 | 5.6, p=0.7 | 21.5, p=0.006 |
| first digit vs uniform on [1,1322], distinct values: p | p=1e-05 | p=1e-05 | p=1e-05 |

| comparison (chi-square homogeneity, permutation p, 20 000 perms) | last digit, all numbers | last digit, distinct values | first digit, all numbers | first digit, distinct values | share even (Fisher p) |
|---|---|---|---|---|---|
| B1 vs B2 | chi2=51.3, p=5e-05 | chi2=6.1, p=0.74 | chi2=31.1, p=0.00015 | chi2=11.1, p=0.20 | 0.590 vs 0.528 (p=0.03) |
| B3 vs B2 | chi2=79.4, p=5e-05 | chi2=6.8, p=0.67 | chi2=42.6, p=5e-05 | chi2=14.9, p=0.06 | 0.531 vs 0.528 (p=0.91) |
| B1 vs B3 | chi2=31.6, p=0.00045 | chi2=9.0, p=0.44 | chi2=4.2, p=0.843 | chi2=9.7, p=0.29 | 0.590 vs 0.531 (p=0.05) |

### Serial structure vs shuffles of the same cipher (stats_03_sequential.py; 20 000 permutations)
| statistic | B1 obs (null mean +- sd), p | B2 obs (null), p | B3 obs (null), p |
|---|---|---|---|
| lag-1 autocorrelation (Pearson) | 0.252 (-0.00189 +- 0.044), p=5e-05 | 0.0456 (-0.00162 +- 0.036), p=0.19 | 0.622 (-0.00143 +- 0.04), p=5e-05 |
| lag-2 autocorrelation (Pearson) | 0.0265 (-0.00181 +- 0.043), p=0.51 | -0.00248 (-0.000951 +- 0.036), p=0.97 | 0.454 (-0.00124 +- 0.04), p=5e-05 |
| lag-1 rank autocorrelation | 0.195 (-0.00218 +- 0.044), p=0.0001 | 0.206 (-0.000873 +- 0.036), p=5e-05 | 0.489 (-0.0015 +- 0.04), p=5e-05 |
| lag-2 rank autocorrelation | 0.000876 (-0.0014 +- 0.044), p=0.96 | 0.0683 (-0.000845 +- 0.036), p=0.058 | 0.332 (-0.00212 +- 0.04), p=5e-05 |
| runs up and down (count) | 317 (346 +- 9.5), p=0.0027 | 486 (506 +- 12), p=0.098 | 332 (410 +- 11), p=5e-05 |
| adjacent pairs with |a-b| <= 5 (one-sided: more) | 10 (15.5 +- 3.8), p=0.95 | 51 (36.4 +- 5.8), p=0.0094 | 26 (26.2 +- 4.9), p=0.55 |
| adjacent pairs with |a-b| <= 10 (one-sided: more) | 32 (28.2 +- 5), p=0.25 | 77 (63.8 +- 7.3), p=0.044 | 76 (49.4 +- 6.5), p=5e-05 |
| adjacent pairs with |a-b| <= 1 | 0 (4.45 +- 2.1), p=1 | 16 (11.8 +- 3.4), p=0.13 | 1 (7.89 +- 2.8), p=1 |
| adjacent pairs with the same last digit | 33 (55.8 +- 7), p=0.001 | 58 (83 +- 8.5), p=0.0039 | 28 (64.7 +- 7.6), p=5e-05 |
| adjacent pairs with the same first digit | 53 (78.3 +- 7.7), p=0.001 | 142 (120 +- 9.5), p=0.021 | 114 (95.2 +- 8.4), p=0.03 |
| trend: Spearman rho(position, value) | 0.0289 (-0.000477 +- 0.044), p=0.5 | 0.0292 (-0.000457 +- 0.037), p=0.42 | 0.169 (-0.000342 +- 0.04), p=0.0004 |
| mean of first half minus second half | -33.7 (-0.088 +- 31), p=0.29 | -13.5 (0.0822 +- 14), p=0.35 | -80.9 (-0.0336 +- 14), p=5e-05 |
| variance of gaps between large numbers | 4.99e+03 (1.79e+03 +- 1.1e+03), p=0.02 | 59.9 (83.8 +- 20), p=0.18 | 312 (90.6 +- 24), p=0.0002 |
- B1: runs test asymptotic z = -3.06 (p = 0.00224); positions of the 10 numbers above 1322: KS vs uniform p = 0.479
- B2: runs test asymptotic z = -1.86 (p = 0.0624); positions of the 77 numbers above 483: KS vs uniform p = 0.759
- B3: runs test asymptotic z = -7.61 (p = 2.71e-14); positions of the 60 numbers above 319: KS vs uniform p = 0.000

Alphabetic runs in the DOI decode (longest non-decreasing run of decoded letters; p = share of 5000 shuffles of the same letters with a run at least as long):
| key | B1 | B2 | B3 |
|---|---|---|---|
| pamphlet numbering (as cipher 2 was decoded) | 17 ('abcdefghiijklmmno' at 188), null 6.3, p=0.0002 | 6 ('efirst' at 203), null 6.4, p=0.88 | 7 ('accsttt' at 150), null 6.5, p=0.44 |
| straight count, self-evident split (Gillogly's Table I) | 8 ('efghiijp' at 192), null 6.3, p=0.098 | 7 ('aceottt' at 100), null 6.4, p=0.39 | 7 ('accsttt' at 150), null 6.7, p=0.51 |
| straight count | 6 ('aabbce' at 84), null 6.3, p=0.86 | 7 ('chiittt' at 587), null 6.9, p=0.62 | 5 ('atttt' at 9), null 6.6, p=1 |

### Serial extras (stats_08_serial_extras.py; 5000 permutations)
| statistic | B2, letter-preserving shuffle: obs (null), p | B2, full shuffle | B1, full shuffle | B3, full shuffle |
|---|---|---|---|---|
| lag1_spearman | 0.206 (-0.0191 +- 0.035), p=0.0002 | 0.206 (-0.00201 +- 0.036), p=0.0002 | 0.195 (-0.00109 +- 0.044), p=0.0002 | 0.489 (-0.000947 +- 0.041), p=0.0002 |
| lag2_spearman | 0.0683 (0.0032 +- 0.034), p=0.027 | 0.0683 (-0.000675 +- 0.036), p=0.027 | 0.000876 (-0.00134 +- 0.044), p=0.48 | 0.332 (-0.000926 +- 0.041), p=0.0002 |
| lag1_pearson | 0.0456 (-0.0365 +- 0.03), p=0.0034 | 0.0456 (-0.0021 +- 0.036), p=0.097 | 0.252 (-0.00105 +- 0.043), p=0.0002 | 0.622 (-0.00138 +- 0.04), p=0.0002 |
| close_le5 | 51 (33.1 +- 5.4), p=0.001 | 51 (36.3 +- 5.9), p=0.012 | 10 (15.5 +- 3.8), p=0.95 | 26 (26.2 +- 4.9), p=0.55 |
| close_le10 | 77 (59.1 +- 6.9), p=0.0068 | 77 (64 +- 7.3), p=0.046 | 32 (28 +- 5.1), p=0.25 | 76 (49.4 +- 6.6), p=0.0002 |
| runs_up_down | 486 (511 +- 11), p=0.016 | 486 (506 +- 12), p=0.05 | 317 (346 +- 9.6), p=0.0012 | 332 (411 +- 11), p=0.0002 |
| longest_ascending_run | 6 (5.64 +- 0.71), p=0.54 | 6 (5.66 +- 0.71), p=0.55 | 7 (5.5 +- 0.71), p=0.073 | 10 (5.56 +- 0.69), p=0.0002 |
| ascending_runs_ge4 | 39 (25.4 +- 4.4), p=0.002 | 39 (24.3 +- 4.5), p=0.001 | 36 (17 +- 3.6), p=0.0002 | 49 (20.1 +- 4.1), p=0.0002 |
| ascending_runs_ge6 | 2 (0.778 +- 0.88), p=0.19 | 2 (0.8 +- 0.89), p=0.19 | 3 (0.569 +- 0.74), p=0.017 | 17 (0.667 +- 0.83), p=0.0002 |
| mean_ascending_run | 2.12 (2.01 +- 0.041), p=0.0052 | 2.12 (1.98 +- 0.04), p=0.0014 | 2.27 (1.99 +- 0.05), p=0.0002 | 2.58 (1.99 +- 0.046), p=0.0002 |

### Homophone-reuse simulation (stats_04_homophone_sim.py; 1000 simulations per model)
Encoder fitted to B2: reuse probability by number of homophones already used d: d=1-1: 0.46 (n=37), d=2-2: 0.15 (n=22), d=3-3: 0.43 (n=33), d=4-5: 0.59 (n=79), d=6-8: 0.83 (n=226), d=9-12: 0.86 (n=178), d=13-99: 0.94 (n=152); overall 0.779; logistic on log d: -1.28 + 1.37 log d; reuse weight count^gamma with gamma = 0.6; new-number weight exp(-n/tau) with tau = 119 (log-likelihood -457.1 vs -702.1 for uniform choice among unused homophones); slip rate 0.0144.

**B1** (observed value | simulated mean +- sd | percentile of the observed value among simulations)
| statistic | observed | DOI_fitted_prose | DOI_random_prose | iid_uniform | iid_loguniform | LONG2906_fitted_prose | LONG2906_random_prose |
|---|---|---|---|---|---|---|---|
| distinct numbers | 298 | 164 +- 6.4 (pct 1.000) | 377 +- 8.2 (pct 0.000) | 476 +- 6 (pct 0.000) | 274 +- 9.7 (pct 0.993) | 169 +- 6.6 (pct 1.000) | 450 +- 7 (pct 0.000) |
| share of types used once | 0.594 | 0.363 +- 0.033 (pct 1.000) | 0.752 +- 0.019 (pct 0.000) | 0.913 +- 0.013 (pct 0.000) | 0.769 +- 0.021 (pct 0.000) | 0.371 +- 0.032 (pct 1.000) | 0.867 +- 0.015 (pct 0.000) |
| count of the commonest number | 8 | 17 +- 3.2 (pct 0.000) | 11.9 +- 4.3 (pct 0.298) | 3.01 +- 0.43 (pct 1.000) | 45.3 +- 6.2 (pct 0.000) | 16.4 +- 3.1 (pct 0.000) | 3.97 +- 0.71 (pct 1.000) |
| share of tokens in the 10 commonest | 0.117 | 0.217 +- 0.017 (pct 0.000) | 0.0974 +- 0.0096 (pct 0.985) | 0.0432 +- 0.0031 (pct 1.000) | 0.311 +- 0.019 (pct 0.000) | 0.21 +- 0.017 (pct 0.000) | 0.0573 +- 0.0051 (pct 1.000) |
| Zipf slope (top 30) | -0.3 | -0.364 +- 0.054 (pct 0.878) | -0.351 +- 0.073 (pct 0.749) | -0.09 +- 0.045 (pct 0.000) | -0.849 +- 0.05 (pct 1.000) | -0.354 +- 0.054 (pct 0.834) | -0.221 +- 0.043 (pct 0.028) |
| lag-1 rank autocorrelation | 0.195 | -0.0127 +- 0.048 (pct 1.000) | -0.00371 +- 0.043 (pct 1.000) | -0.00631 +- 0.047 (pct 1.000) | -0.00113 +- 0.044 (pct 1.000) | 0.0151 +- 0.045 (pct 1.000) | -0.000154 +- 0.043 (pct 1.000) |
| runs-up-and-down z | -3.06 | 0.347 +- 1 (pct 0.001) | -0.0437 +- 0.97 (pct 0.001) | 0.128 +- 1 (pct 0.001) | 0.406 +- 0.99 (pct 0.000) | -0.017 +- 0.95 (pct 0.000) | -0.0116 +- 1 (pct 0.001) |
| adjacent pairs |a-b|<=5 | 10 | 17.7 +- 4.4 (pct 0.047) | 4.38 +- 2.1 (pct 0.994) | 1.92 +- 1.3 (pct 1.000) | 48.1 +- 7.4 (pct 0.000) | 17.1 +- 4.2 (pct 0.051) | 1.9 +- 1.3 (pct 1.000) |
| adjacent same last digit | 33 | 51.7 +- 7.1 (pct 0.001) | 51.8 +- 6.8 (pct 0.004) | 52.4 +- 6.9 (pct 0.003) | 55.1 +- 7.1 (pct 0.000) | 52.4 +- 7.1 (pct 0.006) | 51.7 +- 7 (pct 0.004) |
| share of types ending in 0 | 0.138 | 0.102 +- 0.016 (pct 0.989) | 0.0979 +- 0.012 (pct 0.999) | 0.1 +- 0.013 (pct 0.996) | 0.0979 +- 0.015 (pct 0.992) | 0.102 +- 0.016 (pct 0.988) | 0.0956 +- 0.012 (pct 0.999) |
| share of types even | 0.564 | 0.478 +- 0.025 (pct 1.000) | 0.486 +- 0.02 (pct 1.000) | 0.5 +- 0.022 (pct 0.999) | 0.498 +- 0.025 (pct 0.995) | 0.501 +- 0.025 (pct 0.996) | 0.497 +- 0.02 (pct 1.000) |
| share of numbers <= 100 | 0.437 | 0.437 +- 0.039 (pct 0.509) | 0.0879 +- 0.013 (pct 1.000) | 0.0344 +- 0.0079 (pct 1.000) | 0.579 +- 0.021 (pct 0.000) | 0.426 +- 0.038 (pct 0.605) | 0.038 +- 0.0081 (pct 1.000) |
| median | 123 | 123 +- 15 (pct 0.518) | 631 +- 32 (pct 0.000) | 1.46e+03 +- 62 (pct 0.000) | 53.7 +- 9.4 (pct 1.000) | 128 +- 15 (pct 0.384) | 1.42e+03 +- 58 (pct 0.000) |
- top-10 counts observed: [8, 7, 7, 6, 6, 6, 6, 5, 5, 5]; simulated means: DOI_fitted_prose: [17.0, 14.1, 12.7, 11.6, 10.8, 10.2, 9.7, 9.3, 8.9, 8.6]; DOI_random_prose: [11.9, 5.7, 5.0, 4.6, 4.3, 4.1, 3.9, 3.8, 3.7, 3.5]; iid_uniform: [3.0, 2.7, 2.4, 2.2, 2.1, 2.0, 2.0, 2.0, 2.0, 2.0]; iid_loguniform: [45.3, 27.0, 19.3, 15.2, 12.6, 10.6, 9.3, 8.3, 7.5, 6.8]; LONG2906_fitted_prose: [16.4, 13.5, 12.1, 11.2, 10.5, 9.9, 9.5, 9.1, 8.7, 8.4]; LONG2906_random_prose: [4.0, 3.4, 3.1, 3.0, 3.0, 2.9, 2.8, 2.7, 2.5, 2.4]

**B3** (observed value | simulated mean +- sd | percentile of the observed value among simulations)
| statistic | observed | DOI_fitted_prose | DOI_random_prose | iid_uniform | iid_loguniform | LONG975_fitted_prose | LONG975_random_prose | DOI_fitted_names | LONG975_fitted_names |
|---|---|---|---|---|---|---|---|---|---|
| distinct numbers | 263 | 179 +- 6.8 (pct 1.000) | 429 +- 8.9 (pct 0.000) | 458 +- 8 (pct 0.000) | 256 +- 9.5 (pct 0.796) | 183 +- 6.7 (pct 1.000) | 403 +- 8.6 (pct 0.000) | 179 +- 6.8 (pct 1.000) | 178 +- 6.2 (pct 1.000) |
| share of types used once | 0.471 | 0.34 +- 0.031 (pct 1.000) | 0.725 +- 0.018 (pct 0.000) | 0.717 +- 0.018 (pct 0.000) | 0.668 +- 0.024 (pct 0.000) | 0.34 +- 0.031 (pct 1.000) | 0.658 +- 0.02 (pct 0.000) | 0.349 +- 0.031 (pct 1.000) | 0.327 +- 0.031 (pct 1.000) |
| count of the commonest number | 13 | 19 +- 3.1 (pct 0.018) | 14.2 +- 4.7 (pct 0.416) | 4.42 +- 0.62 (pct 1.000) | 62.5 +- 7.4 (pct 0.000) | 18.5 +- 3.4 (pct 0.031) | 6.65 +- 0.98 (pct 1.000) | 21.8 +- 1.8 (pct 0.000) | 17.5 +- 3.1 (pct 0.053) |
| share of tokens in the 10 commonest | 0.142 | 0.208 +- 0.015 (pct 0.000) | 0.0935 +- 0.0086 (pct 1.000) | 0.0557 +- 0.0036 (pct 1.000) | 0.356 +- 0.018 (pct 0.000) | 0.201 +- 0.016 (pct 0.000) | 0.0804 +- 0.0059 (pct 1.000) | 0.215 +- 0.014 (pct 0.000) | 0.197 +- 0.015 (pct 0.000) |
| Zipf slope (top 30) | -0.328 | -0.355 +- 0.05 (pct 0.706) | -0.362 +- 0.051 (pct 0.739) | -0.168 +- 0.055 (pct 0.003) | -0.869 +- 0.051 (pct 1.000) | -0.346 +- 0.053 (pct 0.616) | -0.25 +- 0.04 (pct 0.023) | -0.381 +- 0.042 (pct 0.904) | -0.327 +- 0.052 (pct 0.477) |
| lag-1 rank autocorrelation | 0.489 | -0.0138 +- 0.043 (pct 1.000) | -0.00329 +- 0.039 (pct 1.000) | -0.00231 +- 0.041 (pct 1.000) | -0.000527 +- 0.041 (pct 1.000) | 0.0169 +- 0.042 (pct 1.000) | -0.00186 +- 0.039 (pct 1.000) | 0.0245 +- 0.039 (pct 1.000) | 0.0125 +- 0.041 (pct 1.000) |
| runs-up-and-down z | -7.61 | 0.399 +- 1 (pct 0.000) | 0.0362 +- 1 (pct 0.000) | 0.0221 +- 1 (pct 0.000) | 0.564 +- 0.99 (pct 0.000) | -0.0162 +- 0.99 (pct 0.000) | 0.0326 +- 1 (pct 0.000) | -0.266 +- 0.93 (pct 0.000) | 0.0128 +- 0.97 (pct 0.000) |
| adjacent pairs |a-b|<=5 | 26 | 20.7 +- 4.8 (pct 0.885) | 5.19 +- 2.3 (pct 1.000) | 6.97 +- 2.7 (pct 1.000) | 76.2 +- 9.5 (pct 0.000) | 20.1 +- 4.7 (pct 0.909) | 6.95 +- 2.7 (pct 1.000) | 23.8 +- 5.3 (pct 0.700) | 19.1 +- 4.8 (pct 0.936) |
| adjacent same last digit | 28 | 61.8 +- 7.5 (pct 0.000) | 61 +- 7.3 (pct 0.000) | 62 +- 7.4 (pct 0.000) | 67.4 +- 7.9 (pct 0.000) | 61.6 +- 7.8 (pct 0.000) | 61.3 +- 7.5 (pct 0.000) | 66.6 +- 8.2 (pct 0.000) | 60 +- 7.8 (pct 0.000) |
| share of types ending in 0 | 0.0837 | 0.104 +- 0.015 (pct 0.088) | 0.0983 +- 0.011 (pct 0.090) | 0.0991 +- 0.0099 (pct 0.062) | 0.0977 +- 0.014 (pct 0.163) | 0.104 +- 0.014 (pct 0.078) | 0.1 +- 0.011 (pct 0.063) | 0.105 +- 0.015 (pct 0.077) | 0.106 +- 0.013 (pct 0.040) |
| share of types even | 0.502 | 0.479 +- 0.024 (pct 0.821) | 0.488 +- 0.018 (pct 0.767) | 0.499 +- 0.017 (pct 0.554) | 0.499 +- 0.024 (pct 0.532) | 0.495 +- 0.023 (pct 0.591) | 0.501 +- 0.018 (pct 0.509) | 0.466 +- 0.023 (pct 0.933) | 0.491 +- 0.023 (pct 0.684) |
| share of numbers <= 100 | 0.539 | 0.428 +- 0.036 (pct 0.999) | 0.0874 +- 0.011 (pct 1.000) | 0.103 +- 0.013 (pct 1.000) | 0.67 +- 0.019 (pct 0.000) | 0.422 +- 0.037 (pct 0.998) | 0.113 +- 0.013 (pct 1.000) | 0.386 +- 0.033 (pct 1.000) | 0.438 +- 0.035 (pct 0.999) |
| median | 94.5 | 127 +- 14 (pct 0.006) | 632 +- 29 (pct 0.000) | 487 +- 19 (pct 0.000) | 31 +- 4.4 (pct 1.000) | 130 +- 15 (pct 0.003) | 484 +- 17 (pct 0.000) | 142 +- 13 (pct 0.000) | 127 +- 15 (pct 0.004) |
- top-10 counts observed: [13, 11, 10, 9, 9, 8, 7, 7, 7, 7]; simulated means: DOI_fitted_prose: [19.0, 16.0, 14.4, 13.3, 12.4, 11.7, 11.1, 10.6, 10.2, 9.8]; DOI_random_prose: [14.2, 6.4, 5.7, 5.2, 4.9, 4.6, 4.4, 4.3, 4.1, 4.0]; iid_uniform: [4.4, 4.0, 3.8, 3.6, 3.3, 3.2, 3.1, 3.0, 3.0, 3.0]; iid_loguniform: [62.5, 36.9, 26.1, 20.5, 16.8, 14.4, 12.5, 11.1, 10.0, 9.1]; LONG975_fitted_prose: [18.5, 15.4, 13.8, 12.8, 12.0, 11.3, 10.8, 10.3, 9.9, 9.6]; LONG975_random_prose: [6.6, 5.8, 5.4, 5.1, 4.9, 4.7, 4.5, 4.3, 4.2, 4.1]; DOI_fitted_names: [21.8, 17.1, 14.8, 13.4, 12.4, 11.7, 11.1, 10.6, 10.2, 9.8]; LONG975_fitted_names: [17.5, 14.9, 13.5, 12.6, 11.8, 11.2, 10.7, 10.3, 9.9, 9.5]

**B2** (observed value | simulated mean +- sd | percentile of the observed value among simulations)
| statistic | observed | DOI_fitted_prose | DOI_random_prose | iid_uniform | iid_loguniform | DOI_fitted_B2text | DOI_random_B2text |
|---|---|---|---|---|---|---|---|
| distinct numbers | 181 | 198 +- 7.4 (pct 0.008) | 498 +- 10 (pct 0.000) | 534 +- 9.5 (pct 0.000) | 295 +- 11 (pct 0.000) | 196 +- 7.2 (pct 0.015) | 481 +- 7.2 (pct 0.000) |
| share of types used once | 0.238 | 0.317 +- 0.029 (pct 0.003) | 0.688 +- 0.017 (pct 0.000) | 0.668 +- 0.018 (pct 0.000) | 0.65 +- 0.023 (pct 0.000) | 0.319 +- 0.03 (pct 0.004) | 0.679 +- 0.017 (pct 0.000) |
| count of the commonest number | 18 | 22.4 +- 3.7 (pct 0.128) | 16.9 +- 5.7 (pct 0.562) | 4.83 +- 0.66 (pct 1.000) | 76.2 +- 8.2 (pct 0.000) | 21.5 +- 3.9 (pct 0.231) | 10.6 +- 1.3 (pct 1.000) |
| share of tokens in the 10 commonest | 0.186 | 0.197 +- 0.015 (pct 0.233) | 0.0887 +- 0.008 (pct 1.000) | 0.0506 +- 0.0037 (pct 1.000) | 0.353 +- 0.016 (pct 0.000) | 0.193 +- 0.014 (pct 0.339) | 0.0924 +- 0.004 (pct 1.000) |
| Zipf slope (top 30) | -0.288 | -0.35 +- 0.051 (pct 0.880) | -0.371 +- 0.06 (pct 0.902) | -0.149 +- 0.035 (pct 0.000) | -0.879 +- 0.045 (pct 1.000) | -0.323 +- 0.049 (pct 0.770) | -0.332 +- 0.03 (pct 0.966) |
| lag-1 rank autocorrelation | 0.206 | -0.0105 +- 0.039 (pct 1.000) | -0.00184 +- 0.036 (pct 1.000) | -0.00301 +- 0.035 (pct 1.000) | -0.000947 +- 0.036 (pct 1.000) | -0.0215 +- 0.039 (pct 1.000) | -0.00503 +- 0.037 (pct 1.000) |
| runs-up-and-down z | -1.86 | 0.426 +- 0.98 (pct 0.007) | 0.0445 +- 0.98 (pct 0.020) | 0.0627 +- 0.99 (pct 0.033) | 0.635 +- 1 (pct 0.009) | 0.499 +- 1 (pct 0.013) | 0.00135 +- 0.98 (pct 0.031) |
| adjacent pairs |a-b|<=5 | 51 | 24.4 +- 5.2 (pct 1.000) | 6.37 +- 2.5 (pct 1.000) | 8.26 +- 2.8 (pct 1.000) | 93.5 +- 10 (pct 0.000) | 23 +- 5 (pct 1.000) | 6.35 +- 2.5 (pct 1.000) |
| adjacent same last digit | 58 | 76.3 +- 8.5 (pct 0.017) | 75.8 +- 8.6 (pct 0.026) | 76.4 +- 8.5 (pct 0.014) | 82.3 +- 8.4 (pct 0.002) | 76.3 +- 8.3 (pct 0.017) | 75.6 +- 8 (pct 0.019) |
| share of types ending in 0 | 0.144 | 0.106 +- 0.014 (pct 0.998) | 0.098 +- 0.01 (pct 1.000) | 0.0997 +- 0.0085 (pct 1.000) | 0.0987 +- 0.013 (pct 1.000) | 0.106 +- 0.014 (pct 0.998) | 0.0977 +- 0.0097 (pct 1.000) |
| share of types even | 0.525 | 0.479 +- 0.021 (pct 0.981) | 0.488 +- 0.017 (pct 0.978) | 0.5 +- 0.015 (pct 0.964) | 0.5 +- 0.022 (pct 0.876) | 0.475 +- 0.022 (pct 0.990) | 0.491 +- 0.017 (pct 0.981) |
| share of numbers <= 100 | 0.531 | 0.417 +- 0.036 (pct 0.998) | 0.088 +- 0.01 (pct 1.000) | 0.0993 +- 0.011 (pct 1.000) | 0.667 +- 0.018 (pct 0.000) | 0.409 +- 0.034 (pct 0.998) | 0.089 +- 0.0098 (pct 1.000) |
| median | 85 | 132 +- 14 (pct 0.000) | 630 +- 26 (pct 0.000) | 504 +- 18 (pct 0.000) | 31.4 +- 4.1 (pct 1.000) | 135 +- 14 (pct 0.000) | 625 +- 26 (pct 0.000) |
- top-10 counts observed: [18, 15, 15, 15, 15, 13, 13, 13, 13, 12]; simulated means: DOI_fitted_prose: [22.4, 18.8, 16.8, 15.5, 14.5, 13.7, 13.0, 12.4, 11.9, 11.5]; DOI_random_prose: [16.9, 7.3, 6.5, 6.0, 5.7, 5.4, 5.2, 5.0, 4.8, 4.7]; iid_uniform: [4.8, 4.3, 4.1, 4.0, 3.9, 3.8, 3.6, 3.5, 3.3, 3.2]; iid_loguniform: [76.2, 44.8, 32.4, 25.3, 20.8, 17.5, 15.2, 13.5, 12.1, 11.0]; DOI_fitted_B2text: [21.5, 18.0, 16.3, 15.1, 14.2, 13.5, 12.9, 12.4, 11.9, 11.5]; DOI_random_B2text: [10.6, 9.1, 8.1, 7.3, 6.7, 6.3, 6.0, 5.7, 5.4, 5.2]

### Calibrated-reuse (best-case) homophonic models (stats_09_calibrated_reuse.py)
**B1**: DOI: reuse probability calibrated to r = 0.41, tau = 119; LONG2906: reuse probability calibrated to r = 0.45, tau = 264
| statistic | observed | DOI | LONG2906 |
|---|---|---|---|
| distinct numbers | 298 | 299 +- 11 (pct 0.473) | 300 +- 11 (pct 0.455) |
| share of types used once | 0.594 | 0.628 +- 0.027 (pct 0.109) | 0.631 +- 0.025 (pct 0.076) |
| count of the commonest number | 8 | 12.7 +- 3.2 (pct 0.098) | 10.3 +- 2.2 (pct 0.196) |
| share of tokens in the 10 commonest | 0.117 | 0.136 +- 0.013 (pct 0.089) | 0.131 +- 0.013 (pct 0.163) |
| Zipf slope (top 30) | -0.3 | -0.378 +- 0.066 (pct 0.880) | -0.342 +- 0.061 (pct 0.740) |
| lag-1 rank autocorrelation | 0.195 | 0.0673 +- 0.051 (pct 0.992) | 0.0352 +- 0.046 (pct 1.000) |
| runs-up-and-down z | -3.06 | 0.234 +- 0.99 (pct 0.001) | -0.0782 +- 0.97 (pct 0.002) |
| adjacent pairs |a-b|<=5 | 10 | 12.1 +- 3.5 (pct 0.339) | 8.6 +- 2.9 (pct 0.754) |
| adjacent same last digit | 33 | 51.8 +- 6.7 (pct 0.002) | 52.1 +- 6.7 (pct 0.002) |
| share of types ending in 0 | 0.138 | 0.0995 +- 0.0098 (pct 1.000) | 0.0984 +- 0.012 (pct 0.999) |
| share of types even | 0.564 | 0.477 +- 0.016 (pct 1.000) | 0.5 +- 0.02 (pct 1.000) |
| share of numbers <= 100 | 0.437 | 0.318 +- 0.023 (pct 1.000) | 0.23 +- 0.024 (pct 1.000) |
| median | 123 | 181 +- 12 (pct 0.000) | 267 +- 24 (pct 0.000) |
- DOI: top-10 counts observed [8, 7, 7, 6, 6, 6, 6, 5, 5, 5], simulated mean [12.7, 9.0, 7.7, 7.0, 6.5, 6.1, 5.8, 5.6, 5.3, 5.1], 2.5% [7.0, 7.0, 6.0, 5.0, 5.0, 5.0, 5.0, 5.0, 4.0, 4.0], 97.5% [20.0, 13.0, 10.0, 9.0, 8.0, 7.0, 7.0, 7.0, 6.0, 6.0]
- LONG2906: top-10 counts observed [8, 7, 7, 6, 6, 6, 6, 5, 5, 5], simulated mean [10.3, 8.5, 7.6, 7.0, 6.5, 6.2, 5.9, 5.6, 5.4, 5.2], 2.5% [7.0, 6.0, 6.0, 5.0, 5.0, 5.0, 5.0, 5.0, 5.0, 4.0], 97.5% [15.0, 11.0, 10.0, 9.0, 8.0, 8.0, 7.0, 7.0, 6.0, 6.0]

**B3**: DOI: reuse probability calibrated to r = 0.59, tau = 119; LONG975: reuse probability calibrated to r = 0.59, tau = 89
| statistic | observed | DOI | LONG975 |
|---|---|---|---|
| distinct numbers | 263 | 263 +- 12 (pct 0.512) | 262 +- 11 (pct 0.562) |
| share of types used once | 0.471 | 0.516 +- 0.029 (pct 0.063) | 0.509 +- 0.028 (pct 0.091) |
| count of the commonest number | 13 | 17.3 +- 3.2 (pct 0.115) | 16.4 +- 3.5 (pct 0.190) |
| share of tokens in the 10 commonest | 0.142 | 0.177 +- 0.018 (pct 0.025) | 0.171 +- 0.016 (pct 0.039) |
| Zipf slope (top 30) | -0.328 | -0.397 +- 0.061 (pct 0.868) | -0.377 +- 0.06 (pct 0.805) |
| lag-1 rank autocorrelation | 0.489 | 0.0387 +- 0.046 (pct 1.000) | 0.0777 +- 0.047 (pct 1.000) |
| runs-up-and-down z | -7.61 | 0.316 +- 1 (pct 0.000) | -0.275 +- 1 (pct 0.000) |
| adjacent pairs |a-b|<=5 | 26 | 17.3 +- 4.4 (pct 0.984) | 20 +- 5 (pct 0.911) |
| adjacent same last digit | 28 | 62 +- 7.5 (pct 0.000) | 62.1 +- 7.6 (pct 0.000) |
| share of types ending in 0 | 0.0837 | 0.099 +- 0.011 (pct 0.083) | 0.0993 +- 0.0098 (pct 0.067) |
| share of types even | 0.502 | 0.476 +- 0.018 (pct 0.909) | 0.493 +- 0.016 (pct 0.705) |
| share of numbers <= 100 | 0.539 | 0.366 +- 0.032 (pct 1.000) | 0.42 +- 0.031 (pct 1.000) |
| median | 94.5 | 155 +- 14 (pct 0.000) | 131 +- 13 (pct 0.000) |
- DOI: top-10 counts observed [13, 11, 10, 9, 9, 8, 7, 7, 7, 7], simulated mean [17.3, 14.2, 12.4, 11.2, 10.4, 9.7, 9.2, 8.7, 8.3, 8.0], 2.5% [12.0, 10.0, 9.0, 9.0, 8.0, 8.0, 7.0, 7.0, 7.0, 7.0], 97.5% [24.0, 19.0, 17.0, 15.0, 13.0, 12.0, 11.0, 11.0, 10.0, 10.0]
- LONG975: top-10 counts observed [13, 11, 10, 9, 9, 8, 7, 7, 7, 7], simulated mean [16.4, 13.5, 11.9, 10.9, 10.1, 9.5, 9.0, 8.6, 8.2, 7.9], 2.5% [11.0, 10.0, 9.0, 8.0, 8.0, 8.0, 7.0, 7.0, 7.0, 7.0], 97.5% [25.0, 19.0, 15.0, 14.0, 13.0, 12.0, 11.0, 10.0, 10.0, 10.0]

### Scan-forward mixed encoder (stats_10_scan_forward.py; 300 simulations per grid point, seed 20260916)
B2: nearest-forward-homophone moves 123 of 761 steps against 97.3 +- 6.9 under letter-preserving shuffles (p = 0.0015); nearest-backward moves 65 against 58.0 +- 6.5 (p = 0.165).

**B2** (q = probability of taking the nearest forward homophone; otherwise the fitted encoder for B2, the calibrated encoder for B1/B3; simulated mean +- sd, percentile of the observed value)
| statistic | observed | DOI_q0.0 | DOI_q0.1 | DOI_q0.2 | DOI_q0.3 | DOI_q0.4 | DOI_q0.5 | DOI_q0.6 | DOI_q0.7 | DOI_q0.8 |
|---|---|---|---|---|---|---|---|---|---|---|
| lag-1 autocorrelation (Pearson) | 0.0456 | -0.0302 +- 0.034 (0.98) | 0.0795 +- 0.047 (0.22) | 0.182 +- 0.052 (0.00) | 0.291 +- 0.049 (0.00) | 0.389 +- 0.055 (0.00) | 0.483 +- 0.042 (0.00) | 0.576 +- 0.038 (0.00) | 0.664 +- 0.033 (0.00) | 0.74 +- 0.029 (0.00) |
| lag-1 autocorrelation (rank) | 0.206 | -0.012 +- 0.035 (1.00) | 0.0838 +- 0.04 (1.00) | 0.178 +- 0.039 (0.77) | 0.276 +- 0.038 (0.03) | 0.368 +- 0.04 (0.00) | 0.462 +- 0.035 (0.00) | 0.555 +- 0.035 (0.00) | 0.647 +- 0.032 (0.00) | 0.731 +- 0.03 (0.00) |
| lag-2 autocorrelation (Pearson) | -0.00248 | 0.0177 +- 0.038 (0.30) | 0.0198 +- 0.036 (0.30) | 0.0418 +- 0.039 (0.12) | 0.0936 +- 0.047 (0.01) | 0.159 +- 0.057 (0.00) | 0.233 +- 0.05 (0.00) | 0.333 +- 0.053 (0.00) | 0.441 +- 0.049 (0.00) | 0.543 +- 0.047 (0.00) |
| lag-2 autocorrelation (rank) | 0.0683 | 0.0196 +- 0.038 (0.90) | 0.025 +- 0.036 (0.87) | 0.0445 +- 0.037 (0.73) | 0.0912 +- 0.041 (0.29) | 0.146 +- 0.046 (0.05) | 0.218 +- 0.044 (0.00) | 0.313 +- 0.047 (0.00) | 0.418 +- 0.046 (0.00) | 0.528 +- 0.047 (0.00) |
| mean length of strictly rising runs | 2.12 | 1.99 +- 0.042 (1.00) | 2.18 +- 0.053 (0.15) | 2.41 +- 0.07 (0.00) | 2.7 +- 0.085 (0.00) | 3.09 +- 0.12 (0.00) | 3.58 +- 0.14 (0.00) | 4.33 +- 0.23 (0.00) | 5.49 +- 0.32 (0.00) | 7.47 +- 0.55 (0.00) |
| rising runs of length >= 4 | 39 | 24.6 +- 4.5 (1.00) | 39.9 +- 4.7 (0.46) | 56.7 +- 5.9 (0.00) | 70.9 +- 5.6 (0.00) | 83 +- 4.9 (0.00) | 90.6 +- 4.7 (0.00) | 92.8 +- 4.3 (0.00) | 87.9 +- 4.2 (0.00) | 76.7 +- 4.3 (0.00) |
| rising runs of length >= 6 | 2 | 0.78 +- 0.86 (0.96) | 3.36 +- 1.7 (0.34) | 7.9 +- 2.6 (0.01) | 16.2 +- 3.4 (0.00) | 26.7 +- 4.4 (0.00) | 37.3 +- 4.3 (0.00) | 49 +- 4.3 (0.00) | 57 +- 3.7 (0.00) | 57.9 +- 3.1 (0.00) |
| longest rising run | 6 | 5.61 +- 0.61 (0.93) | 6.78 +- 0.9 (0.43) | 7.83 +- 1 (0.05) | 9.28 +- 1.3 (0.00) | 11 +- 1.6 (0.00) | 12.9 +- 2 (0.00) | 16 +- 2.6 (0.00) | 20 +- 3.6 (0.00) | 26.1 +- 4.2 (0.00) |
| share of rising steps | 0.53 | 0.499 +- 0.011 (1.00) | 0.541 +- 0.011 (0.15) | 0.586 +- 0.012 (0.00) | 0.63 +- 0.012 (0.00) | 0.676 +- 0.013 (0.00) | 0.721 +- 0.011 (0.00) | 0.77 +- 0.012 (0.00) | 0.818 +- 0.011 (0.00) | 0.867 +- 0.0097 (0.00) |
| runs-up-and-down z | -1.86 | 0.473 +- 1 (0.01) | -1.59 +- 0.95 (0.39) | -4.02 +- 1.1 (0.97) | -6.81 +- 1.1 (1.00) | -10.1 +- 1.2 (1.00) | -13.5 +- 1.2 (1.00) | -17.8 +- 1.3 (1.00) | -22.5 +- 1.2 (1.00) | -27.5 +- 1.1 (1.00) |
| adjacent pairs |a-b| <= 10 | 77 | 43.2 +- 6.5 (1.00) | 68.6 +- 8.3 (0.87) | 93.1 +- 9.5 (0.05) | 119 +- 11 (0.00) | 144 +- 12 (0.00) | 170 +- 12 (0.00) | 196 +- 12 (0.00) | 222 +- 13 (0.00) | 250 +- 12 (0.00) |
| trend (rank correlation of value with position) | 0.0292 | 0.096 +- 0.041 (0.05) | 0.112 +- 0.046 (0.04) | 0.127 +- 0.049 (0.02) | 0.142 +- 0.051 (0.02) | 0.152 +- 0.06 (0.01) | 0.166 +- 0.061 (0.01) | 0.183 +- 0.071 (0.02) | 0.193 +- 0.082 (0.02) | 0.181 +- 0.091 (0.04) |
| mean of first half minus second half | -13.5 | -26.9 +- 15 (0.83) | -33.5 +- 18 (0.88) | -41.6 +- 22 (0.90) | -47.6 +- 24 (0.92) | -59.7 +- 30 (0.94) | -66 +- 38 (0.91) | -84.5 +- 47 (0.94) | -98.4 +- 56 (0.93) | -103 +- 72 (0.89) |
| KS p for positions of the top-decile numbers | 0.759 | 0.292 +- 0.3 (0.90) | 0.217 +- 0.25 (0.94) | 0.189 +- 0.25 (0.94) | 0.15 +- 0.22 (0.96) | 0.12 +- 0.21 (0.97) | 0.0817 +- 0.16 (0.99) | 0.0461 +- 0.11 (1.00) | 0.0317 +- 0.085 (1.00) | 0.0218 +- 0.056 (1.00) |
| distinct numbers | 181 | 194 +- 6.9 (0.06) | 208 +- 7.5 (0.00) | 222 +- 8.9 (0.00) | 236 +- 9.6 (0.00) | 253 +- 10 (0.00) | 269 +- 13 (0.00) | 294 +- 16 (0.00) | 320 +- 16 (0.00) | 349 +- 13 (0.00) |
| median | 85 | 131 +- 14 (0.00) | 142 +- 15 (0.00) | 153 +- 18 (0.00) | 166 +- 22 (0.00) | 186 +- 28 (0.00) | 207 +- 33 (0.00) | 249 +- 48 (0.00) | 316 +- 80 (0.00) | 459 +- 1.4e+02 (0.00) |

**B1** (q = probability of taking the nearest forward homophone; otherwise the fitted encoder for B2, the calibrated encoder for B1/B3; simulated mean +- sd, percentile of the observed value)
| statistic | observed | DOI_q0.0 | DOI_q0.1 | DOI_q0.2 | DOI_q0.3 | DOI_q0.4 | DOI_q0.5 | DOI_q0.6 | DOI_q0.7 | DOI_q0.8 | LONG2906_q0.0 | LONG2906_q0.1 | LONG2906_q0.2 | LONG2906_q0.3 | LONG2906_q0.4 | LONG2906_q0.5 | LONG2906_q0.6 | LONG2906_q0.7 | LONG2906_q0.8 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| lag-1 autocorrelation (Pearson) | 0.252 | 0.0621 +- 0.047 (1.00) | 0.144 +- 0.055 (0.98) | 0.232 +- 0.058 (0.61) | 0.322 +- 0.06 (0.10) | 0.404 +- 0.055 (0.00) | 0.499 +- 0.052 (0.00) | 0.587 +- 0.051 (0.00) | 0.67 +- 0.039 (0.00) | 0.754 +- 0.031 (0.00) | 0.0312 +- 0.047 (1.00) | 0.13 +- 0.056 (0.99) | 0.229 +- 0.061 (0.63) | 0.318 +- 0.061 (0.14) | 0.406 +- 0.058 (0.00) | 0.502 +- 0.061 (0.00) | 0.598 +- 0.054 (0.00) | 0.689 +- 0.048 (0.00) | 0.789 +- 0.042 (0.00) |
| lag-1 autocorrelation (rank) | 0.195 | 0.0688 +- 0.048 (1.00) | 0.147 +- 0.05 (0.83) | 0.232 +- 0.051 (0.25) | 0.312 +- 0.052 (0.01) | 0.391 +- 0.045 (0.00) | 0.478 +- 0.046 (0.00) | 0.562 +- 0.043 (0.00) | 0.643 +- 0.036 (0.00) | 0.73 +- 0.03 (0.00) | 0.0307 +- 0.047 (1.00) | 0.124 +- 0.049 (0.93) | 0.213 +- 0.047 (0.37) | 0.307 +- 0.046 (0.01) | 0.395 +- 0.043 (0.00) | 0.485 +- 0.046 (0.00) | 0.581 +- 0.041 (0.00) | 0.671 +- 0.037 (0.00) | 0.769 +- 0.033 (0.00) |
| lag-2 autocorrelation (Pearson) | 0.0265 | 0.0744 +- 0.05 (0.15) | 0.0779 +- 0.05 (0.14) | 0.105 +- 0.052 (0.07) | 0.139 +- 0.059 (0.03) | 0.185 +- 0.059 (0.00) | 0.261 +- 0.063 (0.00) | 0.349 +- 0.071 (0.00) | 0.45 +- 0.061 (0.00) | 0.568 +- 0.049 (0.00) | 0.0214 +- 0.041 (0.54) | 0.0365 +- 0.051 (0.46) | 0.0653 +- 0.056 (0.24) | 0.113 +- 0.058 (0.07) | 0.172 +- 0.06 (0.01) | 0.254 +- 0.073 (0.00) | 0.36 +- 0.074 (0.00) | 0.476 +- 0.073 (0.00) | 0.624 +- 0.067 (0.00) |
| lag-2 autocorrelation (rank) | 0.000876 | 0.0805 +- 0.05 (0.07) | 0.0825 +- 0.046 (0.04) | 0.107 +- 0.05 (0.01) | 0.136 +- 0.054 (0.01) | 0.178 +- 0.054 (0.00) | 0.245 +- 0.056 (0.00) | 0.322 +- 0.06 (0.00) | 0.415 +- 0.053 (0.00) | 0.531 +- 0.048 (0.00) | 0.0175 +- 0.045 (0.35) | 0.034 +- 0.046 (0.24) | 0.0577 +- 0.05 (0.12) | 0.104 +- 0.049 (0.02) | 0.163 +- 0.049 (0.00) | 0.237 +- 0.056 (0.00) | 0.338 +- 0.056 (0.00) | 0.449 +- 0.058 (0.00) | 0.59 +- 0.054 (0.00) |
| mean length of strictly rising runs | 2.27 | 2 +- 0.053 (1.00) | 2.19 +- 0.065 (0.89) | 2.42 +- 0.081 (0.02) | 2.73 +- 0.11 (0.00) | 3.08 +- 0.14 (0.00) | 3.59 +- 0.17 (0.00) | 4.32 +- 0.24 (0.00) | 5.4 +- 0.35 (0.00) | 7.33 +- 0.55 (0.00) | 2 +- 0.053 (1.00) | 2.21 +- 0.068 (0.82) | 2.45 +- 0.083 (0.01) | 2.79 +- 0.12 (0.00) | 3.18 +- 0.15 (0.00) | 3.75 +- 0.2 (0.00) | 4.57 +- 0.27 (0.00) | 5.83 +- 0.45 (0.00) | 8.11 +- 0.72 (0.00) |
| rising runs of length >= 4 | 36 | 16.8 +- 3.8 (1.00) | 28.1 +- 4.6 (0.97) | 38.7 +- 4.4 (0.31) | 48.6 +- 4.6 (0.00) | 56.4 +- 3.9 (0.00) | 61.7 +- 3.6 (0.00) | 63.1 +- 3.3 (0.00) | 60.6 +- 3.5 (0.00) | 52.2 +- 3.5 (0.00) | 17.5 +- 3.7 (1.00) | 29 +- 4.5 (0.96) | 40.3 +- 4.7 (0.21) | 50.7 +- 4.6 (0.00) | 57.7 +- 4.2 (0.00) | 62.5 +- 3.7 (0.00) | 62.6 +- 3.6 (0.00) | 58.4 +- 3.5 (0.00) | 49.2 +- 3.6 (0.00) |
| rising runs of length >= 6 | 3 | 0.687 +- 0.85 (0.99) | 2.43 +- 1.4 (0.78) | 5.75 +- 2.3 (0.16) | 11.3 +- 3 (0.00) | 17.8 +- 3.4 (0.00) | 25.8 +- 3.6 (0.00) | 33.5 +- 3.5 (0.00) | 38.6 +- 3.1 (0.00) | 39.4 +- 2.9 (0.00) | 0.593 +- 0.74 (1.00) | 2.63 +- 1.5 (0.74) | 6.61 +- 2.5 (0.09) | 12.9 +- 3.3 (0.00) | 19.9 +- 3.3 (0.00) | 28.1 +- 3.5 (0.00) | 35 +- 3.3 (0.00) | 39 +- 2.8 (0.00) | 38.1 +- 2.9 (0.00) |
| longest rising run | 7 | 5.57 +- 0.71 (1.00) | 6.57 +- 0.92 (0.84) | 7.57 +- 1.1 (0.53) | 8.98 +- 1.4 (0.10) | 10.5 +- 1.7 (0.01) | 12.6 +- 2.3 (0.00) | 15.8 +- 2.8 (0.00) | 19.1 +- 3.7 (0.00) | 25.1 +- 4.8 (0.00) | 5.51 +- 0.7 (0.99) | 6.71 +- 1 (0.80) | 7.86 +- 1.2 (0.43) | 9.34 +- 1.5 (0.06) | 10.9 +- 1.8 (0.00) | 13 +- 2.1 (0.00) | 16.1 +- 2.8 (0.00) | 20.3 +- 3.6 (0.00) | 27.9 +- 5.4 (0.00) |
| share of rising steps | 0.561 | 0.501 +- 0.013 (1.00) | 0.545 +- 0.014 (0.89) | 0.588 +- 0.014 (0.02) | 0.634 +- 0.015 (0.00) | 0.676 +- 0.014 (0.00) | 0.722 +- 0.014 (0.00) | 0.769 +- 0.013 (0.00) | 0.816 +- 0.012 (0.00) | 0.864 +- 0.01 (0.00) | 0.5 +- 0.013 (1.00) | 0.548 +- 0.014 (0.82) | 0.593 +- 0.014 (0.01) | 0.642 +- 0.015 (0.00) | 0.686 +- 0.014 (0.00) | 0.734 +- 0.014 (0.00) | 0.782 +- 0.013 (0.00) | 0.829 +- 0.013 (0.00) | 0.877 +- 0.011 (0.00) |
| runs-up-and-down z | -3.06 | 0.147 +- 1.1 (0.00) | -1.44 +- 1.1 (0.09) | -3.48 +- 1 (0.66) | -5.7 +- 1.1 (1.00) | -8.26 +- 1 (1.00) | -11.2 +- 1.1 (1.00) | -14.7 +- 1.1 (1.00) | -18.3 +- 1.1 (1.00) | -22.5 +- 0.99 (1.00) | -0.163 +- 0.97 (0.00) | -1.76 +- 1 (0.08) | -3.9 +- 1.1 (0.77) | -6.48 +- 1.2 (1.00) | -9.04 +- 1.1 (1.00) | -12.2 +- 1.2 (1.00) | -15.7 +- 1.1 (1.00) | -19.5 +- 1.2 (1.00) | -23.7 +- 1.1 (1.00) |
| adjacent pairs |a-b| <= 10 | 32 | 21.6 +- 5.1 (0.97) | 40 +- 6.2 (0.12) | 59.9 +- 7.2 (0.00) | 78.1 +- 8 (0.00) | 96.7 +- 8.2 (0.00) | 115 +- 9.7 (0.00) | 134 +- 8.9 (0.00) | 153 +- 10 (0.00) | 172 +- 10 (0.00) | 15.9 +- 4.1 (1.00) | 35.7 +- 5.7 (0.30) | 55.9 +- 6.9 (0.00) | 76.6 +- 7.8 (0.00) | 95.7 +- 9 (0.00) | 115 +- 8.6 (0.00) | 136 +- 9.7 (0.00) | 155 +- 10 (0.00) | 176 +- 11 (0.00) |
| trend (rank correlation of value with position) | 0.0289 | 0.285 +- 0.048 (0.00) | 0.279 +- 0.052 (0.00) | 0.28 +- 0.055 (0.00) | 0.271 +- 0.063 (0.00) | 0.259 +- 0.067 (0.00) | 0.238 +- 0.072 (0.00) | 0.218 +- 0.09 (0.01) | 0.186 +- 0.098 (0.06) | 0.14 +- 0.12 (0.18) | 0.175 +- 0.05 (0.00) | 0.173 +- 0.057 (0.01) | 0.178 +- 0.061 (0.01) | 0.177 +- 0.069 (0.03) | 0.169 +- 0.077 (0.05) | 0.163 +- 0.089 (0.05) | 0.156 +- 0.098 (0.12) | 0.126 +- 0.11 (0.19) | 0.137 +- 0.13 (0.21) |
| mean of first half minus second half | -33.7 | -135 +- 24 (1.00) | -132 +- 26 (1.00) | -130 +- 29 (1.00) | -128 +- 31 (0.99) | -119 +- 36 (0.99) | -105 +- 45 (0.95) | -101 +- 53 (0.89) | -86.1 +- 63 (0.78) | -67.8 +- 84 (0.68) | -110 +- 34 (0.99) | -106 +- 39 (0.98) | -111 +- 39 (0.97) | -105 +- 48 (0.93) | -99.2 +- 54 (0.90) | -96.3 +- 61 (0.88) | -89.3 +- 69 (0.81) | -77 +- 75 (0.75) | -78.6 +- 98 (0.68) |
| KS p for positions of the top-decile numbers | 0.163 | 0.00627 +- 0.032 (0.99) | 0.0157 +- 0.067 (0.98) | 0.0217 +- 0.071 (0.97) | 0.0315 +- 0.093 (0.93) | 0.0594 +- 0.14 (0.88) | 0.0828 +- 0.16 (0.83) | 0.0676 +- 0.13 (0.86) | 0.0513 +- 0.097 (0.89) | 0.0289 +- 0.066 (0.94) | 0.0435 +- 0.11 (0.92) | 0.0713 +- 0.16 (0.86) | 0.0548 +- 0.13 (0.87) | 0.0575 +- 0.13 (0.89) | 0.0574 +- 0.14 (0.91) | 0.0504 +- 0.12 (0.90) | 0.0367 +- 0.083 (0.94) | 0.0297 +- 0.077 (0.94) | 0.0111 +- 0.036 (0.99) |
| distinct numbers | 298 | 298 +- 11 (0.55) | 300 +- 10 (0.45) | 300 +- 9.7 (0.44) | 299 +- 11 (0.46) | 296 +- 10 (0.58) | 295 +- 12 (0.59) | 293 +- 12 (0.69) | 294 +- 11 (0.66) | 299 +- 11 (0.49) | 297 +- 11 (0.59) | 303 +- 11 (0.36) | 308 +- 11 (0.18) | 310 +- 11 (0.16) | 313 +- 11 (0.10) | 314 +- 11 (0.09) | 314 +- 12 (0.10) | 318 +- 13 (0.07) | 327 +- 15 (0.04) |
| median | 123 | 181 +- 12 (0.00) | 185 +- 14 (0.00) | 189 +- 14 (0.00) | 194 +- 16 (0.00) | 198 +- 19 (0.00) | 212 +- 24 (0.00) | 225 +- 27 (0.00) | 250 +- 34 (0.00) | 301 +- 49 (0.00) | 268 +- 23 (0.00) | 273 +- 24 (0.00) | 279 +- 30 (0.00) | 284 +- 29 (0.00) | 295 +- 36 (0.00) | 302 +- 35 (0.00) | 318 +- 41 (0.00) | 352 +- 54 (0.00) | 408 +- 78 (0.00) |

**B3** (q = probability of taking the nearest forward homophone; otherwise the fitted encoder for B2, the calibrated encoder for B1/B3; simulated mean +- sd, percentile of the observed value)
| statistic | observed | DOI_q0.0 | DOI_q0.1 | DOI_q0.2 | DOI_q0.3 | DOI_q0.4 | DOI_q0.5 | DOI_q0.6 | DOI_q0.7 | DOI_q0.8 | LONG975_q0.0 | LONG975_q0.1 | LONG975_q0.2 | LONG975_q0.3 | LONG975_q0.4 | LONG975_q0.5 | LONG975_q0.6 | LONG975_q0.7 | LONG975_q0.8 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| lag-1 autocorrelation (Pearson) | 0.622 | 0.0308 +- 0.046 (1.00) | 0.121 +- 0.045 (1.00) | 0.221 +- 0.051 (1.00) | 0.31 +- 0.055 (1.00) | 0.402 +- 0.049 (1.00) | 0.495 +- 0.049 (1.00) | 0.585 +- 0.046 (0.81) | 0.666 +- 0.037 (0.11) | 0.745 +- 0.03 (0.00) | 0.0918 +- 0.049 (1.00) | 0.171 +- 0.052 (1.00) | 0.252 +- 0.057 (1.00) | 0.335 +- 0.055 (1.00) | 0.425 +- 0.054 (1.00) | 0.513 +- 0.052 (0.99) | 0.591 +- 0.048 (0.76) | 0.673 +- 0.039 (0.11) | 0.735 +- 0.031 (0.00) |
| lag-1 autocorrelation (rank) | 0.489 | 0.0378 +- 0.046 (1.00) | 0.122 +- 0.044 (1.00) | 0.215 +- 0.044 (1.00) | 0.296 +- 0.043 (1.00) | 0.383 +- 0.044 (0.99) | 0.473 +- 0.041 (0.67) | 0.56 +- 0.039 (0.04) | 0.645 +- 0.033 (0.00) | 0.728 +- 0.029 (0.00) | 0.0804 +- 0.046 (1.00) | 0.162 +- 0.045 (1.00) | 0.241 +- 0.043 (1.00) | 0.321 +- 0.044 (1.00) | 0.405 +- 0.046 (0.96) | 0.491 +- 0.04 (0.47) | 0.571 +- 0.038 (0.01) | 0.656 +- 0.033 (0.00) | 0.726 +- 0.028 (0.00) |
| lag-2 autocorrelation (Pearson) | 0.454 | 0.0526 +- 0.048 (1.00) | 0.0558 +- 0.042 (1.00) | 0.078 +- 0.05 (1.00) | 0.124 +- 0.056 (1.00) | 0.174 +- 0.054 (1.00) | 0.25 +- 0.061 (1.00) | 0.342 +- 0.062 (0.97) | 0.444 +- 0.054 (0.56) | 0.551 +- 0.049 (0.04) | 0.0674 +- 0.046 (1.00) | 0.078 +- 0.05 (1.00) | 0.105 +- 0.052 (1.00) | 0.139 +- 0.054 (1.00) | 0.199 +- 0.06 (1.00) | 0.278 +- 0.062 (1.00) | 0.352 +- 0.066 (0.93) | 0.452 +- 0.058 (0.48) | 0.533 +- 0.051 (0.07) |
| lag-2 autocorrelation (rank) | 0.332 | 0.0532 +- 0.046 (1.00) | 0.0572 +- 0.042 (1.00) | 0.0821 +- 0.047 (1.00) | 0.12 +- 0.051 (1.00) | 0.168 +- 0.053 (1.00) | 0.234 +- 0.054 (0.96) | 0.315 +- 0.052 (0.62) | 0.417 +- 0.047 (0.03) | 0.525 +- 0.046 (0.00) | 0.0576 +- 0.043 (1.00) | 0.0677 +- 0.047 (1.00) | 0.0949 +- 0.052 (1.00) | 0.124 +- 0.046 (1.00) | 0.181 +- 0.054 (1.00) | 0.251 +- 0.054 (0.94) | 0.328 +- 0.053 (0.54) | 0.428 +- 0.053 (0.04) | 0.518 +- 0.047 (0.00) |
| mean length of strictly rising runs | 2.58 | 1.99 +- 0.049 (1.00) | 2.19 +- 0.064 (1.00) | 2.42 +- 0.068 (0.99) | 2.71 +- 0.092 (0.09) | 3.08 +- 0.13 (0.00) | 3.6 +- 0.18 (0.00) | 4.32 +- 0.23 (0.00) | 5.45 +- 0.34 (0.00) | 7.43 +- 0.56 (0.00) | 2 +- 0.047 (1.00) | 2.19 +- 0.061 (1.00) | 2.43 +- 0.073 (0.97) | 2.72 +- 0.098 (0.07) | 3.11 +- 0.11 (0.00) | 3.65 +- 0.17 (0.00) | 4.38 +- 0.24 (0.00) | 5.55 +- 0.33 (0.00) | 7.59 +- 0.61 (0.00) |
| rising runs of length >= 4 | 49 | 19.7 +- 4.1 (1.00) | 33.4 +- 4.8 (1.00) | 46.1 +- 4.9 (0.74) | 57.4 +- 4.7 (0.04) | 67.3 +- 4.4 (0.00) | 73.7 +- 4.4 (0.00) | 75.5 +- 3.6 (0.00) | 71.3 +- 3.7 (0.00) | 61.9 +- 3.8 (0.00) | 20.8 +- 4.3 (1.00) | 34 +- 4.6 (1.00) | 47.1 +- 5 (0.68) | 58.2 +- 4.7 (0.03) | 67.9 +- 4.3 (0.00) | 74 +- 4.2 (0.00) | 75.7 +- 4 (0.00) | 71.8 +- 3.9 (0.00) | 61.7 +- 4 (0.00) |
| rising runs of length >= 6 | 17 | 0.6 +- 0.8 (1.00) | 2.87 +- 1.6 (1.00) | 6.81 +- 2.4 (1.00) | 13.5 +- 3.1 (0.90) | 21.1 +- 3.6 (0.14) | 31 +- 3.7 (0.00) | 39.7 +- 3.9 (0.00) | 45.9 +- 3.4 (0.00) | 46.7 +- 3.2 (0.00) | 0.667 +- 0.81 (1.00) | 2.89 +- 1.7 (1.00) | 7.02 +- 2.4 (1.00) | 13.5 +- 3.3 (0.86) | 22 +- 3.7 (0.12) | 31.7 +- 3.9 (0.00) | 40.5 +- 3.9 (0.00) | 46.5 +- 3.4 (0.00) | 47 +- 3.1 (0.00) |
| longest rising run | 10 | 5.48 +- 0.65 (1.00) | 6.56 +- 0.87 (1.00) | 7.71 +- 1.1 (0.99) | 9.07 +- 1.4 (0.87) | 10.6 +- 1.5 (0.52) | 12.8 +- 2.1 (0.11) | 15.5 +- 2.5 (0.00) | 19.6 +- 3.4 (0.00) | 25.5 +- 4.6 (0.00) | 5.6 +- 0.71 (1.00) | 6.58 +- 0.86 (1.00) | 7.74 +- 1.1 (0.99) | 9.11 +- 1.4 (0.86) | 10.8 +- 1.6 (0.47) | 12.9 +- 2.2 (0.12) | 15.5 +- 2.7 (0.01) | 19.3 +- 3.2 (0.00) | 24.8 +- 4.3 (0.00) |
| share of rising steps | 0.613 | 0.498 +- 0.012 (1.00) | 0.544 +- 0.013 (1.00) | 0.588 +- 0.012 (0.99) | 0.631 +- 0.013 (0.09) | 0.676 +- 0.013 (0.00) | 0.723 +- 0.014 (0.00) | 0.769 +- 0.012 (0.00) | 0.817 +- 0.011 (0.00) | 0.866 +- 0.011 (0.00) | 0.5 +- 0.012 (1.00) | 0.545 +- 0.013 (1.00) | 0.589 +- 0.012 (0.97) | 0.634 +- 0.013 (0.07) | 0.68 +- 0.011 (0.00) | 0.727 +- 0.012 (0.00) | 0.772 +- 0.012 (0.00) | 0.82 +- 0.011 (0.00) | 0.869 +- 0.01 (0.00) |
| runs-up-and-down z | -7.61 | 0.348 +- 0.97 (0.00) | -1.65 +- 1.1 (0.00) | -3.73 +- 1 (0.00) | -6.19 +- 1.1 (0.12) | -8.98 +- 1.1 (0.88) | -12.4 +- 1.1 (1.00) | -16 +- 1.2 (1.00) | -20.1 +- 1.1 (1.00) | -24.7 +- 1.1 (1.00) | -0.206 +- 0.99 (0.00) | -2 +- 0.98 (0.00) | -3.98 +- 1 (0.00) | -6.41 +- 1.1 (0.14) | -9.29 +- 1.1 (0.95) | -12.6 +- 1.1 (1.00) | -16.2 +- 1.2 (1.00) | -20.4 +- 1.1 (1.00) | -24.9 +- 1.1 (1.00) |
| adjacent pairs |a-b| <= 10 | 76 | 29.8 +- 5.7 (1.00) | 51.9 +- 7.3 (1.00) | 74.2 +- 8.2 (0.61) | 94.5 +- 9.5 (0.03) | 116 +- 9.9 (0.00) | 138 +- 10 (0.00) | 160 +- 11 (0.00) | 180 +- 11 (0.00) | 204 +- 10 (0.00) | 37.6 +- 6.3 (1.00) | 58.5 +- 7.7 (0.99) | 79.3 +- 7.7 (0.37) | 101 +- 9.5 (0.01) | 123 +- 11 (0.00) | 145 +- 11 (0.00) | 167 +- 10 (0.00) | 189 +- 12 (0.00) | 211 +- 12 (0.00) |
| trend (rank correlation of value with position) | 0.169 | 0.235 +- 0.046 (0.08) | 0.233 +- 0.047 (0.09) | 0.235 +- 0.058 (0.11) | 0.221 +- 0.065 (0.22) | 0.22 +- 0.072 (0.24) | 0.21 +- 0.08 (0.32) | 0.188 +- 0.086 (0.42) | 0.17 +- 0.098 (0.45) | 0.124 +- 0.11 (0.67) | 0.274 +- 0.044 (0.01) | 0.271 +- 0.047 (0.02) | 0.273 +- 0.055 (0.04) | 0.265 +- 0.059 (0.08) | 0.265 +- 0.067 (0.08) | 0.251 +- 0.077 (0.16) | 0.233 +- 0.082 (0.19) | 0.215 +- 0.098 (0.30) | 0.151 +- 0.1 (0.58) |
| mean of first half minus second half | -80.9 | -103 +- 21 (0.84) | -103 +- 23 (0.82) | -102 +- 29 (0.78) | -94.2 +- 33 (0.65) | -94.3 +- 37 (0.64) | -90.6 +- 46 (0.58) | -84.5 +- 50 (0.54) | -76.3 +- 61 (0.50) | -60.2 +- 74 (0.39) | -105 +- 16 (0.93) | -103 +- 18 (0.87) | -101 +- 21 (0.84) | -98.8 +- 22 (0.82) | -94.9 +- 26 (0.66) | -89.2 +- 31 (0.63) | -85.3 +- 33 (0.54) | -79.5 +- 43 (0.48) | -56.8 +- 46 (0.31) |
| KS p for positions of the top-decile numbers | 8.98e-10 | 0.00968 +- 0.039 (0.00) | 0.0215 +- 0.084 (0.00) | 0.0448 +- 0.12 (0.01) | 0.071 +- 0.15 (0.01) | 0.0951 +- 0.18 (0.01) | 0.0927 +- 0.16 (0.02) | 0.0778 +- 0.14 (0.03) | 0.0435 +- 0.089 (0.03) | 0.0389 +- 0.089 (0.04) | 0.000183 +- 0.0015 (0.17) | 0.0018 +- 0.02 (0.14) | 0.00332 +- 0.052 (0.18) | 0.00228 +- 0.02 (0.24) | 0.00375 +- 0.028 (0.21) | 0.00987 +- 0.059 (0.22) | 0.0151 +- 0.064 (0.18) | 0.0174 +- 0.066 (0.16) | 0.0345 +- 0.088 (0.05) |
| distinct numbers | 263 | 261 +- 11 (0.59) | 270 +- 11 (0.29) | 277 +- 11 (0.11) | 282 +- 12 (0.06) | 288 +- 12 (0.02) | 292 +- 12 (0.01) | 302 +- 13 (0.00) | 310 +- 14 (0.00) | 325 +- 13 (0.00) | 260 +- 11 (0.65) | 268 +- 10 (0.33) | 273 +- 12 (0.20) | 276 +- 11 (0.13) | 278 +- 13 (0.12) | 283 +- 12 (0.05) | 288 +- 13 (0.03) | 293 +- 14 (0.02) | 304 +- 13 (0.00) |
| median | 94.5 | 156 +- 15 (0.00) | 164 +- 16 (0.00) | 171 +- 17 (0.00) | 182 +- 19 (0.00) | 193 +- 22 (0.00) | 208 +- 26 (0.00) | 236 +- 33 (0.00) | 274 +- 50 (0.00) | 344 +- 74 (0.00) | 131 +- 13 (0.00) | 137 +- 13 (0.00) | 144 +- 14 (0.00) | 151 +- 14 (0.00) | 159 +- 16 (0.00) | 173 +- 19 (0.00) | 188 +- 24 (0.00) | 211 +- 33 (0.00) | 250 +- 40 (0.00) |

### Autocorrelation at lags 1-10 and block-level variance (stats_11_acf.py; 2000 shuffles, 300 simulations per model, seed 20260917)
Block ratio = variance of the means of consecutive blocks of 10 / 20 / 50 numbers divided by its value for exchangeable data (about 1). 'detrended' = after removing a least-squares line of value on position (the calibrated encoders drift upward because they spend the early words of the key first).
**B1**
| statistic | lag 1 | lag 2 | lag 3 | lag 4 | lag 5 | lag 6 | lag 7 | lag 8 | lag 9 | lag 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| observed (Pearson) | 0.252 | 0.026 | -0.096 | -0.119 | 0.025 | -0.057 | -0.067 | 0.001 | 0.010 | 0.045 |
| shuffles 97.5 % (Pearson) | 0.094 | 0.087 | 0.085 | 0.089 | 0.089 | 0.092 | 0.091 | 0.088 | 0.090 | 0.088 |
| percentile of observed among shuffles (Pearson) | 1.000 | 0.741 | 0.007 | 0.000 | 0.748 | 0.098 | 0.054 | 0.533 | 0.619 | 0.860 |
| DOI_q0.1 mean +- sd (Pearson) | 0.145 +- 0.050 | 0.075 +- 0.051 | 0.077 +- 0.049 | 0.068 +- 0.056 | 0.074 +- 0.052 | 0.080 +- 0.049 | 0.075 +- 0.047 | 0.074 +- 0.053 | 0.077 +- 0.053 | 0.070 +- 0.054 |
| percentile of observed among DOI_q0.1 (Pearson) | 0.97 | 0.17 | 0.00 | 0.00 | 0.17 | 0.00 | 0.00 | 0.07 | 0.11 | 0.34 |
| DOI_q0.2 mean +- sd (Pearson) | 0.232 +- 0.054 | 0.100 +- 0.055 | 0.075 +- 0.050 | 0.073 +- 0.052 | 0.076 +- 0.053 | 0.083 +- 0.048 | 0.077 +- 0.049 | 0.081 +- 0.053 | 0.076 +- 0.050 | 0.072 +- 0.055 |
| percentile of observed among DOI_q0.2 (Pearson) | 0.67 | 0.09 | 0.00 | 0.00 | 0.18 | 0.00 | 0.00 | 0.08 | 0.10 | 0.34 |
| observed (rank) | 0.195 | 0.001 | -0.081 | -0.060 | 0.041 | -0.059 | -0.145 | -0.041 | -0.009 | 0.087 |
| shuffles 97.5 % (rank) | 0.083 | 0.084 | 0.085 | 0.080 | 0.087 | 0.087 | 0.082 | 0.078 | 0.083 | 0.085 |
| percentile of observed among shuffles (rank) | 1.000 | 0.516 | 0.036 | 0.097 | 0.830 | 0.097 | 0.001 | 0.176 | 0.457 | 0.978 |
| DOI_q0.1 mean +- sd (rank) | 0.147 +- 0.046 | 0.080 +- 0.051 | 0.081 +- 0.051 | 0.075 +- 0.050 | 0.076 +- 0.050 | 0.083 +- 0.044 | 0.079 +- 0.045 | 0.074 +- 0.049 | 0.078 +- 0.051 | 0.075 +- 0.049 |
| percentile of observed among DOI_q0.1 (rank) | 0.84 | 0.06 | 0.00 | 0.00 | 0.24 | 0.00 | 0.00 | 0.01 | 0.05 | 0.61 |
| DOI_q0.2 mean +- sd (rank) | 0.231 +- 0.049 | 0.108 +- 0.051 | 0.084 +- 0.052 | 0.082 +- 0.054 | 0.081 +- 0.053 | 0.086 +- 0.050 | 0.082 +- 0.053 | 0.084 +- 0.053 | 0.079 +- 0.051 | 0.077 +- 0.054 |
| percentile of observed among DOI_q0.2 (rank) | 0.24 | 0.02 | 0.00 | 0.00 | 0.24 | 0.00 | 0.00 | 0.02 | 0.04 | 0.56 |
| observed (Pearson, detrended) | 0.251 | 0.024 | -0.099 | -0.122 | 0.022 | -0.060 | -0.070 | -0.001 | 0.007 | 0.042 |
| shuffles 97.5 % (Pearson, detrended) | 0.092 | 0.084 | 0.084 | 0.088 | 0.087 | 0.090 | 0.087 | 0.086 | 0.086 | 0.086 |
| percentile of observed among shuffles (Pearson, detrended) | 1.000 | 0.733 | 0.007 | 0.000 | 0.737 | 0.093 | 0.054 | 0.529 | 0.613 | 0.854 |
| DOI_q0.1 mean +- sd (Pearson, detrended) | 0.075 +- 0.050 | -0.001 +- 0.047 | 0.001 +- 0.048 | -0.008 +- 0.052 | -0.002 +- 0.049 | 0.005 +- 0.047 | -0.000 +- 0.045 | -0.001 +- 0.051 | 0.002 +- 0.049 | -0.005 +- 0.049 |
| percentile of observed among DOI_q0.1 (Pearson, detrended) | 1.00 | 0.72 | 0.02 | 0.01 | 0.70 | 0.07 | 0.06 | 0.49 | 0.53 | 0.85 |
| DOI_q0.2 mean +- sd (Pearson, detrended) | 0.168 +- 0.054 | 0.025 +- 0.053 | -0.002 +- 0.048 | -0.004 +- 0.048 | -0.000 +- 0.049 | 0.006 +- 0.044 | 0.001 +- 0.046 | 0.004 +- 0.052 | -0.000 +- 0.045 | -0.005 +- 0.049 |
| percentile of observed among DOI_q0.2 (Pearson, detrended) | 0.93 | 0.52 | 0.01 | 0.00 | 0.65 | 0.06 | 0.06 | 0.44 | 0.57 | 0.81 |
- block ratio (blocks of [10, 20, 50]): observed [1.03, 1.16, 1.39]; shuffles [1.0, 1.0, 1.01] +- [0.18, 0.27, 0.47], upper-tail p [0.4108, 0.2534, 0.1959]; DOI_q0.1: [1.83, 2.65, 5.21] +- [0.3, 0.59, 1.56] (pct [0.0, 0.0, 0.0]); DOI_q0.2: [2.08, 2.92, 5.35] +- [0.36, 0.67, 1.7] (pct [0.0, 0.0, 0.0])
- block ratio, detrended (blocks of [10, 20, 50]): observed [1.01, 1.12, 1.16]; shuffles [0.98, 0.96, 0.92] +- [0.18, 0.26, 0.44], upper-tail p [0.4193, 0.2609, 0.2544]; DOI_q0.1: [1.14, 1.16, 1.23] +- [0.22, 0.33, 0.66] (pct [0.27, 0.47, 0.53]); DOI_q0.2: [1.39, 1.42, 1.36] +- [0.3, 0.45, 0.73] (pct [0.08, 0.28, 0.47])

**B2**
| statistic | lag 1 | lag 2 | lag 3 | lag 4 | lag 5 | lag 6 | lag 7 | lag 8 | lag 9 | lag 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| observed (Pearson) | 0.046 | -0.002 | 0.003 | -0.050 | 0.009 | -0.026 | 0.063 | -0.032 | -0.057 | -0.013 |
| shuffles 97.5 % (Pearson) | 0.075 | 0.072 | 0.069 | 0.070 | 0.070 | 0.073 | 0.072 | 0.070 | 0.075 | 0.069 |
| percentile of observed among shuffles (Pearson) | 0.898 | 0.482 | 0.581 | 0.080 | 0.619 | 0.260 | 0.957 | 0.205 | 0.059 | 0.399 |
| DOI_q0.1 mean +- sd (Pearson) | 0.080 +- 0.050 | 0.019 +- 0.038 | 0.003 +- 0.036 | -0.000 +- 0.036 | 0.010 +- 0.035 | 0.020 +- 0.035 | 0.010 +- 0.040 | 0.013 +- 0.038 | 0.003 +- 0.036 | -0.000 +- 0.039 |
| percentile of observed among DOI_q0.1 (Pearson) | 0.25 | 0.28 | 0.53 | 0.09 | 0.49 | 0.10 | 0.92 | 0.12 | 0.05 | 0.40 |
| observed (rank) | 0.206 | 0.068 | 0.028 | 0.012 | 0.013 | -0.020 | 0.009 | -0.030 | -0.038 | 0.012 |
| shuffles 97.5 % (rank) | 0.070 | 0.069 | 0.067 | 0.070 | 0.068 | 0.075 | 0.069 | 0.067 | 0.072 | 0.069 |
| percentile of observed among shuffles (rank) | 1.000 | 0.973 | 0.806 | 0.650 | 0.664 | 0.312 | 0.616 | 0.220 | 0.148 | 0.658 |
| DOI_q0.1 mean +- sd (rank) | 0.084 +- 0.042 | 0.025 +- 0.038 | 0.011 +- 0.037 | 0.011 +- 0.038 | 0.014 +- 0.038 | 0.020 +- 0.037 | 0.018 +- 0.039 | 0.017 +- 0.037 | 0.011 +- 0.038 | 0.011 +- 0.041 |
| percentile of observed among DOI_q0.1 (rank) | 1.00 | 0.87 | 0.67 | 0.51 | 0.49 | 0.14 | 0.42 | 0.09 | 0.09 | 0.52 |
| observed (Pearson, detrended) | 0.044 | -0.004 | 0.002 | -0.052 | 0.008 | -0.027 | 0.062 | -0.034 | -0.059 | -0.014 |
| shuffles 97.5 % (Pearson, detrended) | 0.073 | 0.071 | 0.068 | 0.069 | 0.069 | 0.072 | 0.070 | 0.070 | 0.074 | 0.067 |
| percentile of observed among shuffles (Pearson, detrended) | 0.897 | 0.479 | 0.580 | 0.078 | 0.610 | 0.258 | 0.955 | 0.204 | 0.057 | 0.394 |
| DOI_q0.1 mean +- sd (Pearson, detrended) | 0.070 +- 0.049 | 0.008 +- 0.038 | -0.008 +- 0.036 | -0.011 +- 0.035 | -0.000 +- 0.034 | 0.009 +- 0.035 | -0.000 +- 0.041 | 0.002 +- 0.038 | -0.008 +- 0.036 | -0.011 +- 0.038 |
| percentile of observed among DOI_q0.1 (Pearson, detrended) | 0.32 | 0.38 | 0.63 | 0.12 | 0.60 | 0.15 | 0.95 | 0.17 | 0.06 | 0.48 |
- block ratio (blocks of [10, 20, 50]): observed [1.05, 1.06, 0.95]; shuffles [1.0, 1.0, 1.0] +- [0.16, 0.22, 0.37], upper-tail p [0.3523, 0.3553, 0.5147]; DOI_q0.1: [1.23, 1.33, 1.68] +- [0.2, 0.31, 0.64] (pct [0.18, 0.21, 0.11])
- block ratio, detrended (blocks of [10, 20, 50]): observed [1.04, 1.03, 0.84]; shuffles [0.99, 0.97, 0.94] +- [0.15, 0.21, 0.35], upper-tail p [0.3568, 0.3648, 0.5527]; DOI_q0.1: [1.13, 1.12, 1.11] +- [0.18, 0.26, 0.43] (pct [0.36, 0.39, 0.26])

**B3**
| statistic | lag 1 | lag 2 | lag 3 | lag 4 | lag 5 | lag 6 | lag 7 | lag 8 | lag 9 | lag 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| observed (Pearson) | 0.622 | 0.454 | 0.319 | 0.229 | 0.198 | 0.159 | 0.121 | 0.044 | -0.034 | -0.065 |
| shuffles 97.5 % (Pearson) | 0.082 | 0.081 | 0.082 | 0.081 | 0.081 | 0.082 | 0.084 | 0.083 | 0.086 | 0.089 |
| percentile of observed among shuffles (Pearson) | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.999 | 0.860 | 0.209 | 0.046 |
| DOI_q0.3 mean +- sd (Pearson) | 0.313 +- 0.051 | 0.121 +- 0.052 | 0.064 +- 0.051 | 0.049 +- 0.048 | 0.051 +- 0.051 | 0.055 +- 0.056 | 0.051 +- 0.054 | 0.048 +- 0.053 | 0.045 +- 0.053 | 0.038 +- 0.053 |
| percentile of observed among DOI_q0.3 (Pearson) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.96 | 0.91 | 0.49 | 0.07 | 0.02 |
| DOI_q0.6 mean +- sd (Pearson) | 0.577 +- 0.039 | 0.334 +- 0.056 | 0.193 +- 0.062 | 0.118 +- 0.065 | 0.076 +- 0.065 | 0.055 +- 0.066 | 0.040 +- 0.065 | 0.033 +- 0.062 | 0.029 +- 0.059 | 0.025 +- 0.058 |
| percentile of observed among DOI_q0.6 (Pearson) | 0.86 | 0.98 | 0.98 | 0.95 | 0.98 | 0.94 | 0.90 | 0.58 | 0.13 | 0.07 |
| observed (rank) | 0.489 | 0.332 | 0.247 | 0.183 | 0.130 | 0.072 | 0.014 | -0.076 | -0.088 | -0.078 |
| shuffles 97.5 % (rank) | 0.076 | 0.079 | 0.077 | 0.076 | 0.076 | 0.079 | 0.078 | 0.075 | 0.076 | 0.079 |
| percentile of observed among shuffles (rank) | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.964 | 0.637 | 0.031 | 0.015 | 0.029 |
| DOI_q0.3 mean +- sd (rank) | 0.299 +- 0.043 | 0.118 +- 0.049 | 0.068 +- 0.050 | 0.058 +- 0.050 | 0.057 +- 0.052 | 0.058 +- 0.054 | 0.056 +- 0.052 | 0.054 +- 0.053 | 0.052 +- 0.053 | 0.047 +- 0.052 |
| percentile of observed among DOI_q0.3 (rank) | 1.00 | 1.00 | 1.00 | 1.00 | 0.93 | 0.60 | 0.22 | 0.01 | 0.00 | 0.01 |
| DOI_q0.6 mean +- sd (rank) | 0.552 +- 0.033 | 0.307 +- 0.049 | 0.173 +- 0.057 | 0.106 +- 0.062 | 0.070 +- 0.064 | 0.053 +- 0.066 | 0.042 +- 0.064 | 0.038 +- 0.061 | 0.034 +- 0.059 | 0.032 +- 0.059 |
| percentile of observed among DOI_q0.6 (rank) | 0.03 | 0.68 | 0.89 | 0.88 | 0.81 | 0.63 | 0.34 | 0.02 | 0.02 | 0.04 |
| observed (Pearson, detrended) | 0.594 | 0.414 | 0.270 | 0.175 | 0.143 | 0.102 | 0.063 | -0.018 | -0.100 | -0.133 |
| shuffles 97.5 % (Pearson, detrended) | 0.080 | 0.080 | 0.081 | 0.079 | 0.078 | 0.081 | 0.083 | 0.082 | 0.084 | 0.088 |
| percentile of observed among shuffles (Pearson, detrended) | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.994 | 0.938 | 0.362 | 0.004 | 0.000 |
| DOI_q0.3 mean +- sd (Pearson, detrended) | 0.274 +- 0.050 | 0.070 +- 0.048 | 0.011 +- 0.044 | -0.005 +- 0.044 | -0.003 +- 0.044 | 0.002 +- 0.048 | -0.002 +- 0.047 | -0.005 +- 0.046 | -0.008 +- 0.045 | -0.015 +- 0.045 |
| percentile of observed among DOI_q0.3 (Pearson, detrended) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.98 | 0.93 | 0.40 | 0.01 | 0.00 |
| DOI_q0.6 mean +- sd (Pearson, detrended) | 0.562 +- 0.039 | 0.310 +- 0.054 | 0.164 +- 0.057 | 0.086 +- 0.058 | 0.043 +- 0.058 | 0.021 +- 0.059 | 0.006 +- 0.060 | -0.001 +- 0.058 | -0.006 +- 0.055 | -0.010 +- 0.054 |
| percentile of observed among DOI_q0.6 (Pearson, detrended) | 0.79 | 0.98 | 0.97 | 0.93 | 0.96 | 0.91 | 0.83 | 0.41 | 0.05 | 0.01 |
- block ratio (blocks of [10, 20, 50]): observed [3.47, 4.23, 4.49]; shuffles [1.0, 1.0, 1.0] +- [0.17, 0.25, 0.41], upper-tail p [0.0005, 0.0005, 0.0005]; DOI_q0.3: [2.1, 2.65, 4.37] +- [0.38, 0.7, 1.68] (pct [1.0, 0.97, 0.57]); DOI_q0.6: [3.22, 3.82, 4.97] +- [0.51, 0.98, 2.13] (pct [0.72, 0.66, 0.47])
- block ratio, detrended (blocks of [10, 20, 50]): observed [3.27, 3.51, 2.24]; shuffles [0.98, 0.97, 0.91] +- [0.17, 0.24, 0.39], upper-tail p [0.0005, 0.0005, 0.007]; DOI_q0.3: [1.65, 1.66, 1.62] +- [0.3, 0.45, 0.72] (pct [1.0, 1.0, 0.81]); DOI_q0.6: [2.97, 3.23, 3.23] +- [0.47, 0.81, 1.38] (pct [0.76, 0.65, 0.23])

### Letter frequencies of the DOI decode (stats_05_letterfreq.py; 4000 locally shuffled keys)
| key | cipher | decoded (out of range) | KL to English | KL to key initials | chi2 vs re-weighted-key expectation (p, MC) | KL to English under the null (mean) | p(closer to English than null) | letters with z > 3 |
|---|---|---|---|---|---|---|---|---|
| resolved | B1 | 508 (12) | 0.253 | 0.042 | 28.7 (p=0.191) | 0.275 | 0.265 | [] |
| resolved | B3 | 618 (0) | 0.246 | 0.063 | 22.9 (p=0.696) | 0.273 | 0.199 | ['q'] |
| resolved | B2 | 762 (0) | 0.039 | 0.408 | 560.7 (p=0.000) | 0.289 | 0.000 | ['e', 'n', 'u', 'v', 'x'] |
| straight | B1 | 510 (10) | 0.254 | 0.045 | 14.9 (p=0.833) | 0.283 | 0.201 | [] |
| straight | B3 | 618 (0) | 0.315 | 0.094 | 41.2 (p=0.058) | 0.283 | 0.836 | [] |
| straight | B2 | 762 (0) | 0.175 | 0.208 | 186.1 (p=0.000) | 0.299 | 0.001 | ['n', 'u'] |

- B1 decode (pamphlet numbering), commonest letters: t 0.185, a 0.142, o 0.089, s 0.063, c 0.059, i 0.055, b 0.051, w 0.043, f 0.041, e 0.039
- B3 decode (pamphlet numbering), commonest letters: t 0.215, a 0.126, o 0.086, w 0.060, e 0.057, s 0.053, c 0.050, r 0.050, b 0.044, h 0.040
- B2 decode (pamphlet numbering), commonest letters: e 0.135, t 0.098, n 0.089, o 0.083, i 0.068, d 0.064, s 0.059, a 0.056, r 0.052, h 0.049
- English (Norvig): e 0.125, t 0.093, a 0.080, o 0.076, i 0.076, n 0.072, s 0.065, r 0.063, h 0.051, l 0.041
- key initials (pamphlet numbering): t 0.194, a 0.127, o 0.111, h 0.060, i 0.050, s 0.047, p 0.047, f 0.046, w 0.045, c 0.040

### Capacity of cipher 3 (stats_06_capacity.py; 20 000 Monte-Carlo lists of 30 entries)
| scenario | description | mean letters | sd | 1st pct | 5th pct | P(total <= 618) |
|---|---|---|---|---|---|---|
| S0_bare_minimum | associate: first name + surname; relative: first name only (surname assumed shared); residence: county name only, no word 'county', no state | 792 | 19 | 747 | 760 | 0.0000 |
| S1_relative_surname | as S0 but the relative also gets a surname (a married sister or a mother would need one) | 978 | 21 | 929 | 944 | 0.0000 |
| S2_county_word | as S1 plus the word 'county' (6 letters) after each county name, as B2 writes 'county of bedford' | 1158 | 21 | 1110 | 1124 | 0.0000 |
| S3_county_and_state | as S2 plus 'virginia' (8 letters) for each residence | 1398 | 21 | 1350 | 1364 | 0.0000 |
| S4_initials_only | extreme abbreviation: associate = initial + surname; relative = initial + surname; residence = county name | 691 | 18 | 651 | 662 | 0.0000 |
| S5_initials_no_relative_surname | associate = initial + surname; relative = first name only; residence = county name | 640 | 18 | 599 | 611 | 0.1152 |
| S6_surnames_and_counties_only | floor: associate surname + relative surname + county name, nothing else | 631 | 17 | 590 | 602 | 0.2397 |

- name lengths, amherst_1810_heads: n = 864, first name mean 6.08, surname mean 6.37, full name mean 12.45 (sd 2.24)
- name lengths, bedford_1850_men_30plus: n = 426, first name mean 5.80, surname mean 6.19, full name mean 11.98 (sd 2.07)
- name lengths, botetourt_1850_index_30plus: n = 3299, first name mean 5.86, surname mean 6.42, full name mean 12.28 (sd 2.29)
- name lengths, bedford_1850_all: n = 2366, first name mean 5.49, surname mean 6.20
- name lengths, botetourt_1850_all: n = 10855, first name mean 5.87, surname mean 6.42
- 78 Virginia counties/cities formed by 1822: mean name length 8.46 letters, shortest 3