## Task 4 probability tables (generated by code/render_tables.py)

Simulations: 1,000,000 exact-letter shuffles/draws, 200,000 for the substitution statistics; seed 20260914.

### (i) i.i.d. letters drawn from the DOI-initial distribution: exact probabilities (n = 520)

| key (letter distribution) | statistic | L=10 | L=12 | L=14 | L=17 | L=20 |
|---|---|---|---|---|---|---|
| gillogly_table1 | nondecreasing, no breaks | 4.54e-03 | 1.85e-04 | 7.18e-06 | 5.27e-08 | 3.79e-10 |
| gillogly_table1 | nondecreasing, with B1 break fraction 0.019 | 3.76e-03 | 1.47e-04 | 5.50e-06 | 3.81e-08 | 2.58e-10 |
| gillogly_table1 | step01, no breaks | 4.68e-05 | 1.71e-06 | 6.31e-08 | 4.49e-10 | 3.20e-12 |
| gillogly_table1 | step01, with B1 break fraction 0.019 | 3.87e-05 | 1.36e-06 | 4.83e-08 | 3.24e-10 | 2.18e-12 |
| pamphlet_encoder_key | nondecreasing, no breaks | 4.43e-03 | 1.79e-04 | 6.85e-06 | 4.91e-08 | 3.45e-10 |
| pamphlet_encoder_key | nondecreasing, with B1 break fraction 0.021 | 3.60e-03 | 1.39e-04 | 5.10e-06 | 3.43e-08 | 2.26e-10 |
| pamphlet_encoder_key | step01, no breaks | 4.40e-05 | 1.58e-06 | 5.73e-08 | 3.98e-10 | 2.76e-12 |
| pamphlet_encoder_key | step01, with B1 break fraction 0.021 | 3.57e-05 | 1.23e-06 | 4.27e-08 | 2.78e-10 | 1.81e-12 |

Uniform-26-letter model (Gillogly's arithmetic): P(+0/+1 run >= 14 in 520) = 1.16e-12, >= 17: 4.85e-16, >= 20: 2.01e-19; non-decreasing >= 14: 1.06e-07; Gillogly's 495/13^13 = 1.63e-12.

Expected number of maximal non-decreasing runs of length >= L (i.i.d. DOI letters, Gillogly distribution): L>=5: 9.9, L>=8: 0.129, L>=10: 0.0057, L>=12: 0.000231, L>=14: 8.92e-06, L>=17: 6.53e-08

### Monte-Carlo p-values for B1 (P of a run at least as long as observed; MC standard error; count/N)

| null model | key | statistic | observed | P(>= observed) | P(>= 14) | P(>= 17) | P(>= 20) |
|---|---|---|---|---|---|---|---|
| (i) i.i.d. letters (B1 breaks) | gillogly_table1 | nondecreasing | 14 | 8.00e-06 ± 2.8e-06 (8/1,000,000) | 8.00e-06 ± 2.8e-06 (8/1,000,000) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (i) i.i.d. letters (B1 breaks) | gillogly_table1 | step01 | 14 | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (i) i.i.d. letters (B1 breaks) | pamphlet_encoder_key | nondecreasing | 17 | 0/1,000,000 (<3.0e-06) | 4.00e-06 ± 2.0e-06 (4/1,000,000) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (i) i.i.d. letters (B1 breaks) | pamphlet_encoder_key | step01 | 17 | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (ii) permutation of B1 numbers | gillogly_table1 | S1_longest_nondecreasing_exact | 14 | 4.00e-06 ± 2.0e-06 (4/1,000,000) | 4.00e-06 ± 2.0e-06 (4/1,000,000) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (ii) permutation of B1 numbers | gillogly_table1 | S2_longest_step01_exact | 14 | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (ii) permutation of B1 numbers | gillogly_table1 | S3_nondecreasing_le2_subs | 20 | 0/200,000 (<1.5e-05) | 8.55e-04 ± 6.5e-05 (171/200,000) | 1.50e-05 ± 8.7e-06 (3/200,000) | 0/200,000 (<1.5e-05) |
| (ii) permutation of B1 numbers | gillogly_table1 | S4_nondecreasing_unlimited_subs | 22 | 5.81e-03 ± 1.7e-04 (1163/200,000) | 5.49e-01 ± 1.1e-03 (109871/200,000) | 1.19e-01 ± 7.2e-04 (23806/200,000) | 1.99e-02 ± 3.1e-04 (3990/200,000) |
| (ii) permutation of B1 numbers | gillogly_table1 | S5_step01_le2_subs | 20 | 0/200,000 (<1.5e-05) | 1.50e-05 ± 8.7e-06 (3/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (ii) permutation of B1 numbers | gillogly_table1 | S6_nondecreasing_le1_sub | 17 | 0/200,000 (<1.5e-05) | 7.50e-05 ± 1.9e-05 (15/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (ii) permutation of B1 numbers | gillogly_table1 | S7_step01_le1_sub | 17 | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (ii) permutation of B1 numbers | pamphlet_encoder_key | S1_longest_nondecreasing_exact | 17 | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (ii) permutation of B1 numbers | pamphlet_encoder_key | S2_longest_step01_exact | 17 | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (ii) permutation of B1 numbers | pamphlet_encoder_key | S3_nondecreasing_le2_subs | 22 | 0/200,000 (<1.5e-05) | 7.95e-04 ± 6.3e-05 (159/200,000) | 1.50e-05 ± 8.7e-06 (3/200,000) | 0/200,000 (<1.5e-05) |
| (ii) permutation of B1 numbers | pamphlet_encoder_key | S4_nondecreasing_unlimited_subs | 22 | 4.42e-03 ± 1.5e-04 (884/200,000) | 4.96e-01 ± 1.1e-03 (99289/200,000) | 9.82e-02 ± 6.7e-04 (19648/200,000) | 1.49e-02 ± 2.7e-04 (2982/200,000) |
| (ii) permutation of B1 numbers | pamphlet_encoder_key | S5_step01_le2_subs | 21 | 0/200,000 (<1.5e-05) | 1.50e-05 ± 8.7e-06 (3/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (ii) permutation of B1 numbers | pamphlet_encoder_key | S6_nondecreasing_le1_sub | 20 | 0/200,000 (<1.5e-05) | 6.00e-05 ± 1.7e-05 (12/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (ii) permutation of B1 numbers | pamphlet_encoder_key | S7_step01_le1_sub | 20 | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (iii) uniform numbers 1..key | gillogly_table1 | S1_longest_nondecreasing_exact | 14 | 7.00e-06 ± 2.6e-06 (7/1,000,000) | 7.00e-06 ± 2.6e-06 (7/1,000,000) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (iii) uniform numbers 1..key | gillogly_table1 | S2_longest_step01_exact | 14 | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (iii) uniform numbers 1..key | gillogly_table1 | S3_nondecreasing_le2_subs | 20 | 0/200,000 (<1.5e-05) | 1.36e-03 ± 8.2e-05 (272/200,000) | 2.00e-05 ± 1.0e-05 (4/200,000) | 0/200,000 (<1.5e-05) |
| (iii) uniform numbers 1..key | gillogly_table1 | S4_nondecreasing_unlimited_subs | 22 | 2.71e-03 ± 1.2e-04 (542/200,000) | 4.67e-01 ± 1.1e-03 (93387/200,000) | 7.76e-02 ± 6.0e-04 (15512/200,000) | 1.01e-02 ± 2.2e-04 (2020/200,000) |
| (iii) uniform numbers 1..key | gillogly_table1 | S5_step01_le2_subs | 20 | 0/200,000 (<1.5e-05) | 1.00e-05 ± 7.1e-06 (2/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (iii) uniform numbers 1..key | gillogly_table1 | S6_nondecreasing_le1_sub | 17 | 0/200,000 (<1.5e-05) | 1.40e-04 ± 2.6e-05 (28/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (iii) uniform numbers 1..key | gillogly_table1 | S7_step01_le1_sub | 17 | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (iii) uniform numbers 1..key | pamphlet_encoder_key | S1_longest_nondecreasing_exact | 17 | 0/1,000,000 (<3.0e-06) | 7.00e-06 ± 2.6e-06 (7/1,000,000) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (iii) uniform numbers 1..key | pamphlet_encoder_key | S2_longest_step01_exact | 17 | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) | 0/1,000,000 (<3.0e-06) |
| (iii) uniform numbers 1..key | pamphlet_encoder_key | S3_nondecreasing_le2_subs | 22 | 0/200,000 (<1.5e-05) | 1.29e-03 ± 8.0e-05 (259/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (iii) uniform numbers 1..key | pamphlet_encoder_key | S4_nondecreasing_unlimited_subs | 22 | 2.17e-03 ± 1.0e-04 (435/200,000) | 4.49e-01 ± 1.1e-03 (89885/200,000) | 7.14e-02 ± 5.8e-04 (14289/200,000) | 9.35e-03 ± 2.2e-04 (1870/200,000) |
| (iii) uniform numbers 1..key | pamphlet_encoder_key | S5_step01_le2_subs | 21 | 0/200,000 (<1.5e-05) | 2.00e-05 ± 1.0e-05 (4/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (iii) uniform numbers 1..key | pamphlet_encoder_key | S6_nondecreasing_le1_sub | 20 | 0/200,000 (<1.5e-05) | 1.25e-04 ± 2.5e-05 (25/200,000) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| (iii) uniform numbers 1..key | pamphlet_encoder_key | S7_step01_le1_sub | 20 | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |

### Number of long runs in B1 (supplement, code/gillogly_multi_runs.py; 200,000 simulations)

| key | relation | L | observed # runs >= L | P(>= observed), permutation | P(>= observed), uniform numbers |
|---|---|---|---|---|---|
| gillogly_table1 | nondecreasing | 8 | 1 | 8.05e-02 ± 6.1e-04 (16091/200,000) | 9.70e-02 ± 6.6e-04 (19406/200,000) |
| gillogly_table1 | nondecreasing | 10 | 1 | 3.31e-03 ± 1.3e-04 (661/200,000) | 4.49e-03 ± 1.5e-04 (897/200,000) |
| gillogly_table1 | nondecreasing | 11 | 1 | 5.40e-04 ± 5.2e-05 (108/200,000) | 8.90e-04 ± 6.7e-05 (178/200,000) |
| gillogly_table1 | step01 | 8 | 1 | 6.85e-04 ± 5.9e-05 (137/200,000) | 1.48e-03 ± 8.6e-05 (297/200,000) |
| gillogly_table1 | step01 | 10 | 1 | 2.50e-05 ± 1.1e-05 (5/200,000) | 5.50e-05 ± 1.7e-05 (11/200,000) |
| pamphlet_encoder_key | nondecreasing | 8 | 3 | 5.00e-05 ± 1.6e-05 (10/200,000) | 1.60e-04 ± 2.8e-05 (32/200,000) |
| pamphlet_encoder_key | nondecreasing | 10 | 3 | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| pamphlet_encoder_key | nondecreasing | 11 | 3 | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| pamphlet_encoder_key | step01 | 8 | 3 | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |
| pamphlet_encoder_key | step01 | 10 | 3 | 0/200,000 (<1.5e-05) | 0/200,000 (<1.5e-05) |