DineshAI commited on
Commit
69f42eb
·
verified ·
1 Parent(s): c128857

Add cpu-upgrade evidence for 1f50506f5c18

Browse files
evidence/raw/claim_1.json ADDED
@@ -0,0 +1,729 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "alpha_grid": [
3
+ {
4
+ "alpha": 0.0,
5
+ "finite": true,
6
+ "log10_gamma_one": 4.313884197138987,
7
+ "m": 5
8
+ },
9
+ {
10
+ "alpha": 0.25,
11
+ "finite": true,
12
+ "log10_gamma_one": 5.6515022642973225,
13
+ "m": 5
14
+ },
15
+ {
16
+ "alpha": 0.5,
17
+ "finite": true,
18
+ "log10_gamma_one": 8.326738398613992,
19
+ "m": 5
20
+ },
21
+ {
22
+ "alpha": 0.75,
23
+ "finite": true,
24
+ "log10_gamma_one": 16.352446801564007,
25
+ "m": 5
26
+ },
27
+ {
28
+ "alpha": 0.9,
29
+ "finite": true,
30
+ "log10_gamma_one": 40.42957201041405,
31
+ "m": 5
32
+ },
33
+ {
34
+ "alpha": 0.99,
35
+ "finite": true,
36
+ "log10_gamma_one": 401.5864501431642,
37
+ "m": 5
38
+ },
39
+ {
40
+ "alpha": 0.0,
41
+ "finite": true,
42
+ "log10_gamma_one": 5.171885542816172,
43
+ "m": 11
44
+ },
45
+ {
46
+ "alpha": 0.25,
47
+ "finite": true,
48
+ "log10_gamma_one": 6.795504058533569,
49
+ "m": 11
50
+ },
51
+ {
52
+ "alpha": 0.5,
53
+ "finite": true,
54
+ "log10_gamma_one": 10.042741089968363,
55
+ "m": 11
56
+ },
57
+ {
58
+ "alpha": 0.75,
59
+ "finite": true,
60
+ "log10_gamma_one": 19.784452184272748,
61
+ "m": 11
62
+ },
63
+ {
64
+ "alpha": 0.9,
65
+ "finite": true,
66
+ "log10_gamma_one": 49.00958546718591,
67
+ "m": 11
68
+ },
69
+ {
70
+ "alpha": 0.99,
71
+ "finite": true,
72
+ "log10_gamma_one": 487.38658471088263,
73
+ "m": 11
74
+ },
75
+ {
76
+ "alpha": 0.0,
77
+ "finite": true,
78
+ "log10_gamma_one": 5.837227729538287,
79
+ "m": 21
80
+ },
81
+ {
82
+ "alpha": 0.25,
83
+ "finite": true,
84
+ "log10_gamma_one": 7.682626974163055,
85
+ "m": 21
86
+ },
87
+ {
88
+ "alpha": 0.5,
89
+ "finite": true,
90
+ "log10_gamma_one": 11.373425463412593,
91
+ "m": 21
92
+ },
93
+ {
94
+ "alpha": 0.75,
95
+ "finite": true,
96
+ "log10_gamma_one": 22.445820931161204,
97
+ "m": 21
98
+ },
99
+ {
100
+ "alpha": 0.9,
101
+ "finite": true,
102
+ "log10_gamma_one": 55.66300733440705,
103
+ "m": 21
104
+ },
105
+ {
106
+ "alpha": 0.99,
107
+ "finite": true,
108
+ "log10_gamma_one": 553.920803383094,
109
+ "m": 21
110
+ }
111
+ ],
112
+ "asymptotic": [
113
+ {
114
+ "log2_core": 12.458439672888293,
115
+ "m": 11,
116
+ "passes": true,
117
+ "quarter_m_log2_m": 9.513436951252567
118
+ },
119
+ {
120
+ "log2_core": 18.792330539983986,
121
+ "m": 13,
122
+ "passes": true,
123
+ "quarter_m_log2_m": 12.026429083958549
124
+ },
125
+ {
126
+ "log2_core": 25.707209517986314,
127
+ "m": 15,
128
+ "passes": true,
129
+ "quarter_m_log2_m": 14.650839733531946
130
+ },
131
+ {
132
+ "log2_core": 33.10523833636097,
133
+ "m": 17,
134
+ "passes": true,
135
+ "quarter_m_log2_m": 17.371717075313942
136
+ },
137
+ {
138
+ "log2_core": 40.916879345776586,
139
+ "m": 19,
140
+ "passes": true,
141
+ "quarter_m_log2_m": 20.17765568885703
142
+ },
143
+ {
144
+ "log2_core": 49.09013928996028,
145
+ "m": 21,
146
+ "passes": true,
147
+ "quarter_m_log2_m": 23.059666469588493
148
+ },
149
+ {
150
+ "log2_core": 57.58466107996799,
151
+ "m": 23,
152
+ "passes": true,
153
+ "quarter_m_log2_m": 26.010481247327824
154
+ },
155
+ {
156
+ "log2_core": 66.36820471046335,
157
+ "m": 25,
158
+ "passes": true,
159
+ "quarter_m_log2_m": 29.02410118609203
160
+ },
161
+ {
162
+ "log2_core": 75.41441903597263,
163
+ "m": 27,
164
+ "passes": true,
165
+ "quarter_m_log2_m": 32.09549063960341
166
+ },
167
+ {
168
+ "log2_core": 84.70136160855533,
169
+ "m": 29,
170
+ "passes": true,
171
+ "quarter_m_log2_m": 35.22036221467489
172
+ },
173
+ {
174
+ "log2_core": 94.21047667272192,
175
+ "m": 31,
176
+ "passes": true,
177
+ "quarter_m_log2_m": 38.395021405498284
178
+ },
179
+ {
180
+ "log2_core": 103.92586664158667,
181
+ "m": 33,
182
+ "passes": true,
183
+ "quarter_m_log2_m": 41.61625148470724
184
+ },
185
+ {
186
+ "log2_core": 113.83375869155317,
187
+ "m": 35,
188
+ "passes": true,
189
+ "quarter_m_log2_m": 44.88122639826845
190
+ },
191
+ {
192
+ "log2_core": 123.92210521228678,
193
+ "m": 37,
194
+ "passes": true,
195
+ "quarter_m_log2_m": 48.18744363206779
196
+ },
197
+ {
198
+ "log2_core": 134.18027857992055,
199
+ "m": 39,
200
+ "passes": true,
201
+ "quarter_m_log2_m": 51.53267163390692
202
+ },
203
+ {
204
+ "log2_core": 144.59883395707564,
205
+ "m": 41,
206
+ "passes": true,
207
+ "quarter_m_log2_m": 54.91490804733536
208
+ },
209
+ {
210
+ "log2_core": 155.16932215993597,
211
+ "m": 43,
212
+ "passes": true,
213
+ "quarter_m_log2_m": 58.33234611304755
214
+ },
215
+ {
216
+ "log2_core": 165.8841400393715,
217
+ "m": 45,
218
+ "passes": true,
219
+ "quarter_m_log2_m": 61.78334733370884
220
+ },
221
+ {
222
+ "log2_core": 176.73640942092672,
223
+ "m": 47,
224
+ "passes": true,
225
+ "quarter_m_log2_m": 65.26641900721224
226
+ },
227
+ {
228
+ "log2_core": 187.71987809773032,
229
+ "m": 49,
230
+ "passes": true,
231
+ "quarter_m_log2_m": 68.7801955904113
232
+ },
233
+ {
234
+ "log2_core": 198.82883807194523,
235
+ "m": 51,
236
+ "passes": true,
237
+ "quarter_m_log2_m": 72.32342311013656
238
+ },
239
+ {
240
+ "log2_core": 210.05805744428807,
241
+ "m": 53,
242
+ "passes": true,
243
+ "quarter_m_log2_m": 75.89494602296239
244
+ },
245
+ {
246
+ "log2_core": 221.4027232171107,
247
+ "m": 55,
248
+ "passes": true,
249
+ "quarter_m_log2_m": 79.49369606096407
250
+ },
251
+ {
252
+ "log2_core": 232.85839290882325,
253
+ "m": 57,
254
+ "passes": true,
255
+ "quarter_m_log2_m": 83.11868270184756
256
+ },
257
+ {
258
+ "log2_core": 244.42095334544382,
259
+ "m": 59,
260
+ "passes": true,
261
+ "quarter_m_log2_m": 86.76898497808716
262
+ },
263
+ {
264
+ "log2_core": 256.0865853458393,
265
+ "m": 61,
266
+ "passes": true,
267
+ "quarter_m_log2_m": 90.44374439783402
268
+ },
269
+ {
270
+ "log2_core": 267.85173328317336,
271
+ "m": 63,
272
+ "passes": true,
273
+ "quarter_m_log2_m": 94.14215879512369
274
+ },
275
+ {
276
+ "log2_core": 279.7130787088705,
277
+ "m": 65,
278
+ "passes": true,
279
+ "quarter_m_log2_m": 97.86347696171238
280
+ },
281
+ {
282
+ "log2_core": 291.6675173831063,
283
+ "m": 67,
284
+ "passes": true,
285
+ "quarter_m_log2_m": 101.60699394016768
286
+ },
287
+ {
288
+ "log2_core": 303.7121391789864,
289
+ "m": 69,
290
+ "passes": true,
291
+ "quarter_m_log2_m": 105.37204687942342
292
+ },
293
+ {
294
+ "log2_core": 315.84421042457353,
295
+ "m": 71,
296
+ "passes": true,
297
+ "quarter_m_log2_m": 109.1580113712081
298
+ },
299
+ {
300
+ "log2_core": 328.06115832392027,
301
+ "m": 73,
302
+ "passes": true,
303
+ "quarter_m_log2_m": 112.96429819956032
304
+ },
305
+ {
306
+ "log2_core": 340.36055715984105,
307
+ "m": 75,
308
+ "passes": true,
309
+ "quarter_m_log2_m": 116.79035044679776
310
+ },
311
+ {
312
+ "log2_core": 352.74011603075905,
313
+ "m": 77,
314
+ "passes": true,
315
+ "quarter_m_log2_m": 120.63564090837684
316
+ },
317
+ {
318
+ "log2_core": 365.1976679141512,
319
+ "m": 79,
320
+ "passes": true,
321
+ "quarter_m_log2_m": 124.49966977649778
322
+ },
323
+ {
324
+ "log2_core": 377.7311598819162,
325
+ "m": 81,
326
+ "passes": true,
327
+ "quarter_m_log2_m": 128.38196255841365
328
+ },
329
+ {
330
+ "log2_core": 390.33864431987195,
331
+ "m": 83,
332
+ "passes": true,
333
+ "quarter_m_log2_m": 132.2820682004487
334
+ },
335
+ {
336
+ "log2_core": 403.01827102578784,
337
+ "m": 85,
338
+ "passes": true,
339
+ "quarter_m_log2_m": 136.19955739292615
340
+ },
341
+ {
342
+ "log2_core": 415.76828007874303,
343
+ "m": 87,
344
+ "passes": true,
345
+ "quarter_m_log2_m": 140.13402103470983
346
+ },
347
+ {
348
+ "log2_core": 428.5869953879295,
349
+ "m": 89,
350
+ "passes": true,
351
+ "quarter_m_log2_m": 144.08506883900236
352
+ },
353
+ {
354
+ "log2_core": 441.47281884185344,
355
+ "m": 91,
356
+ "passes": true,
357
+ "quarter_m_log2_m": 148.05232806452034
358
+ },
359
+ {
360
+ "log2_core": 454.4242249896641,
361
+ "m": 93,
362
+ "passes": true,
363
+ "quarter_m_log2_m": 152.03544235826172
364
+ },
365
+ {
366
+ "log2_core": 467.43975619546063,
367
+ "m": 95,
368
+ "passes": true,
369
+ "quarter_m_log2_m": 156.03407069786002
370
+ },
371
+ {
372
+ "log2_core": 480.51801821413534,
373
+ "m": 97,
374
+ "passes": true,
375
+ "quarter_m_log2_m": 160.04788642303785
376
+ },
377
+ {
378
+ "log2_core": 493.65767614389046,
379
+ "m": 99,
380
+ "passes": true,
381
+ "quarter_m_log2_m": 164.07657634697034
382
+ },
383
+ {
384
+ "log2_core": 506.857450716172,
385
+ "m": 101,
386
+ "passes": true,
387
+ "quarter_m_log2_m": 168.1198399394828
388
+ }
389
+ ],
390
+ "claim": 1,
391
+ "gates": {
392
+ "all_structural_checks_pass": true,
393
+ "alpha_domain_certificate_finite": true,
394
+ "asymptotic_certificate_passes": true,
395
+ "direct_trajectories_have_no_shortcuts": true,
396
+ "direct_trajectories_stay_non_nash": true,
397
+ "direct_trajectories_survive_full_budget": true,
398
+ "negative_controls_activate": true
399
+ },
400
+ "limitations": "Finite right-censored numerical audit of a theorem, not a proof replacement. Both m=5 trajectories remain non-Nash through 2,000,000 updates; the universal regularizer quantifier is audited through the proof certificate plus two canonical FTRL regularizers.",
401
+ "negative_controls": {
402
+ "alpha_one_rejected": true,
403
+ "even_m_rejected": true,
404
+ "non_permutation_initialization_differs": true
405
+ },
406
+ "result_sha256": "55c94bd08adb809b85617bd7a2e2bb973ed1509391ac7200b1f1e47ed58a9eb8",
407
+ "runtime_seconds": 225.03165634302422,
408
+ "source_pdf_sha256": "9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218",
409
+ "status": "VERIFIED",
410
+ "structural": [
411
+ {
412
+ "even_row_shift": true,
413
+ "extra_row_column_negative": true,
414
+ "m": 5,
415
+ "matrix_sha256": "64c34943af681e8fb23b913f016955f05c8f3b48fd8266a41901edd00a66ab42",
416
+ "max_entry": 9,
417
+ "odd_column_shift": true,
418
+ "period_count": 7,
419
+ "period_labels": [
420
+ 3,
421
+ 9
422
+ ],
423
+ "positive_entries_exact": true
424
+ },
425
+ {
426
+ "even_row_shift": true,
427
+ "extra_row_column_negative": true,
428
+ "m": 7,
429
+ "matrix_sha256": "bfde8d870b91a7b586ed1f800a6190d84ad3c7f5fe55f1087853f544d0a85175",
430
+ "max_entry": 13,
431
+ "odd_column_shift": true,
432
+ "period_count": 11,
433
+ "period_labels": [
434
+ 3,
435
+ 13
436
+ ],
437
+ "positive_entries_exact": true
438
+ },
439
+ {
440
+ "even_row_shift": true,
441
+ "extra_row_column_negative": true,
442
+ "m": 9,
443
+ "matrix_sha256": "00d60a5c2f07b36ba4180ee88420d23e9943374e5ba9b4c5567c6e205a2a5d54",
444
+ "max_entry": 17,
445
+ "odd_column_shift": true,
446
+ "period_count": 15,
447
+ "period_labels": [
448
+ 3,
449
+ 17
450
+ ],
451
+ "positive_entries_exact": true
452
+ },
453
+ {
454
+ "even_row_shift": true,
455
+ "extra_row_column_negative": true,
456
+ "m": 11,
457
+ "matrix_sha256": "4fe2312a7b45ee23c3d7c00e1d203f544123531e9313489d47a697a915e95288",
458
+ "max_entry": 21,
459
+ "odd_column_shift": true,
460
+ "period_count": 19,
461
+ "period_labels": [
462
+ 3,
463
+ 21
464
+ ],
465
+ "positive_entries_exact": true
466
+ },
467
+ {
468
+ "even_row_shift": true,
469
+ "extra_row_column_negative": true,
470
+ "m": 13,
471
+ "matrix_sha256": "026fcae2763b627effea9123dc4888a9f99f5d6e2852348b73c81290607e1e6c",
472
+ "max_entry": 25,
473
+ "odd_column_shift": true,
474
+ "period_count": 23,
475
+ "period_labels": [
476
+ 3,
477
+ 25
478
+ ],
479
+ "positive_entries_exact": true
480
+ },
481
+ {
482
+ "even_row_shift": true,
483
+ "extra_row_column_negative": true,
484
+ "m": 15,
485
+ "matrix_sha256": "4a2d7e36087e90e1c0b51cebd46828c55c42b00c4eb9d714ca51b8fdd0f252c2",
486
+ "max_entry": 29,
487
+ "odd_column_shift": true,
488
+ "period_count": 27,
489
+ "period_labels": [
490
+ 3,
491
+ 29
492
+ ],
493
+ "positive_entries_exact": true
494
+ },
495
+ {
496
+ "even_row_shift": true,
497
+ "extra_row_column_negative": true,
498
+ "m": 17,
499
+ "matrix_sha256": "1fbe76ca9b80b7ca8bc572e1dbe002c54777455e200739d0399b68f94cfbdfab",
500
+ "max_entry": 33,
501
+ "odd_column_shift": true,
502
+ "period_count": 31,
503
+ "period_labels": [
504
+ 3,
505
+ 33
506
+ ],
507
+ "positive_entries_exact": true
508
+ },
509
+ {
510
+ "even_row_shift": true,
511
+ "extra_row_column_negative": true,
512
+ "m": 19,
513
+ "matrix_sha256": "4037b114be6f471221d1ffb569fe5e46c2dcb3e81c63c5ed8af13148fdd1198a",
514
+ "max_entry": 37,
515
+ "odd_column_shift": true,
516
+ "period_count": 35,
517
+ "period_labels": [
518
+ 3,
519
+ 37
520
+ ],
521
+ "positive_entries_exact": true
522
+ },
523
+ {
524
+ "even_row_shift": true,
525
+ "extra_row_column_negative": true,
526
+ "m": 21,
527
+ "matrix_sha256": "c98534adf4573aa23a7bb854204b980c132c3a1d8acdda7b2555edfe673bee7f",
528
+ "max_entry": 41,
529
+ "odd_column_shift": true,
530
+ "period_count": 39,
531
+ "period_labels": [
532
+ 3,
533
+ 41
534
+ ],
535
+ "positive_entries_exact": true
536
+ },
537
+ {
538
+ "even_row_shift": true,
539
+ "extra_row_column_negative": true,
540
+ "m": 23,
541
+ "matrix_sha256": "fb1b9015aec30f727588448aecb70af76e23a9a3ff435d2829314b8691683b9f",
542
+ "max_entry": 45,
543
+ "odd_column_shift": true,
544
+ "period_count": 43,
545
+ "period_labels": [
546
+ 3,
547
+ 45
548
+ ],
549
+ "positive_entries_exact": true
550
+ },
551
+ {
552
+ "even_row_shift": true,
553
+ "extra_row_column_negative": true,
554
+ "m": 25,
555
+ "matrix_sha256": "a4ab1b74ecd50ca71ba0ab0c887b8ccb7864dd02b633f416c67d82ebfd0ec800",
556
+ "max_entry": 49,
557
+ "odd_column_shift": true,
558
+ "period_count": 47,
559
+ "period_labels": [
560
+ 3,
561
+ 49
562
+ ],
563
+ "positive_entries_exact": true
564
+ },
565
+ {
566
+ "even_row_shift": true,
567
+ "extra_row_column_negative": true,
568
+ "m": 27,
569
+ "matrix_sha256": "153aac373a33dbbc1f89a01a241086491610951cf31c68d60159a0f38a2a6b35",
570
+ "max_entry": 53,
571
+ "odd_column_shift": true,
572
+ "period_count": 51,
573
+ "period_labels": [
574
+ 3,
575
+ 53
576
+ ],
577
+ "positive_entries_exact": true
578
+ },
579
+ {
580
+ "even_row_shift": true,
581
+ "extra_row_column_negative": true,
582
+ "m": 29,
583
+ "matrix_sha256": "c02476a8e7ed9f1e87cca89457e99d0e590394fa3b3e81ef6597a82013671dab",
584
+ "max_entry": 57,
585
+ "odd_column_shift": true,
586
+ "period_count": 55,
587
+ "period_labels": [
588
+ 3,
589
+ 57
590
+ ],
591
+ "positive_entries_exact": true
592
+ },
593
+ {
594
+ "even_row_shift": true,
595
+ "extra_row_column_negative": true,
596
+ "m": 31,
597
+ "matrix_sha256": "ee79dd31121b747bb69c4419e7c6fb47d254ca8f8b42028b62494836f61f4a21",
598
+ "max_entry": 61,
599
+ "odd_column_shift": true,
600
+ "period_count": 59,
601
+ "period_labels": [
602
+ 3,
603
+ 61
604
+ ],
605
+ "positive_entries_exact": true
606
+ },
607
+ {
608
+ "even_row_shift": true,
609
+ "extra_row_column_negative": true,
610
+ "m": 33,
611
+ "matrix_sha256": "36b3b59495cca1a8baa512f40370f3ef8256307d075c15ee65e3e0f00689c8fe",
612
+ "max_entry": 65,
613
+ "odd_column_shift": true,
614
+ "period_count": 63,
615
+ "period_labels": [
616
+ 3,
617
+ 65
618
+ ],
619
+ "positive_entries_exact": true
620
+ },
621
+ {
622
+ "even_row_shift": true,
623
+ "extra_row_column_negative": true,
624
+ "m": 35,
625
+ "matrix_sha256": "b1f3146d3ba5b3629b83ac2fc49f0e4bc4a922042f1ea060337f883cd4254324",
626
+ "max_entry": 69,
627
+ "odd_column_shift": true,
628
+ "period_count": 67,
629
+ "period_labels": [
630
+ 3,
631
+ 69
632
+ ],
633
+ "positive_entries_exact": true
634
+ },
635
+ {
636
+ "even_row_shift": true,
637
+ "extra_row_column_negative": true,
638
+ "m": 37,
639
+ "matrix_sha256": "64e4d7c875dcaf99c4c3d61868f43ca53196c5f2f765e926d8e1546dfd25411b",
640
+ "max_entry": 73,
641
+ "odd_column_shift": true,
642
+ "period_count": 71,
643
+ "period_labels": [
644
+ 3,
645
+ 73
646
+ ],
647
+ "positive_entries_exact": true
648
+ },
649
+ {
650
+ "even_row_shift": true,
651
+ "extra_row_column_negative": true,
652
+ "m": 39,
653
+ "matrix_sha256": "25dd22866e6751fad15333ad404dfe32ad39d22b8607450560202043a29623a4",
654
+ "max_entry": 77,
655
+ "odd_column_shift": true,
656
+ "period_count": 75,
657
+ "period_labels": [
658
+ 3,
659
+ 77
660
+ ],
661
+ "positive_entries_exact": true
662
+ },
663
+ {
664
+ "even_row_shift": true,
665
+ "extra_row_column_negative": true,
666
+ "m": 41,
667
+ "matrix_sha256": "d3d7f688f7b6ce057f38ad4f2abc71c35c4910e66199963ced4b598f9025d7e7",
668
+ "max_entry": 81,
669
+ "odd_column_shift": true,
670
+ "period_count": 79,
671
+ "period_labels": [
672
+ 3,
673
+ 81
674
+ ],
675
+ "positive_entries_exact": true
676
+ }
677
+ ],
678
+ "trajectories": [
679
+ {
680
+ "alpha": 0.0,
681
+ "epsilon": 0.025,
682
+ "gamma": 103005,
683
+ "gamma_one": 20601,
684
+ "last_period": 6,
685
+ "m": 5,
686
+ "minimum_preterminal_nash_gap": 0.1824255238063568,
687
+ "no_shortcuts": true,
688
+ "period_boundaries": {
689
+ "3": 1,
690
+ "4": 2,
691
+ "5": 103007,
692
+ "6": 515037
693
+ },
694
+ "reached_target_period": false,
695
+ "regularizer": "entropy",
696
+ "regularizer_range": 1.6094379124341003,
697
+ "right_censored": true,
698
+ "rounds": 2000001,
699
+ "shortcut_failures": [],
700
+ "target_period": 9,
701
+ "transition_count": 3
702
+ },
703
+ {
704
+ "alpha": 0.0,
705
+ "epsilon": 0.025,
706
+ "gamma": 25605,
707
+ "gamma_one": 5121,
708
+ "last_period": 7,
709
+ "m": 5,
710
+ "minimum_preterminal_nash_gap": 0.25,
711
+ "no_shortcuts": true,
712
+ "period_boundaries": {
713
+ "3": 1,
714
+ "4": 2,
715
+ "5": 25609,
716
+ "6": 128039,
717
+ "7": 742622
718
+ },
719
+ "reached_target_period": false,
720
+ "regularizer": "euclidean",
721
+ "regularizer_range": 0.4,
722
+ "right_censored": true,
723
+ "rounds": 2000001,
724
+ "shortcut_failures": [],
725
+ "target_period": 9,
726
+ "transition_count": 4
727
+ }
728
+ ]
729
+ }
evidence/raw/claim_2.json ADDED
@@ -0,0 +1,1080 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "claim": 2,
3
+ "gates": {
4
+ "all_factorial_recurrence_checks_pass": true,
5
+ "all_generated_paths_are_induced": true,
6
+ "all_recurrence_sequences_dominate_factorials": true,
7
+ "generated_lengths_show_exponential_trend": true,
8
+ "negative_control_activates": true,
9
+ "paper_figure_path_is_induced": true
10
+ },
11
+ "limitations": "The exponential snake-existence lemma is cited by the paper; this audit independently checks finite induced paths and the full game/recurrence mechanism, not the asymptotic combinatorics proof.",
12
+ "log2_length_vs_n_slope": 0.7649105649857733,
13
+ "negative_controls": {
14
+ "gray_code_has_shortcut": true
15
+ },
16
+ "result_sha256": "e8f2d32160fb43829919a63d9cdaf5602e45fec015aec9f6aa4fb8391178f8e8",
17
+ "runtime_seconds": 11.333668414037675,
18
+ "search": {
19
+ "restarts_per_dimension": 4096,
20
+ "strategy": "uniform random feasible continuation"
21
+ },
22
+ "seed": 17353,
23
+ "snakes": [
24
+ {
25
+ "dwell_times": [
26
+ 1,
27
+ 1,
28
+ 3,
29
+ 11,
30
+ 47,
31
+ 246,
32
+ 1522,
33
+ 10891
34
+ ],
35
+ "factorial_lower_bound_passes": true,
36
+ "factorial_recurrence_passes": true,
37
+ "fraction_of_cube": 0.5,
38
+ "induced": true,
39
+ "length": 8,
40
+ "log10_factorial_iteration_lower_bound": 3.702430536445525,
41
+ "n": 4,
42
+ "path": [
43
+ 0,
44
+ 8,
45
+ 12,
46
+ 14,
47
+ 6,
48
+ 7,
49
+ 3,
50
+ 11
51
+ ],
52
+ "terminal_dwell_time": 10891
53
+ },
54
+ {
55
+ "dwell_times": [
56
+ 1,
57
+ 1,
58
+ 3,
59
+ 11,
60
+ 47,
61
+ 246,
62
+ 1522,
63
+ 10891,
64
+ 88605,
65
+ 808091,
66
+ 8167985,
67
+ 90644982,
68
+ 1095818866,
69
+ 14335480321
70
+ ],
71
+ "factorial_lower_bound_passes": true,
72
+ "factorial_recurrence_passes": true,
73
+ "fraction_of_cube": 0.4375,
74
+ "induced": true,
75
+ "length": 14,
76
+ "log10_factorial_iteration_lower_bound": 9.794280316389479,
77
+ "n": 5,
78
+ "path": [
79
+ 0,
80
+ 16,
81
+ 18,
82
+ 26,
83
+ 27,
84
+ 11,
85
+ 3,
86
+ 7,
87
+ 23,
88
+ 21,
89
+ 29,
90
+ 13,
91
+ 12,
92
+ 14
93
+ ],
94
+ "terminal_dwell_time": 14335480321
95
+ },
96
+ {
97
+ "dwell_times": [
98
+ 1,
99
+ 1,
100
+ 3,
101
+ 11,
102
+ 47,
103
+ 246,
104
+ 1522,
105
+ 10891,
106
+ 88605,
107
+ 808091,
108
+ 8167985,
109
+ 90644982,
110
+ 1095818866,
111
+ 14335480321,
112
+ 201784362603,
113
+ 3041010172709,
114
+ 48856850395487,
115
+ 833593122323316,
116
+ 15053331168013564,
117
+ 286843843107838855,
118
+ 5751881320933253037,
119
+ 121075517772158485565,
120
+ 2669398215717619835975,
121
+ 61516947583302283250580,
122
+ 1479070387447708913306236,
123
+ 37038155542315719586030801
124
+ ],
125
+ "factorial_lower_bound_passes": true,
126
+ "factorial_recurrence_passes": true,
127
+ "fraction_of_cube": 0.40625,
128
+ "induced": true,
129
+ "length": 26,
130
+ "log10_factorial_iteration_lower_bound": 25.19064567883507,
131
+ "n": 6,
132
+ "path": [
133
+ 0,
134
+ 2,
135
+ 18,
136
+ 26,
137
+ 58,
138
+ 42,
139
+ 46,
140
+ 14,
141
+ 12,
142
+ 28,
143
+ 20,
144
+ 52,
145
+ 48,
146
+ 49,
147
+ 51,
148
+ 35,
149
+ 39,
150
+ 7,
151
+ 23,
152
+ 31,
153
+ 63,
154
+ 61,
155
+ 45,
156
+ 41,
157
+ 9,
158
+ 25
159
+ ],
160
+ "terminal_dwell_time": 37038155542315719586030801
161
+ },
162
+ {
163
+ "dwell_times": [
164
+ 1,
165
+ 1,
166
+ 3,
167
+ 11,
168
+ 47,
169
+ 246,
170
+ 1522,
171
+ 10891,
172
+ 88605,
173
+ 808091,
174
+ 8167985,
175
+ 90644982,
176
+ 1095818866,
177
+ 14335480321,
178
+ 201784362603,
179
+ 3041010172709,
180
+ 48856850395487,
181
+ 833593122323316,
182
+ 15053331168013564,
183
+ 286843843107838855,
184
+ 5751881320933253037,
185
+ 121075517772158485565,
186
+ 2669398215717619835975,
187
+ 61516947583302283250580,
188
+ 1479070387447708913306236,
189
+ 37038155542315719586030801,
190
+ 964468444786602192027418563,
191
+ 26077624641777385830035685239,
192
+ 731136479217018891646576027157,
193
+ 21228998480982934807557494481630,
194
+ 637600126375940425206801432450370,
195
+ 19786806836966976953009407463572639,
196
+ 633814687734255516126357006043571277,
197
+ 20935650272065800791773812291781741079,
198
+ 712445286310762821102433388373128245061,
199
+ 24956500883583670223319616865357521469502,
200
+ 899145843258646897657794792675336452616082,
201
+ 33293331765143631350152303447945059587040781,
202
+ 1266045040452983146878699534331563778983589311,
203
+ 49409024952276318919394083319864565726328691969,
204
+ 1977626143964071887531655837607726108392900565795,
205
+ 81132047633413402236850519990215308631851110186140,
206
+ 3409522360676595301357430978093575964282722371799780,
207
+ 146690544146777200438940396559102400327339109365187851,
208
+ 6457791487158510526076029414534498299733833079994099145
209
+ ],
210
+ "factorial_lower_bound_passes": true,
211
+ "factorial_recurrence_passes": true,
212
+ "fraction_of_cube": 0.3515625,
213
+ "induced": true,
214
+ "length": 45,
215
+ "log10_factorial_iteration_lower_bound": 54.424599347339274,
216
+ "n": 7,
217
+ "path": [
218
+ 0,
219
+ 2,
220
+ 66,
221
+ 98,
222
+ 99,
223
+ 107,
224
+ 75,
225
+ 73,
226
+ 65,
227
+ 81,
228
+ 83,
229
+ 87,
230
+ 23,
231
+ 7,
232
+ 15,
233
+ 13,
234
+ 29,
235
+ 28,
236
+ 20,
237
+ 84,
238
+ 68,
239
+ 76,
240
+ 78,
241
+ 94,
242
+ 90,
243
+ 26,
244
+ 27,
245
+ 59,
246
+ 63,
247
+ 127,
248
+ 125,
249
+ 117,
250
+ 53,
251
+ 37,
252
+ 33,
253
+ 41,
254
+ 40,
255
+ 44,
256
+ 46,
257
+ 38,
258
+ 54,
259
+ 50,
260
+ 48,
261
+ 112,
262
+ 120
263
+ ],
264
+ "terminal_dwell_time": 6457791487158510526076029414534498299733833079994099145
265
+ },
266
+ {
267
+ "dwell_times": [
268
+ 1,
269
+ 1,
270
+ 3,
271
+ 11,
272
+ 47,
273
+ 246,
274
+ 1522,
275
+ 10891,
276
+ 88605,
277
+ 808091,
278
+ 8167985,
279
+ 90644982,
280
+ 1095818866,
281
+ 14335480321,
282
+ 201784362603,
283
+ 3041010172709,
284
+ 48856850395487,
285
+ 833593122323316,
286
+ 15053331168013564,
287
+ 286843843107838855,
288
+ 5751881320933253037,
289
+ 121075517772158485565,
290
+ 2669398215717619835975,
291
+ 61516947583302283250580,
292
+ 1479070387447708913306236,
293
+ 37038155542315719586030801,
294
+ 964468444786602192027418563,
295
+ 26077624641777385830035685239,
296
+ 731136479217018891646576027157,
297
+ 21228998480982934807557494481630,
298
+ 637600126375940425206801432450370,
299
+ 19786806836966976953009407463572639,
300
+ 633814687734255516126357006043571277,
301
+ 20935650272065800791773812291781741079,
302
+ 712445286310762821102433388373128245061,
303
+ 24956500883583670223319616865357521469502,
304
+ 899145843258646897657794792675336452616082,
305
+ 33293331765143631350152303447945059587040781,
306
+ 1266045040452983146878699534331563778983589311,
307
+ 49409024952276318919394083319864565726328691969,
308
+ 1977626143964071887531655837607726108392900565795,
309
+ 81132047633413402236850519990215308631851110186140,
310
+ 3409522360676595301357430978093575964282722371799780,
311
+ 146690544146777200438940396559102400327339109365187851,
312
+ 6457791487158510526076029414534498299733833079994099145,
313
+ 290747226332931827804326731444185939315409570379287479057,
314
+ 13380826793228951673462751356674185105293532712522411962979,
315
+ 629189459815521143100997609887527358176881409274086404212132,
316
+ 30214468440063594943485396824728759448901898793332523127536764,
317
+ 1481137852272212605652616150215346309871633689180936268867352317,
318
+ 74087093701073715341315798401330560989020972916577587526338980503,
319
+ 3779922287410963819535431376625752713404829873703114243961320328155,
320
+ 196630015824305393488564898771257129323032489565496508927476651528865,
321
+ 10425169279824066210743309250016310612398304659659804427891999610854638,
322
+ 563155697038587305296278885593274843767937637733291763035279823090517954,
323
+ 30983984726448980806339786825109565623653587110119062830242464163716317395,
324
+ 1735666103746653678984120406704674751099802389084752289308433263145759563289,
325
+ 98963941473040829113088704431481358557403909369102803034982259673471060208947,
326
+ 5741643708380562150760026227064345145433990714480606186523951082328536947778265,
327
+ 338855911751740994194249556103417240043026871780332234631861752317867034604694406,
328
+ 20337094613136110735191760322155130753053903928621824238092397839485571513860548634,
329
+ 1240901528348539237515510743929955032126648571308860723853069758708030544025124717481,
330
+ 76956226110547302485636625646684235318086899717298047312086659491171850270534357587363,
331
+ 4849482807635149609391652036785141494946193394418055923552196408586917164239555391466125,
332
+ 310443835577564777417482329711799800963674493512657581544440631941030912853364110920866311,
333
+ 20183697554441974926190196205484455362253882409127070400523690590972265861050665748257609348,
334
+ 1332434405472177099272665195921861934218274596265121724541215095604708399269361987835438791180,
335
+ 89293284014686818095614260111947730894098998686864380519133555719940415719732646328072647894495,
336
+ 6073275436959078640770078490959789565247561183523572386600113215284269822269526383408552967325749,
337
+ 419145278250415340429179284196539004138915378924508742053509141749393284563832069140012707044044949,
338
+ 29346241420526699714856613299335680916814630900487971735233072562941091735946598798253947788511240207,
339
+ 2084002196842046707589614189558660110331636570858433468092833273535705792464566039151026726974065473268
340
+ ],
341
+ "factorial_lower_bound_passes": true,
342
+ "factorial_recurrence_passes": true,
343
+ "fraction_of_cube": 0.28125,
344
+ "induced": true,
345
+ "length": 72,
346
+ "log10_factorial_iteration_lower_bound": 101.92966338439903,
347
+ "n": 8,
348
+ "path": [
349
+ 0,
350
+ 128,
351
+ 129,
352
+ 131,
353
+ 163,
354
+ 171,
355
+ 175,
356
+ 47,
357
+ 111,
358
+ 103,
359
+ 119,
360
+ 55,
361
+ 23,
362
+ 151,
363
+ 159,
364
+ 155,
365
+ 153,
366
+ 152,
367
+ 24,
368
+ 88,
369
+ 92,
370
+ 93,
371
+ 95,
372
+ 91,
373
+ 75,
374
+ 203,
375
+ 201,
376
+ 200,
377
+ 232,
378
+ 236,
379
+ 108,
380
+ 44,
381
+ 40,
382
+ 42,
383
+ 58,
384
+ 62,
385
+ 30,
386
+ 14,
387
+ 142,
388
+ 140,
389
+ 141,
390
+ 13,
391
+ 5,
392
+ 69,
393
+ 197,
394
+ 199,
395
+ 198,
396
+ 70,
397
+ 66,
398
+ 98,
399
+ 114,
400
+ 242,
401
+ 210,
402
+ 218,
403
+ 222,
404
+ 254,
405
+ 255,
406
+ 251,
407
+ 249,
408
+ 241,
409
+ 209,
410
+ 81,
411
+ 17,
412
+ 49,
413
+ 57,
414
+ 61,
415
+ 189,
416
+ 181,
417
+ 165,
418
+ 164,
419
+ 166,
420
+ 182
421
+ ],
422
+ "terminal_dwell_time": 2084002196842046707589614189558660110331636570858433468092833273535705792464566039151026726974065473268
423
+ },
424
+ {
425
+ "dwell_times": [
426
+ 1,
427
+ 1,
428
+ 3,
429
+ 11,
430
+ 47,
431
+ 246,
432
+ 1522,
433
+ 10891,
434
+ 88605,
435
+ 808091,
436
+ 8167985,
437
+ 90644982,
438
+ 1095818866,
439
+ 14335480321,
440
+ 201784362603,
441
+ 3041010172709,
442
+ 48856850395487,
443
+ 833593122323316,
444
+ 15053331168013564,
445
+ 286843843107838855,
446
+ 5751881320933253037,
447
+ 121075517772158485565,
448
+ 2669398215717619835975,
449
+ 61516947583302283250580,
450
+ 1479070387447708913306236,
451
+ 37038155542315719586030801,
452
+ 964468444786602192027418563,
453
+ 26077624641777385830035685239,
454
+ 731136479217018891646576027157,
455
+ 21228998480982934807557494481630,
456
+ 637600126375940425206801432450370,
457
+ 19786806836966976953009407463572639,
458
+ 633814687734255516126357006043571277,
459
+ 20935650272065800791773812291781741079,
460
+ 712445286310762821102433388373128245061,
461
+ 24956500883583670223319616865357521469502,
462
+ 899145843258646897657794792675336452616082,
463
+ 33293331765143631350152303447945059587040781,
464
+ 1266045040452983146878699534331563778983589311,
465
+ 49409024952276318919394083319864565726328691969,
466
+ 1977626143964071887531655837607726108392900565795,
467
+ 81132047633413402236850519990215308631851110186140,
468
+ 3409522360676595301357430978093575964282722371799780,
469
+ 146690544146777200438940396559102400327339109365187851,
470
+ 6457791487158510526076029414534498299733833079994099145,
471
+ 290747226332931827804326731444185939315409570379287479057,
472
+ 13380826793228951673462751356674185105293532712522411962979,
473
+ 629189459815521143100997609887527358176881409274086404212132,
474
+ 30214468440063594943485396824728759448901898793332523127536764,
475
+ 1481137852272212605652616150215346309871633689180936268867352317,
476
+ 74087093701073715341315798401330560989020972916577587526338980503,
477
+ 3779922287410963819535431376625752713404829873703114243961320328155,
478
+ 196630015824305393488564898771257129323032489565496508927476651528865,
479
+ 10425169279824066210743309250016310612398304659659804427891999610854638,
480
+ 563155697038587305296278885593274843767937637733291763035279823090517954,
481
+ 30983984726448980806339786825109565623653587110119062830242464163716317395,
482
+ 1735666103746653678984120406704674751099802389084752289308433263145759563289,
483
+ 98963941473040829113088704431481358557403909369102803034982259673471060208947,
484
+ 5741643708380562150760026227064345145433990714480606186523951082328536947778265,
485
+ 338855911751740994194249556103417240043026871780332234631861752317867034604694406,
486
+ 20337094613136110735191760322155130753053903928621824238092397839485571513860548634,
487
+ 1240901528348539237515510743929955032126648571308860723853069758708030544025124717481,
488
+ 76956226110547302485636625646684235318086899717298047312086659491171850270534357587363,
489
+ 4849482807635149609391652036785141494946193394418055923552196408586917164239555391466125,
490
+ 310443835577564777417482329711799800963674493512657581544440631941030912853364110920866311,
491
+ 20183697554441974926190196205484455362253882409127070400523690590972265861050665748257609348,
492
+ 1332434405472177099272665195921861934218274596265121724541215095604708399269361987835438791180,
493
+ 89293284014686818095614260111947730894098998686864380519133555719940415719732646328072647894495,
494
+ 6073275436959078640770078490959789565247561183523572386600113215284269822269526383408552967325749,
495
+ 419145278250415340429179284196539004138915378924508742053509141749393284563832069140012707044044949,
496
+ 29346241420526699714856613299335680916814630900487971735233072562941091735946598798253947788511240207,
497
+ 2084002196842046707589614189558660110331636570858433468092833273535705792464566039151026726974065473268,
498
+ 150077498340751953538497019001334323204762878180348638499243451880775576529509500075300336125203279931644,
499
+ 10957740961925103466120746326910897095049612925822946832588738477607293344995514784715593034747683939148249,
500
+ 811022879334466341381904719474063658688952031554311075338244920503906627605933936693134379120820886471745403,
501
+ 60837671607038539922588290156328504790233678445072893941284465272602603177343456696073116496694737337515279887,
502
+ 4624473914936339849132475544016808059365093637482608468104624757857672176819943870579332453965680764347052449821,
503
+ 356145318163934473355379011615611872453927649119293556745551952062067414006653987989373594002593388242461008975838,
504
+ 27783958479676832147098928751529380482510476803621114611680927024692184324183651266017295981490672750676671870254178,
505
+ 2195288804374809875351976076929606287127016867314263294452233182762769564655735088760036314151083883664689263252643415,
506
+ 175650883683979441961528113455586384268623918584469156267459263335127231461398425435191687171750112779187430847493204693,
507
+ 14229916511060569309869104118338726980006203131124363589478950273437961280901161288008624318751647416206187723277774863839,
508
+ 1167028777006630523215641624831367306248586345558420355736187197154378632423138578177799202098968207268958532430506236988957,
509
+ 96877616212767903476414595542187185541391493284919983975765078957931526475455909388064244843552999732085238678746700678984814,
510
+ 8138886614998266006277317162544435293822382821721084242298850429597927222122075099903995591411541500721942794664838551837829890,
511
+ 691902225661120722524054145413178076754450088210505002845534708639040729054249303316147383823559213697588460417259107246383932341,
512
+ 59511729126440379958101465895396274278102352636036337263898534375785756573256311536472319827838105792300142269032971729557117330519,
513
+ 5178212239348180089626588901194130914038107756196375535782562634803116692297193810675393631763939074553140273294146794615525977127497,
514
+ 455742180652865265449493146289997592940039519249855629666707962840022865127409103859514821978175639144522610000961887975924441820552059,
515
+ 40566231598440949705154010338985243028589563720540247684455386144854535112693535766138186227249952173869320006113922845159318568437479404,
516
+ 3651416526528511153056023231045138532550790347868143272677090615380815613977321307023522432781521341824080408463762657866104437821276958900,
517
+ 332319464967472479569367453569570852087476891441150797564953122600068485383498221112728615131641490713763392868637702211397708145154580204611,
518
+ 30577041737791115838302240636792860676422415169162158791993032242705440133349514179032296831958343679808305292986667579882119624895697183042513,
519
+ 2843997160513249435033175012568757108386631027129799634706395011217565958602260739596022879652158250501380406681580165240703143800996740596750857,
520
+ 267366306478561473048001738901309725961754090709056042819063080807003250539910129043590956094351050279125326732397763614353119973996006658371201419,
521
+ 25402642780303927240209481707183657311159191887195404717578187480788676611312221474067738173488842866840766608493261568304208465143266442952178477700,
522
+ 2438921042638572811468918761729471786413133144824277088337165808695612511435161396000140223919646555273010678330303441599006518922096480144939213407292,
523
+ 236600740934721013677437809735679593592001628233548013877907024333969036173805088553207045195470355389371376017424242677598041183378183647433229267201701,
524
+ 23189311265278655422263914354539315948378808084825414704052316861372398307796876441379095828123494056540730404573924584503394473438401415385336329330532143,
525
+ 2295978390600847914418737667589845077444697468102274255104817099692659197769197253262765514857785952763545933429341283064240311840831338827887643332822312403,
526
+ 229621025932426152544775116865790850370738252598968616345434742974294831770498708042467357595738334686214730941668748610469662134888147517453834812297747488073,
527
+ 23194019360964631078659252824876108242868511107591478594725592977713090517077136485554889921296914977105127960874481846244520664680271589796149178274723381887150,
528
+ 2366019572654987857981883302884224610046787980941216951080722197175332702842064407520262788224425596823801052802864297909680211909131815157116443347433800532354850,
529
+ 243723207706843858805531922672142445937983408481575545244628058028958509994770932908143706553972273020606462449310287952666455963145505421805477424475391578740749547,
530
+ 25349579391463151305599140216618845602596747701207663026876845194584976235688216794919802052097722490977512855889132026430832226610008865552502472057327013743234724289,
531
+ 2661949536117294941579537973023325999704493949797765917766453643912948050005166315226902203593637546340251255263789065070913407870828490622145548859990335827664249194843,
532
+ 282191998041802834894491022415104022066351346001485338359652435603340097587440125607908494263771078979083153300903468733951945483869169024843337130814895791015522887617745,
533
+ 30197205496285580981134004352811904578240150015173760556044734726603777953549441651607337294020756420286587011016319593854129174453557132889685837922580190251655957918555510,
534
+ 3261580360247281731350752032571633611091509233722526126730472078080267120585139503759781203157285193080871126334520343796306372125931574907402614032432823706919397204154294986,
535
+ 355542453810498068733042636106565709934312351277001653199901940962542880337894204604058217978305365782707700679734178395621346524451300447946594589336166508438272944885023160689,
536
+ 39112931217322790687675682783280636541476404828559782523085681619946545707752522201110840171533528313751290952825809107209345109214381481761817834020401609467555572307717044261531,
537
+ 4341890877379409172399447880785341173764918266333150808446906420993010745499655081333431728252354838007379636104415393783488195360765518770769933624291004549705714916972428310225781,
538
+ 486330887936128102304502836839859099736176139498488755799087060242087505384224982383210179297715061278950017979164203698503320520051987805629962723515095769464706911530343067214296383,
539
+ 54959731872117116279291355317244156670842371266593746674174813065752304578791468771401238467641310006885881029462353662425609435631417423163910565776425199899476816910378287882423889492,
540
+ 6265895725196325684507844524080542070545128322677109328184963166218059791944866442401736359283307352173462776229917563874092705752714949018893201915657796977279526404378006218641788805212,
541
+ 720632963787555401390113560504883761735877062041655493049232618904439353237986829886408887052234892305246882652442311646601162466922508150870468431281068856892270065851190356947334546378743,
542
+ 83599689208750376116005729590002818583118101706008599395628969514945677914587130520628050154192745422049124713543490235651467810795584663537081139625749586680068012717902264594803315572731101
543
+ ],
544
+ "factorial_lower_bound_passes": true,
545
+ "factorial_recurrence_passes": true,
546
+ "fraction_of_cube": 0.228515625,
547
+ "induced": true,
548
+ "length": 117,
549
+ "log10_factorial_iteration_lower_bound": 190.53059777072724,
550
+ "n": 9,
551
+ "path": [
552
+ 0,
553
+ 16,
554
+ 20,
555
+ 84,
556
+ 86,
557
+ 214,
558
+ 198,
559
+ 194,
560
+ 66,
561
+ 67,
562
+ 3,
563
+ 7,
564
+ 5,
565
+ 69,
566
+ 325,
567
+ 333,
568
+ 329,
569
+ 457,
570
+ 473,
571
+ 505,
572
+ 509,
573
+ 511,
574
+ 447,
575
+ 443,
576
+ 411,
577
+ 395,
578
+ 399,
579
+ 391,
580
+ 389,
581
+ 388,
582
+ 404,
583
+ 400,
584
+ 432,
585
+ 433,
586
+ 417,
587
+ 425,
588
+ 424,
589
+ 426,
590
+ 430,
591
+ 422,
592
+ 438,
593
+ 310,
594
+ 318,
595
+ 286,
596
+ 414,
597
+ 478,
598
+ 476,
599
+ 460,
600
+ 204,
601
+ 205,
602
+ 141,
603
+ 157,
604
+ 153,
605
+ 185,
606
+ 57,
607
+ 61,
608
+ 63,
609
+ 55,
610
+ 51,
611
+ 115,
612
+ 114,
613
+ 242,
614
+ 240,
615
+ 241,
616
+ 245,
617
+ 247,
618
+ 231,
619
+ 487,
620
+ 483,
621
+ 499,
622
+ 467,
623
+ 339,
624
+ 343,
625
+ 279,
626
+ 277,
627
+ 273,
628
+ 281,
629
+ 280,
630
+ 344,
631
+ 346,
632
+ 378,
633
+ 379,
634
+ 363,
635
+ 367,
636
+ 366,
637
+ 110,
638
+ 78,
639
+ 79,
640
+ 95,
641
+ 223,
642
+ 219,
643
+ 203,
644
+ 235,
645
+ 234,
646
+ 232,
647
+ 104,
648
+ 360,
649
+ 352,
650
+ 353,
651
+ 369,
652
+ 373,
653
+ 372,
654
+ 380,
655
+ 124,
656
+ 252,
657
+ 254,
658
+ 190,
659
+ 186,
660
+ 154,
661
+ 138,
662
+ 10,
663
+ 266,
664
+ 258,
665
+ 290,
666
+ 34,
667
+ 162,
668
+ 163
669
+ ],
670
+ "terminal_dwell_time": 83599689208750376116005729590002818583118101706008599395628969514945677914587130520628050154192745422049124713543490235651467810795584663537081139625749586680068012717902264594803315572731101
671
+ },
672
+ {
673
+ "dwell_times": [
674
+ 1,
675
+ 1,
676
+ 3,
677
+ 11,
678
+ 47,
679
+ 246,
680
+ 1522,
681
+ 10891,
682
+ 88605,
683
+ 808091,
684
+ 8167985,
685
+ 90644982,
686
+ 1095818866,
687
+ 14335480321,
688
+ 201784362603,
689
+ 3041010172709,
690
+ 48856850395487,
691
+ 833593122323316,
692
+ 15053331168013564,
693
+ 286843843107838855,
694
+ 5751881320933253037,
695
+ 121075517772158485565,
696
+ 2669398215717619835975,
697
+ 61516947583302283250580,
698
+ 1479070387447708913306236,
699
+ 37038155542315719586030801,
700
+ 964468444786602192027418563,
701
+ 26077624641777385830035685239,
702
+ 731136479217018891646576027157,
703
+ 21228998480982934807557494481630,
704
+ 637600126375940425206801432450370,
705
+ 19786806836966976953009407463572639,
706
+ 633814687734255516126357006043571277,
707
+ 20935650272065800791773812291781741079,
708
+ 712445286310762821102433388373128245061,
709
+ 24956500883583670223319616865357521469502,
710
+ 899145843258646897657794792675336452616082,
711
+ 33293331765143631350152303447945059587040781,
712
+ 1266045040452983146878699534331563778983589311,
713
+ 49409024952276318919394083319864565726328691969,
714
+ 1977626143964071887531655837607726108392900565795,
715
+ 81132047633413402236850519990215308631851110186140,
716
+ 3409522360676595301357430978093575964282722371799780,
717
+ 146690544146777200438940396559102400327339109365187851,
718
+ 6457791487158510526076029414534498299733833079994099145,
719
+ 290747226332931827804326731444185939315409570379287479057,
720
+ 13380826793228951673462751356674185105293532712522411962979,
721
+ 629189459815521143100997609887527358176881409274086404212132,
722
+ 30214468440063594943485396824728759448901898793332523127536764,
723
+ 1481137852272212605652616150215346309871633689180936268867352317,
724
+ 74087093701073715341315798401330560989020972916577587526338980503,
725
+ 3779922287410963819535431376625752713404829873703114243961320328155,
726
+ 196630015824305393488564898771257129323032489565496508927476651528865,
727
+ 10425169279824066210743309250016310612398304659659804427891999610854638,
728
+ 563155697038587305296278885593274843767937637733291763035279823090517954,
729
+ 30983984726448980806339786825109565623653587110119062830242464163716317395,
730
+ 1735666103746653678984120406704674751099802389084752289308433263145759563289,
731
+ 98963941473040829113088704431481358557403909369102803034982259673471060208947,
732
+ 5741643708380562150760026227064345145433990714480606186523951082328536947778265,
733
+ 338855911751740994194249556103417240043026871780332234631861752317867034604694406,
734
+ 20337094613136110735191760322155130753053903928621824238092397839485571513860548634,
735
+ 1240901528348539237515510743929955032126648571308860723853069758708030544025124717481,
736
+ 76956226110547302485636625646684235318086899717298047312086659491171850270534357587363,
737
+ 4849482807635149609391652036785141494946193394418055923552196408586917164239555391466125,
738
+ 310443835577564777417482329711799800963674493512657581544440631941030912853364110920866311,
739
+ 20183697554441974926190196205484455362253882409127070400523690590972265861050665748257609348,
740
+ 1332434405472177099272665195921861934218274596265121724541215095604708399269361987835438791180,
741
+ 89293284014686818095614260111947730894098998686864380519133555719940415719732646328072647894495,
742
+ 6073275436959078640770078490959789565247561183523572386600113215284269822269526383408552967325749,
743
+ 419145278250415340429179284196539004138915378924508742053509141749393284563832069140012707044044949,
744
+ 29346241420526699714856613299335680916814630900487971735233072562941091735946598798253947788511240207,
745
+ 2084002196842046707589614189558660110331636570858433468092833273535705792464566039151026726974065473268,
746
+ 150077498340751953538497019001334323204762878180348638499243451880775576529509500075300336125203279931644,
747
+ 10957740961925103466120746326910897095049612925822946832588738477607293344995514784715593034747683939148249,
748
+ 811022879334466341381904719474063658688952031554311075338244920503906627605933936693134379120820886471745403,
749
+ 60837671607038539922588290156328504790233678445072893941284465272602603177343456696073116496694737337515279887,
750
+ 4624473914936339849132475544016808059365093637482608468104624757857672176819943870579332453965680764347052449821,
751
+ 356145318163934473355379011615611872453927649119293556745551952062067414006653987989373594002593388242461008975838,
752
+ 27783958479676832147098928751529380482510476803621114611680927024692184324183651266017295981490672750676671870254178,
753
+ 2195288804374809875351976076929606287127016867314263294452233182762769564655735088760036314151083883664689263252643415,
754
+ 175650883683979441961528113455586384268623918584469156267459263335127231461398425435191687171750112779187430847493204693,
755
+ 14229916511060569309869104118338726980006203131124363589478950273437961280901161288008624318751647416206187723277774863839,
756
+ 1167028777006630523215641624831367306248586345558420355736187197154378632423138578177799202098968207268958532430506236988957,
757
+ 96877616212767903476414595542187185541391493284919983975765078957931526475455909388064244843552999732085238678746700678984814,
758
+ 8138886614998266006277317162544435293822382821721084242298850429597927222122075099903995591411541500721942794664838551837829890,
759
+ 691902225661120722524054145413178076754450088210505002845534708639040729054249303316147383823559213697588460417259107246383932341,
760
+ 59511729126440379958101465895396274278102352636036337263898534375785756573256311536472319827838105792300142269032971729557117330519,
761
+ 5178212239348180089626588901194130914038107756196375535782562634803116692297193810675393631763939074553140273294146794615525977127497,
762
+ 455742180652865265449493146289997592940039519249855629666707962840022865127409103859514821978175639144522610000961887975924441820552059,
763
+ 40566231598440949705154010338985243028589563720540247684455386144854535112693535766138186227249952173869320006113922845159318568437479404,
764
+ 3651416526528511153056023231045138532550790347868143272677090615380815613977321307023522432781521341824080408463762657866104437821276958900,
765
+ 332319464967472479569367453569570852087476891441150797564953122600068485383498221112728615131641490713763392868637702211397708145154580204611,
766
+ 30577041737791115838302240636792860676422415169162158791993032242705440133349514179032296831958343679808305292986667579882119624895697183042513,
767
+ 2843997160513249435033175012568757108386631027129799634706395011217565958602260739596022879652158250501380406681580165240703143800996740596750857,
768
+ 267366306478561473048001738901309725961754090709056042819063080807003250539910129043590956094351050279125326732397763614353119973996006658371201419,
769
+ 25402642780303927240209481707183657311159191887195404717578187480788676611312221474067738173488842866840766608493261568304208465143266442952178477700,
770
+ 2438921042638572811468918761729471786413133144824277088337165808695612511435161396000140223919646555273010678330303441599006518922096480144939213407292,
771
+ 236600740934721013677437809735679593592001628233548013877907024333969036173805088553207045195470355389371376017424242677598041183378183647433229267201701,
772
+ 23189311265278655422263914354539315948378808084825414704052316861372398307796876441379095828123494056540730404573924584503394473438401415385336329330532143,
773
+ 2295978390600847914418737667589845077444697468102274255104817099692659197769197253262765514857785952763545933429341283064240311840831338827887643332822312403,
774
+ 229621025932426152544775116865790850370738252598968616345434742974294831770498708042467357595738334686214730941668748610469662134888147517453834812297747488073,
775
+ 23194019360964631078659252824876108242868511107591478594725592977713090517077136485554889921296914977105127960874481846244520664680271589796149178274723381887150,
776
+ 2366019572654987857981883302884224610046787980941216951080722197175332702842064407520262788224425596823801052802864297909680211909131815157116443347433800532354850,
777
+ 243723207706843858805531922672142445937983408481575545244628058028958509994770932908143706553972273020606462449310287952666455963145505421805477424475391578740749547,
778
+ 25349579391463151305599140216618845602596747701207663026876845194584976235688216794919802052097722490977512855889132026430832226610008865552502472057327013743234724289,
779
+ 2661949536117294941579537973023325999704493949797765917766453643912948050005166315226902203593637546340251255263789065070913407870828490622145548859990335827664249194843,
780
+ 282191998041802834894491022415104022066351346001485338359652435603340097587440125607908494263771078979083153300903468733951945483869169024843337130814895791015522887617745,
781
+ 30197205496285580981134004352811904578240150015173760556044734726603777953549441651607337294020756420286587011016319593854129174453557132889685837922580190251655957918555510,
782
+ 3261580360247281731350752032571633611091509233722526126730472078080267120585139503759781203157285193080871126334520343796306372125931574907402614032432823706919397204154294986,
783
+ 355542453810498068733042636106565709934312351277001653199901940962542880337894204604058217978305365782707700679734178395621346524451300447946594589336166508438272944885023160689,
784
+ 39112931217322790687675682783280636541476404828559782523085681619946545707752522201110840171533528313751290952825809107209345109214381481761817834020401609467555572307717044261531,
785
+ 4341890877379409172399447880785341173764918266333150808446906420993010745499655081333431728252354838007379636104415393783488195360765518770769933624291004549705714916972428310225781,
786
+ 486330887936128102304502836839859099736176139498488755799087060242087505384224982383210179297715061278950017979164203698503320520051987805629962723515095769464706911530343067214296383,
787
+ 54959731872117116279291355317244156670842371266593746674174813065752304578791468771401238467641310006885881029462353662425609435631417423163910565776425199899476816910378287882423889492,
788
+ 6265895725196325684507844524080542070545128322677109328184963166218059791944866442401736359283307352173462776229917563874092705752714949018893201915657796977279526404378006218641788805212,
789
+ 720632963787555401390113560504883761735877062041655493049232618904439353237986829886408887052234892305246882652442311646601162466922508150870468431281068856892270065851190356947334546378743,
790
+ 83599689208750376116005729590002818583118101706008599395628969514945677914587130520628050154192745422049124713543490235651467810795584663537081139625749586680068012717902264594803315572731101,
791
+ 9781884215427810217060440913656409477130292992301528688702689121486076906783886354978264185041313168436042532816782593796842889045097804247021806756526064020807426003305635637634741286001422893,
792
+ 1154345930843790249426956401182707607479982983051099485881044975537382171974254503497513265600612231711278510816744041124051324135560483569491207400218592450116746440493031070336815665980231111959,
793
+ 137376946933993012991856733911740533917373590588416400575454495677055577819412496871710038252583849053164004607784451035661465625912972955308589684790911084638518847621179264481542852349023640643668,
794
+ 16486387894410260690037902259304953485986973661262353421281301985591911056265956168782806629111855297346953891468826580330551477614271916652149092010899971493695776234159012623499718698854955664402748,
795
+ 1994990302388684999729177294107238476641515550889119481845925801636638078260784658181292671786150848322734814321809087929954904035153733683241761657244674076557282481616784814223021203296557743651214497,
796
+ 243405302124967322429054327669305596283675078322052247724556811637638389813463961305504642132611259634546942341184960616459852132851300345782450453854898888606208238081192591470883201343734090161637618291,
797
+ 29940847014296338083080934163742114762897482328072434062157217991075150205675060109330517734243043379391034504929438674882921806047966905631624010778128996923665149081011688217177529134339009461200749830119,
798
+ 3712908418588473128996967957868033427411251641042977209540138921545759649938991045940390558937793097102118309822180509968314323468507735620593710829706601621251663999195854963588203324262759363345782882198117,
799
+ 464143491175581970861144041100424756547648689538649459650715899252989871110531213318528600445851406202647466097106261716479189990434878434892874772480638096561187434015968048896868633341975976141238481828474974,
800
+ 58485792553136476135421687990241676942658449192209440267179224133813396367284692332225321944739217799899370840093694547821669711653470128453198598938301141654773410614306134126536036832705400185855230071759087106,
801
+ 7428159767798644411934072672496611086327480048225186162688343432069242116729848507477511467556171672231366110390541503226779463977614060641784607364541119103616232671235554035623476988227402434395631988862875012943,
802
+ 950862932357869190998817988447329687469258107805123854098003318438567630350156693917866800260498578113128672647621010808484764461839119554630423131752013877486188287142319999017759578726795176505938986005626074731549,
803
+ 122668745969789187690620411903452570009095882497157930981597636140221793066982048411408948624352924729204756211942938301697980395256799771170887130597368976906006952508450564481842353754662058405654852980858612372135911,
804
+ 15947887780519129529554479922854965720260846909047136502456391710668229106975093272632466794515680394907464206546161896146047260753802271076217077937970414095822546866856248356061906919170985282256801760814308898021016501,
805
+ 2089295960565812246663972177185935341480428286907162424178602623949814893882838569591956397904548970091572143866524490870505282693962919499365787057750684036823336923057420070795543454267687170388654319356431450231891975966,
806
+ 275803013731604335444471387329419756720403681056922460004333785322507114088718822109917770915333940706025970080669312048546811351379693959090199356593897695303242806469468228627919359209364931091624694473431341992709185556978,
807
+ 36683889999595137502862598075656930518513485964482164235324371700188193167734778535464093119956982349292344514140109341676319006616538069422899772292802843306474704243077441999843030820197491886227144202818366814564432265979165,
808
+ 4915917047011584762082682615894852395256752244873786412826199795619129330378242097056831437947751660911458953572991147842524595516612100267591780883803701538499396373785298438775140805139311903166512749812154155334386204639488047,
809
+ 663685483147266619102335303909337988266555594772596665648462615013927573914751313236982595445739025926489513686451027316383108785176647776090943114832681950235179873666660521250570610592580734807383862104453647933756573497962341969,
810
+ 90266141349272134423919907605883492029306311148939394186911026903567881080096957584138837434399945174618873267785019118845773102497253639009611390006933947593769552943709194511789668642100758818686510122126261670310015794768284816403,
811
+ 12367125013649523611578569097133421057942665933938730642839007490875307841168811075129064627909446471404711272140152340798243842155556261500914722408985213176274602649201796856803477261370745591118336934804855487552205076057442481310524,
812
+ 1706753513109064378152240353822309610438119151582922498910249997209610353893982621770571901659259106087280816104644726843996762334337703594476202873325801417326007873120798477841492270976058435019688637915531766729629217947372187196757636,
813
+ 237251104783487837031000568548879431197094143470543059903622861743717275462839813609657470911657104619390316881303685843679444975946063726200297760836374492143064299454328448395935430223666737573212528536517073830203671492561667366765677499,
814
+ 33216861332935227937631520495617271460627913742157189587924811212527496919055455063777217864955108047501555830236867777321691787689568237677202084955038021839696309196371334802015927101168041323885532512878167172825918345592278058857503309977,
815
+ 4683814686681520660512584821928253804927240611413350618675160259774486629548749962499883105426921725790066148592267184566233420044368127269900506047712728197884358466090809648509463527881432860935431909913289927403969324980131840609280138681537,
816
+ 665134900663354687273018275352704340976762038171525910319088596142998129111353935770987425694889859365054389213146791253884584056521375839042460846260812543627707351722391555952784156883963436680241885809388208040124642755685421646780801821993971,
817
+ 95118974372295206082397335819817375922159193780216219302144113837140022113132752745895986820495545241579681385706948003715837233111080606482072996266845233677684771932239356836449860406963791311552937834287604429106578823278821676415428248852507748,
818
+ 13697797411294299239552477955097525114641835479547243733224664440409805293165874383400818818158242876373873332438312722545520408184846800652066804855827459886670056249354221255610392522470578566304713646441152687979017688823716739102909864459420462844,
819
+ 1986275738928229279047424509122375246240545110707919233581455929473203401888451650862611783852488520931232824118378315232907221899273682023832617474712285452117786992556145424305725241389224161478037008503342047411114723028920300250444226844429647507469,
820
+ 290009955015797629406400878700823466791426606311275827539158584004384043523655914730627463294311725538553638817698448614052866121034509179131376326769104945680627917415987721487749656760061597132988679530072198430462927960112112107683765189213539269816583,
821
+ 42633449567942171923880571546285434800906749979933076055574360511978717562924043486508637703981435572316849756468404611663592099941313873511300026927409791877616969399353611518955877755257919862518269029041714799131259482803469772971739754874065429430647819,
822
+ 6310040532312655113797181657444428114944890724780602047562277512219180546689655913721455431732716937243121831860901777646566012791605133825397627866722260716886413310732741503270911016529152785629335057627632583691417630518958059542423484733770361176385728689,
823
+ 940238670777877145347062784627957417336839119109908870890044859363594577960798475063308868213490043241525120973709265767458014642687777741172225572655241085420990840793518338843972765507739519408658335825435505486959380000007313623452540150797720427173370476014,
824
+ 141042110367203833652515345446189746934977778027190976332535080896623909902332059158253483563475986864622931529976140648424073561328510966003488701102970147484938467056901045856773012574273152329317891438787301255459168862170960467736462949832609767445261921177858,
825
+ 21298298861485093397142003494706115482403647896279904970754383331661942428209397143587616990497522093745798074320676984168820799264950296693328467773621819311923929592442284234053629644803080432234369891435594573296731648872321963649051767102864348072281994651583299,
826
+ 3237482462746058867816582163295858590103294219693138096985233253268156772153605950646780318793284722405873079173292378345299324268436200008408658454554512824262512924623137721020463893132077853909543065273569230381193513235495038524435519560740643724388179435656262697,
827
+ 495356114158769529081336263213381120616494341137698015688902504723441425702214425350352704368961621177352309213171337010467473110489956260639801886361516609278922578453887464296540176026087132688134811491114177057374666028420135315147706054187963846541080548719733499075,
828
+ 76288078921871100244716285126920595369019388038327575308009730232304364867641582670333366514202970356333911797321624238740594274377959596064224343770660220837482679985162847383409646542789518443347491178881616225205340990599658678006189143483533443811075830097526921392265,
829
+ 11825147567705874093007006268801430876077032092026905820823655307596144636942456573455638243541812701427074993965452795617352605587474446140745713860354113045289268198820731766105046403537837890989617865888798272396285156231256195600487469052220092027761789112114898819590118,
830
+ 1844799305403554820271348032538625728295042176783011934528613449110898832514205966645987466188708242523328101131334366092132042382883344927128385232201512072706467838070149416818238162636411420705104989233361507737873668311953837006627646750032077943952561656987386877716784378,
831
+ 289645315600569556509179895562317845210751028400828838910100826606956380985097478117944466425397744160913360865078256617678936420171177833391248881870191633930527643611266734080978381540277355554797630868762573367665171500999627131608118780907535801745805450624144265803523106809,
832
+ 45765804587907293121105634925592031975282509369129101545817819682926848119277363537869499544907626620088209605680714258927197048662780670738879558102139405688617235704154389724447725699627958676417949767604220864719522844369081381805903204305786804887109232502919410695636618426323,
833
+ 7277052562967709349182727742391457012522466043507687671397751450177120254675009862028920613421228577316256015202322942563287022843718955164890259124811608748055916762202626263806035227546364487964219849400274669795071456956626551322059328763790764462508937386118224459531591144794141,
834
+ 1164374174034621599143622873333791896861157325907751385334913112447688959477815227103130107766702738126040962644036462148948870987287335940002090262629664294499293587851448019812701073692793768538360888283574456069262597376566966615819972177521812258223460515924831067087548909348565943,
835
+ 187471518782491652784187366809430651217979026186724486352718295029916018808848163592145458327501706594384029656692359972819352252357506301191845562136588352557474875512849027442301695461490053645528619733061110077473958954585476311161914133794204689984469204952363185940179137562271431716,
836
+ 30371550371171865882792661753302201350307680051089351993183248661033531564030101144805042831020178551564171065904271932711441169424779767298963173965513858837443607555780447125222788276795919800781639047491626212981834938893526825007839719315539185580263944787210025870409480295295524607148,
837
+ 4950750174742742210878537143066076161579785889045558079841550039871699218956351030733103818840362746877027278116354679435058683383823645483786291847508563029909989747187825920495498231653559313726576133772208106762325785580345567598313882940655628509762708939166194604776647942694766023209295,
838
+ 811953399043806428913247156134962129967627536016753811095235646039818722488813608279436922462957126103512528757886081719778561159166987241973883840317877108797866734976393612512891906864422409524567933332256792230218814140115492304221928887843568597513258383706290505774507706553875204375492933,
839
+ 133977261404931238673086422541543153223092517348285438238703669093028731084142107914126367686670196422663949442879289711880651017475739493680280457650797076022669082808885015457707817232558229590062100802184741099853121415954265797549676313070701711284745102669679727002790377424084614414305342085,
840
+ 22241037316246071731877655252835045257380738030797392182190167210532487411064390088964688000362973546594621045720493256181428071211993323873041282447910449505670037343796029880991762668574181671847494364709557861123016461924453468914504246659486872786133822607621737664074731018792526645205375714911,
841
+ 3714387204123747564021922027798304377055587063146823096781478812113351565599562959342316662430548864758303549313466895037497501761048623063312184350998419160495925939945958706464817456719882190150732041753783165972853384785921108164022363125346443637529795145292979810186984610735259970379748909127348,
842
+ 624039290518152249140345932326914903805955436532529757472693709681090779066181518231024403356173375684535659557306529571721395254554873984216952134767662210738270197205918874350969840442620236483827049913570635289838291079818060744708249162397878771827649977800756684225140035296860087627789546942019708,
843
+ 105466354350794561887150289111500652378337552914534142149527237820649764193997640033331960028820168532033622847127204094879993226411513538816813283517049412142591418107298404451887666922767742039271084693963292245703279292838545037174622432169762252316047235884432562853458170104480396659540197723326955049,
844
+ 17929904256684551375507070365989607376568717795083884134351858516156595056832592891819418119096827630883191688156788722213562608833687632693448212851839030976023586150651159032648808103982795368523551667903189836080847257507216163843566672405033062718582404069485243044539595997381464176915428772299529897387,
845
+ 3066119090533021058714949384385546160604440691917658129272146205517325292470005789901166686388267793267237143948503328752792788527052929417995753465842776658491120744440382072396204585620942869321323938092353799027026824678529437249143676742849628783405786774551613146024988159201555437135056461242711244425471,
846
+ 527390412851896881337967687677672756955017910664166385531182707248820008652217019610434436390775296988508016945042297049346819579041236617426801836524124816150464053636641876482784456269873165934499835255835179652533897363099882327178123141801772551227650892605120356816114901948508241248074248179475145059700333,
847
+ 91241607437002316765896038608485693250098200401525483156981946490993551549363713248546046728857342955102878419072658758617718815436207404132971108037467121786129287302343884825597583134025705848665136225746046092184302538898591915768188501090045236362303719730765280000065472022469880131266995746982360928681789022,
848
+ 15876567066521347020698839726988333654358453219813262179732406101757409700716118715916586061550905181406627832514197644697952919000972483630799710717581986327881417584627859529725184506381086453934861481790667535883365308055072114257978169416900357818905207403182343347177282179986574055865217055581294602049902271010,
849
+ 2778490475182553012629852619584435199535334993449430533291889233085068668410721374298319027847030246426005465323973312410522974155948278057948955771338598190837767209227308996970705566697491456357600159639062130364317178202822566322383617104829808637250858430245558038234116074694798202700478387153312106531075564590471,
850
+ 489030199671805332623824962747571404346836212807185241208439873168560303726091573108471569738622564120936538119090857469125436031350069250930034589598553058959515419192763112129956195196850287774379689755311086407919175437399870511801487097082908793024488522454054731734087612117668678951830149593545914366568451549499685,
851
+ 86561123741143100954425185128833677328821843488640252324816664336373211122400902634155915088043408570367135685762753962117203335002228676669109366418614283254650822782071776974482484791290241764478481232109698002237791589418738694740662310854424857888923693368377816708052624632415831235724159938175087416693083632694532655,
852
+ 15408369040246573624543239301621863724747343643924773876555353865629664268149862128609923803166625729851120529476074445805627853384167576764449771480413525069776028857083426876194215255653834502122141422615080396035284604156677631455809382509697640240082055005671212597863791286268060426062068690326674784351695961996809702925,
853
+ 2758184616549386816236966581040218243122290071854676132569266753561407253841155914685857436796941019131893094886346030450441971956288343738985310251653168257491732220357794099956782882394746532133684989597402798625958065152734315013482352560824114488492301613892673240594384349529768053583167309965069944574307741356063881733198,
854
+ 496488638858899582052373047531747441194419502100231890566037494304235208701784481854947616315343251608142737193478105062003965661858194807779942430679552898601138903954455265796381172265739994735468018104319645795430748523491794541864220642757735261696606827964607383883421563748337345997689381451032884205162342055027754655644258,
855
+ 89867201731516234028810406790663697170637217600270232588683759072300399874482417879110126174297018590487235209511071430301133418498618876087738028078056121495359557869769535240201030784051241771649973840200099886178407337048472658277648457294966658193292758583330642417087139062211759287554289280979608471920097580250345660175667269,
856
+ 16356327188366441658119624876352667750876144855949554916967943381292515026124392998168079369173308046777066977444004707140546456753099170108491851899803084237468081527770828726096602307514798027469350587512384672212999721594413225245300445923593654356691942376945977212409662951427619699091974158585718172402987452310635100548384617287,
857
+ 2993297739914605231295782554728829568693823311974566186927169826728453997814894048777587112950144859995777671404125187756344531363825073195200403410140483792480012315131872314962566345833107848844569546640219444477887105746956467249076010238865029927333608990947884846232028663308854431313858220266631730911072185613430360488848380096281,
858
+ 550783139974987003088218067365442633899663556967162543344434169230544788490192775773411327850121087642233650337890181665209037562456824033960874097880702956598955542054061567729066817417783906430566902466334048231890813845460531958577810593743779509555982101030311839141722877272253640810432059872996845423186085282617827087085796990210795,
859
+ 101897874103245292949610605050152511402809869703190066600404519669462113346602553652645736661299555895721829589816575539843398192269973696674746584325695903109111654917587228796143329441631714378369643419757760056173958864255423230918395395752963003321812170973719718403992431600307208445057977459633681222687535291221800896415243174474478220,
860
+ 18953555349987269513583685925972300938797041378803931051582164568949412796909717255414133408411990744485130534653203982827450885182444983085681510472394334973622171955723863346584168709405433767249061955080387563351082710676250632929328176721273176250188173634178612605084700616461370246634697653011621789312559137616783862288768186077759060052,
861
+ 3544416745328424405156087202428741737050853744962736160081627521297221170085915775836718595645497946345696902821817072533530002824991409378584754738635250356976281062714054372434158933346745701957873227277091547859014138808133302506047352996323164789390304834877777696492488563123378060666917639872381916037693705294724994459834908558390607007091,
862
+ 666369301126310545097391341268240810355665202764078007187600256850594733484682215047776491908527501458340178898418967920457474940398454988729311567931614765352673469463460815752139785416157487281523118346281491720437399718860036300259150379962150898069243836083271971054532503755201465960996189432056884100898725588494439587607248636967121359962657,
863
+ 125947342227720130897870384956928434884228246549421155207166330179485240813868358794318534040481761498101036169889790816623524177431971099819692570593270744695119513235450557236449586629731475552092681437169347812732339798095978353716463535442557233013458720672823557985529738805599497577796583167592887609373320496354730489204620483472860388155151673,
864
+ 23930661373614392821408265400881947707748153538976014764078462518546830223064130454259014003396132494408869991611506765203698341039945156228023350300419599850746549374743598122714159608796756854619227565426596956675572760285636571600276824284398323537376462771949001219098758245239810267020873734073856072994438863631998696840246239311147532703257700251,
865
+ 4570882266158159449898476879711993226662338038274196438411776111729467167139690781167522852578591809004250083216585495131341526572964293534538188809319789398533642423432149831287153282959919664672608505732241653574801012399329847101206565011456118497058591959239430789707711825371446835907350736676071374376578573119422405027843651767566238129839430441796,
866
+ 877633325097370825195334898853892281715605124638125944633476516041214463902578773889401838809489491343657303677760823575356307056864280396776098145072441722938545726333244435737549900379033882409513204879499099924684809685510442958521917401358422995651536087050210976360285117868917961891106193579731750510002742057200582604030656360293924904092332719314108,
867
+ 169387802500111366138419359800386585173203824726890970799717150305886213486495605746484430149358448553133259010293323495098991362363116327402986180520192631160245708437946851944252402234616182252781263072325812730210633853743852496825055962101810672995028304252910549623761564671744026391712260204615444956193072061326888542068105758326903936594779078698584373,
868
+ 32862111294416037464590849150108479821351389750167739641180382115503094891805945167664651508342786833465564378198029614008535985595808262337931298577100396584880541022606173246696769293758736144892403676874977093298918436310463256990627957215642001414387305224323064454964096301397669811657227082724946328179362510551498936499556943525291886314069913438637195807
869
+ ],
870
+ "factorial_lower_bound_passes": true,
871
+ "factorial_recurrence_passes": true,
872
+ "fraction_of_cube": 0.1904296875,
873
+ "induced": true,
874
+ "length": 195,
875
+ "log10_factorial_iteration_lower_bound": 361.1235834688244,
876
+ "n": 10,
877
+ "path": [
878
+ 0,
879
+ 16,
880
+ 24,
881
+ 56,
882
+ 40,
883
+ 552,
884
+ 556,
885
+ 812,
886
+ 804,
887
+ 805,
888
+ 801,
889
+ 545,
890
+ 609,
891
+ 617,
892
+ 873,
893
+ 889,
894
+ 881,
895
+ 885,
896
+ 629,
897
+ 565,
898
+ 533,
899
+ 789,
900
+ 791,
901
+ 823,
902
+ 831,
903
+ 830,
904
+ 894,
905
+ 892,
906
+ 636,
907
+ 124,
908
+ 125,
909
+ 109,
910
+ 365,
911
+ 301,
912
+ 303,
913
+ 302,
914
+ 298,
915
+ 314,
916
+ 315,
917
+ 307,
918
+ 291,
919
+ 35,
920
+ 163,
921
+ 227,
922
+ 231,
923
+ 247,
924
+ 246,
925
+ 502,
926
+ 500,
927
+ 508,
928
+ 509,
929
+ 1021,
930
+ 989,
931
+ 733,
932
+ 221,
933
+ 205,
934
+ 204,
935
+ 206,
936
+ 222,
937
+ 478,
938
+ 474,
939
+ 472,
940
+ 984,
941
+ 968,
942
+ 969,
943
+ 457,
944
+ 459,
945
+ 491,
946
+ 490,
947
+ 488,
948
+ 232,
949
+ 744,
950
+ 736,
951
+ 752,
952
+ 1008,
953
+ 944,
954
+ 952,
955
+ 696,
956
+ 697,
957
+ 569,
958
+ 537,
959
+ 793,
960
+ 777,
961
+ 265,
962
+ 9,
963
+ 11,
964
+ 10,
965
+ 138,
966
+ 136,
967
+ 648,
968
+ 652,
969
+ 654,
970
+ 526,
971
+ 782,
972
+ 783,
973
+ 911,
974
+ 909,
975
+ 901,
976
+ 389,
977
+ 405,
978
+ 413,
979
+ 285,
980
+ 29,
981
+ 31,
982
+ 23,
983
+ 151,
984
+ 135,
985
+ 647,
986
+ 519,
987
+ 515,
988
+ 514,
989
+ 770,
990
+ 834,
991
+ 962,
992
+ 706,
993
+ 194,
994
+ 210,
995
+ 146,
996
+ 178,
997
+ 186,
998
+ 250,
999
+ 762,
1000
+ 634,
1001
+ 618,
1002
+ 586,
1003
+ 587,
1004
+ 843,
1005
+ 859,
1006
+ 858,
1007
+ 794,
1008
+ 538,
1009
+ 666,
1010
+ 667,
1011
+ 155,
1012
+ 411,
1013
+ 403,
1014
+ 387,
1015
+ 386,
1016
+ 390,
1017
+ 422,
1018
+ 166,
1019
+ 174,
1020
+ 172,
1021
+ 188,
1022
+ 189,
1023
+ 181,
1024
+ 177,
1025
+ 241,
1026
+ 209,
1027
+ 193,
1028
+ 129,
1029
+ 641,
1030
+ 657,
1031
+ 913,
1032
+ 977,
1033
+ 979,
1034
+ 983,
1035
+ 727,
1036
+ 599,
1037
+ 598,
1038
+ 854,
1039
+ 852,
1040
+ 340,
1041
+ 348,
1042
+ 332,
1043
+ 328,
1044
+ 320,
1045
+ 321,
1046
+ 323,
1047
+ 339,
1048
+ 83,
1049
+ 115,
1050
+ 627,
1051
+ 563,
1052
+ 562,
1053
+ 818,
1054
+ 882,
1055
+ 370,
1056
+ 354,
1057
+ 358,
1058
+ 359,
1059
+ 375,
1060
+ 383,
1061
+ 351,
1062
+ 335,
1063
+ 79,
1064
+ 71,
1065
+ 69,
1066
+ 5,
1067
+ 37,
1068
+ 36,
1069
+ 100,
1070
+ 612,
1071
+ 614,
1072
+ 550
1073
+ ],
1074
+ "terminal_dwell_time": 32862111294416037464590849150108479821351389750167739641180382115503094891805945167664651508342786833465564378198029614008535985595808262337931298577100396584880541022606173246696769293758736144892403676874977093298918436310463256990627957215642001414387305224323064454964096301397669811657227082724946328179362510551498936499556943525291886314069913438637195807
1075
+ }
1076
+ ],
1077
+ "source_note": "The minimal equality sequence for the displayed k>=2 recurrence has T1=1, T2=1, T3=3. The proof text later says T3=2!; that value is sufficient as a factorial lower-bound base but is not equality in the displayed recurrence. This audit follows the displayed recurrence.",
1078
+ "source_pdf_sha256": "9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218",
1079
+ "status": "VERIFIED"
1080
+ }
evidence/raw/claim_3.json ADDED
@@ -0,0 +1,2373 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "cases": [
3
+ {
4
+ "converged_round": 1,
5
+ "epsilon": 0.25,
6
+ "final_nash_gap": 0.0,
7
+ "matrix": [
8
+ 0,
9
+ 0,
10
+ 0,
11
+ 0
12
+ ],
13
+ "min_accepted_improvement": null,
14
+ "update_bound": 0,
15
+ "updates": 0
16
+ },
17
+ {
18
+ "converged_round": 1,
19
+ "epsilon": 0.5,
20
+ "final_nash_gap": 0.0,
21
+ "matrix": [
22
+ 0,
23
+ 0,
24
+ 0,
25
+ 0
26
+ ],
27
+ "min_accepted_improvement": null,
28
+ "update_bound": 0,
29
+ "updates": 0
30
+ },
31
+ {
32
+ "converged_round": 1,
33
+ "epsilon": 0.25,
34
+ "final_nash_gap": 0.25,
35
+ "matrix": [
36
+ 0,
37
+ 0,
38
+ 0,
39
+ 1
40
+ ],
41
+ "min_accepted_improvement": null,
42
+ "update_bound": 4,
43
+ "updates": 0
44
+ },
45
+ {
46
+ "converged_round": 1,
47
+ "epsilon": 0.5,
48
+ "final_nash_gap": 0.25,
49
+ "matrix": [
50
+ 0,
51
+ 0,
52
+ 0,
53
+ 1
54
+ ],
55
+ "min_accepted_improvement": null,
56
+ "update_bound": 2,
57
+ "updates": 0
58
+ },
59
+ {
60
+ "converged_round": 1,
61
+ "epsilon": 0.25,
62
+ "final_nash_gap": 0.5,
63
+ "matrix": [
64
+ 0,
65
+ 0,
66
+ 0,
67
+ 2
68
+ ],
69
+ "min_accepted_improvement": null,
70
+ "update_bound": 8,
71
+ "updates": 0
72
+ },
73
+ {
74
+ "converged_round": 1,
75
+ "epsilon": 0.5,
76
+ "final_nash_gap": 0.5,
77
+ "matrix": [
78
+ 0,
79
+ 0,
80
+ 0,
81
+ 2
82
+ ],
83
+ "min_accepted_improvement": null,
84
+ "update_bound": 4,
85
+ "updates": 0
86
+ },
87
+ {
88
+ "converged_round": 1,
89
+ "epsilon": 0.25,
90
+ "final_nash_gap": 0.25,
91
+ "matrix": [
92
+ 0,
93
+ 0,
94
+ 1,
95
+ 0
96
+ ],
97
+ "min_accepted_improvement": null,
98
+ "update_bound": 4,
99
+ "updates": 0
100
+ },
101
+ {
102
+ "converged_round": 1,
103
+ "epsilon": 0.5,
104
+ "final_nash_gap": 0.25,
105
+ "matrix": [
106
+ 0,
107
+ 0,
108
+ 1,
109
+ 0
110
+ ],
111
+ "min_accepted_improvement": null,
112
+ "update_bound": 2,
113
+ "updates": 0
114
+ },
115
+ {
116
+ "converged_round": 1,
117
+ "epsilon": 0.25,
118
+ "final_nash_gap": 0.5,
119
+ "matrix": [
120
+ 0,
121
+ 0,
122
+ 1,
123
+ 1
124
+ ],
125
+ "min_accepted_improvement": null,
126
+ "update_bound": 4,
127
+ "updates": 0
128
+ },
129
+ {
130
+ "converged_round": 1,
131
+ "epsilon": 0.5,
132
+ "final_nash_gap": 0.5,
133
+ "matrix": [
134
+ 0,
135
+ 0,
136
+ 1,
137
+ 1
138
+ ],
139
+ "min_accepted_improvement": null,
140
+ "update_bound": 2,
141
+ "updates": 0
142
+ },
143
+ {
144
+ "converged_round": 2,
145
+ "epsilon": 0.25,
146
+ "final_nash_gap": 0.4087872380968218,
147
+ "matrix": [
148
+ 0,
149
+ 0,
150
+ 1,
151
+ 2
152
+ ],
153
+ "min_accepted_improvement": 0.4763617142904655,
154
+ "update_bound": 8,
155
+ "updates": 1
156
+ },
157
+ {
158
+ "converged_round": 1,
159
+ "epsilon": 0.5,
160
+ "final_nash_gap": 0.75,
161
+ "matrix": [
162
+ 0,
163
+ 0,
164
+ 1,
165
+ 2
166
+ ],
167
+ "min_accepted_improvement": null,
168
+ "update_bound": 4,
169
+ "updates": 0
170
+ },
171
+ {
172
+ "converged_round": 1,
173
+ "epsilon": 0.25,
174
+ "final_nash_gap": 0.5,
175
+ "matrix": [
176
+ 0,
177
+ 0,
178
+ 2,
179
+ 0
180
+ ],
181
+ "min_accepted_improvement": null,
182
+ "update_bound": 8,
183
+ "updates": 0
184
+ },
185
+ {
186
+ "converged_round": 1,
187
+ "epsilon": 0.5,
188
+ "final_nash_gap": 0.5,
189
+ "matrix": [
190
+ 0,
191
+ 0,
192
+ 2,
193
+ 0
194
+ ],
195
+ "min_accepted_improvement": null,
196
+ "update_bound": 4,
197
+ "updates": 0
198
+ },
199
+ {
200
+ "converged_round": 2,
201
+ "epsilon": 0.25,
202
+ "final_nash_gap": 0.4087872380968218,
203
+ "matrix": [
204
+ 0,
205
+ 0,
206
+ 2,
207
+ 1
208
+ ],
209
+ "min_accepted_improvement": 0.4763617142904655,
210
+ "update_bound": 8,
211
+ "updates": 1
212
+ },
213
+ {
214
+ "converged_round": 1,
215
+ "epsilon": 0.5,
216
+ "final_nash_gap": 0.75,
217
+ "matrix": [
218
+ 0,
219
+ 0,
220
+ 2,
221
+ 1
222
+ ],
223
+ "min_accepted_improvement": null,
224
+ "update_bound": 4,
225
+ "updates": 0
226
+ },
227
+ {
228
+ "converged_round": 2,
229
+ "epsilon": 0.25,
230
+ "final_nash_gap": 0.23840584404423537,
231
+ "matrix": [
232
+ 0,
233
+ 0,
234
+ 2,
235
+ 2
236
+ ],
237
+ "min_accepted_improvement": 0.7615941559557646,
238
+ "update_bound": 8,
239
+ "updates": 1
240
+ },
241
+ {
242
+ "converged_round": 1,
243
+ "epsilon": 0.5,
244
+ "final_nash_gap": 1.0,
245
+ "matrix": [
246
+ 0,
247
+ 0,
248
+ 2,
249
+ 2
250
+ ],
251
+ "min_accepted_improvement": null,
252
+ "update_bound": 4,
253
+ "updates": 0
254
+ },
255
+ {
256
+ "converged_round": 1,
257
+ "epsilon": 0.25,
258
+ "final_nash_gap": 0.25,
259
+ "matrix": [
260
+ 0,
261
+ 1,
262
+ 0,
263
+ 0
264
+ ],
265
+ "min_accepted_improvement": null,
266
+ "update_bound": 4,
267
+ "updates": 0
268
+ },
269
+ {
270
+ "converged_round": 1,
271
+ "epsilon": 0.5,
272
+ "final_nash_gap": 0.25,
273
+ "matrix": [
274
+ 0,
275
+ 1,
276
+ 0,
277
+ 0
278
+ ],
279
+ "min_accepted_improvement": null,
280
+ "update_bound": 2,
281
+ "updates": 0
282
+ },
283
+ {
284
+ "converged_round": 1,
285
+ "epsilon": 0.25,
286
+ "final_nash_gap": 0.5,
287
+ "matrix": [
288
+ 0,
289
+ 1,
290
+ 0,
291
+ 1
292
+ ],
293
+ "min_accepted_improvement": null,
294
+ "update_bound": 4,
295
+ "updates": 0
296
+ },
297
+ {
298
+ "converged_round": 1,
299
+ "epsilon": 0.5,
300
+ "final_nash_gap": 0.5,
301
+ "matrix": [
302
+ 0,
303
+ 1,
304
+ 0,
305
+ 1
306
+ ],
307
+ "min_accepted_improvement": null,
308
+ "update_bound": 2,
309
+ "updates": 0
310
+ },
311
+ {
312
+ "converged_round": 3,
313
+ "epsilon": 0.25,
314
+ "final_nash_gap": 0.4464790992674148,
315
+ "matrix": [
316
+ 0,
317
+ 1,
318
+ 0,
319
+ 2
320
+ ],
321
+ "min_accepted_improvement": 0.5894372978022444,
322
+ "update_bound": 8,
323
+ "updates": 1
324
+ },
325
+ {
326
+ "converged_round": 1,
327
+ "epsilon": 0.5,
328
+ "final_nash_gap": 0.75,
329
+ "matrix": [
330
+ 0,
331
+ 1,
332
+ 0,
333
+ 2
334
+ ],
335
+ "min_accepted_improvement": null,
336
+ "update_bound": 4,
337
+ "updates": 0
338
+ },
339
+ {
340
+ "converged_round": 1,
341
+ "epsilon": 0.25,
342
+ "final_nash_gap": 0.0,
343
+ "matrix": [
344
+ 0,
345
+ 1,
346
+ 1,
347
+ 0
348
+ ],
349
+ "min_accepted_improvement": null,
350
+ "update_bound": 4,
351
+ "updates": 0
352
+ },
353
+ {
354
+ "converged_round": 1,
355
+ "epsilon": 0.5,
356
+ "final_nash_gap": 0.0,
357
+ "matrix": [
358
+ 0,
359
+ 1,
360
+ 1,
361
+ 0
362
+ ],
363
+ "min_accepted_improvement": null,
364
+ "update_bound": 2,
365
+ "updates": 0
366
+ },
367
+ {
368
+ "converged_round": 1,
369
+ "epsilon": 0.25,
370
+ "final_nash_gap": 0.25,
371
+ "matrix": [
372
+ 0,
373
+ 1,
374
+ 1,
375
+ 1
376
+ ],
377
+ "min_accepted_improvement": null,
378
+ "update_bound": 4,
379
+ "updates": 0
380
+ },
381
+ {
382
+ "converged_round": 1,
383
+ "epsilon": 0.5,
384
+ "final_nash_gap": 0.25,
385
+ "matrix": [
386
+ 0,
387
+ 1,
388
+ 1,
389
+ 1
390
+ ],
391
+ "min_accepted_improvement": null,
392
+ "update_bound": 2,
393
+ "updates": 0
394
+ },
395
+ {
396
+ "converged_round": 1,
397
+ "epsilon": 0.25,
398
+ "final_nash_gap": 0.5,
399
+ "matrix": [
400
+ 0,
401
+ 1,
402
+ 1,
403
+ 2
404
+ ],
405
+ "min_accepted_improvement": null,
406
+ "update_bound": 8,
407
+ "updates": 0
408
+ },
409
+ {
410
+ "converged_round": 1,
411
+ "epsilon": 0.5,
412
+ "final_nash_gap": 0.5,
413
+ "matrix": [
414
+ 0,
415
+ 1,
416
+ 1,
417
+ 2
418
+ ],
419
+ "min_accepted_improvement": null,
420
+ "update_bound": 4,
421
+ "updates": 0
422
+ },
423
+ {
424
+ "converged_round": 1,
425
+ "epsilon": 0.25,
426
+ "final_nash_gap": 0.25,
427
+ "matrix": [
428
+ 0,
429
+ 1,
430
+ 2,
431
+ 0
432
+ ],
433
+ "min_accepted_improvement": null,
434
+ "update_bound": 8,
435
+ "updates": 0
436
+ },
437
+ {
438
+ "converged_round": 1,
439
+ "epsilon": 0.5,
440
+ "final_nash_gap": 0.25,
441
+ "matrix": [
442
+ 0,
443
+ 1,
444
+ 2,
445
+ 0
446
+ ],
447
+ "min_accepted_improvement": null,
448
+ "update_bound": 4,
449
+ "updates": 0
450
+ },
451
+ {
452
+ "converged_round": 1,
453
+ "epsilon": 0.25,
454
+ "final_nash_gap": 0.5,
455
+ "matrix": [
456
+ 0,
457
+ 1,
458
+ 2,
459
+ 1
460
+ ],
461
+ "min_accepted_improvement": null,
462
+ "update_bound": 8,
463
+ "updates": 0
464
+ },
465
+ {
466
+ "converged_round": 1,
467
+ "epsilon": 0.5,
468
+ "final_nash_gap": 0.5,
469
+ "matrix": [
470
+ 0,
471
+ 1,
472
+ 2,
473
+ 1
474
+ ],
475
+ "min_accepted_improvement": null,
476
+ "update_bound": 4,
477
+ "updates": 0
478
+ },
479
+ {
480
+ "converged_round": 2,
481
+ "epsilon": 0.25,
482
+ "final_nash_gap": 0.2736382857095345,
483
+ "matrix": [
484
+ 0,
485
+ 1,
486
+ 2,
487
+ 2
488
+ ],
489
+ "min_accepted_improvement": 0.4763617142904655,
490
+ "update_bound": 8,
491
+ "updates": 1
492
+ },
493
+ {
494
+ "converged_round": 1,
495
+ "epsilon": 0.5,
496
+ "final_nash_gap": 0.75,
497
+ "matrix": [
498
+ 0,
499
+ 1,
500
+ 2,
501
+ 2
502
+ ],
503
+ "min_accepted_improvement": null,
504
+ "update_bound": 4,
505
+ "updates": 0
506
+ },
507
+ {
508
+ "converged_round": 1,
509
+ "epsilon": 0.25,
510
+ "final_nash_gap": 0.5,
511
+ "matrix": [
512
+ 0,
513
+ 2,
514
+ 0,
515
+ 0
516
+ ],
517
+ "min_accepted_improvement": null,
518
+ "update_bound": 8,
519
+ "updates": 0
520
+ },
521
+ {
522
+ "converged_round": 1,
523
+ "epsilon": 0.5,
524
+ "final_nash_gap": 0.5,
525
+ "matrix": [
526
+ 0,
527
+ 2,
528
+ 0,
529
+ 0
530
+ ],
531
+ "min_accepted_improvement": null,
532
+ "update_bound": 4,
533
+ "updates": 0
534
+ },
535
+ {
536
+ "converged_round": 3,
537
+ "epsilon": 0.25,
538
+ "final_nash_gap": 0.4464790992674148,
539
+ "matrix": [
540
+ 0,
541
+ 2,
542
+ 0,
543
+ 1
544
+ ],
545
+ "min_accepted_improvement": 0.5894372978022444,
546
+ "update_bound": 8,
547
+ "updates": 1
548
+ },
549
+ {
550
+ "converged_round": 1,
551
+ "epsilon": 0.5,
552
+ "final_nash_gap": 0.75,
553
+ "matrix": [
554
+ 0,
555
+ 2,
556
+ 0,
557
+ 1
558
+ ],
559
+ "min_accepted_improvement": null,
560
+ "update_bound": 4,
561
+ "updates": 0
562
+ },
563
+ {
564
+ "converged_round": 3,
565
+ "epsilon": 0.25,
566
+ "final_nash_gap": 0.11161443841433938,
567
+ "matrix": [
568
+ 0,
569
+ 2,
570
+ 0,
571
+ 2
572
+ ],
573
+ "min_accepted_improvement": 0.8883855615856606,
574
+ "update_bound": 8,
575
+ "updates": 1
576
+ },
577
+ {
578
+ "converged_round": 1,
579
+ "epsilon": 0.5,
580
+ "final_nash_gap": 1.0,
581
+ "matrix": [
582
+ 0,
583
+ 2,
584
+ 0,
585
+ 2
586
+ ],
587
+ "min_accepted_improvement": null,
588
+ "update_bound": 4,
589
+ "updates": 0
590
+ },
591
+ {
592
+ "converged_round": 1,
593
+ "epsilon": 0.25,
594
+ "final_nash_gap": 0.25,
595
+ "matrix": [
596
+ 0,
597
+ 2,
598
+ 1,
599
+ 0
600
+ ],
601
+ "min_accepted_improvement": null,
602
+ "update_bound": 8,
603
+ "updates": 0
604
+ },
605
+ {
606
+ "converged_round": 1,
607
+ "epsilon": 0.5,
608
+ "final_nash_gap": 0.25,
609
+ "matrix": [
610
+ 0,
611
+ 2,
612
+ 1,
613
+ 0
614
+ ],
615
+ "min_accepted_improvement": null,
616
+ "update_bound": 4,
617
+ "updates": 0
618
+ },
619
+ {
620
+ "converged_round": 1,
621
+ "epsilon": 0.25,
622
+ "final_nash_gap": 0.5,
623
+ "matrix": [
624
+ 0,
625
+ 2,
626
+ 1,
627
+ 1
628
+ ],
629
+ "min_accepted_improvement": null,
630
+ "update_bound": 8,
631
+ "updates": 0
632
+ },
633
+ {
634
+ "converged_round": 1,
635
+ "epsilon": 0.5,
636
+ "final_nash_gap": 0.5,
637
+ "matrix": [
638
+ 0,
639
+ 2,
640
+ 1,
641
+ 1
642
+ ],
643
+ "min_accepted_improvement": null,
644
+ "update_bound": 4,
645
+ "updates": 0
646
+ },
647
+ {
648
+ "converged_round": 3,
649
+ "epsilon": 0.25,
650
+ "final_nash_gap": 0.1605627021977556,
651
+ "matrix": [
652
+ 0,
653
+ 2,
654
+ 1,
655
+ 2
656
+ ],
657
+ "min_accepted_improvement": 0.5894372978022444,
658
+ "update_bound": 8,
659
+ "updates": 1
660
+ },
661
+ {
662
+ "converged_round": 1,
663
+ "epsilon": 0.5,
664
+ "final_nash_gap": 0.75,
665
+ "matrix": [
666
+ 0,
667
+ 2,
668
+ 1,
669
+ 2
670
+ ],
671
+ "min_accepted_improvement": null,
672
+ "update_bound": 4,
673
+ "updates": 0
674
+ },
675
+ {
676
+ "converged_round": 1,
677
+ "epsilon": 0.25,
678
+ "final_nash_gap": 0.0,
679
+ "matrix": [
680
+ 0,
681
+ 2,
682
+ 2,
683
+ 0
684
+ ],
685
+ "min_accepted_improvement": null,
686
+ "update_bound": 8,
687
+ "updates": 0
688
+ },
689
+ {
690
+ "converged_round": 1,
691
+ "epsilon": 0.5,
692
+ "final_nash_gap": 0.0,
693
+ "matrix": [
694
+ 0,
695
+ 2,
696
+ 2,
697
+ 0
698
+ ],
699
+ "min_accepted_improvement": null,
700
+ "update_bound": 4,
701
+ "updates": 0
702
+ },
703
+ {
704
+ "converged_round": 1,
705
+ "epsilon": 0.25,
706
+ "final_nash_gap": 0.25,
707
+ "matrix": [
708
+ 0,
709
+ 2,
710
+ 2,
711
+ 1
712
+ ],
713
+ "min_accepted_improvement": null,
714
+ "update_bound": 8,
715
+ "updates": 0
716
+ },
717
+ {
718
+ "converged_round": 1,
719
+ "epsilon": 0.5,
720
+ "final_nash_gap": 0.25,
721
+ "matrix": [
722
+ 0,
723
+ 2,
724
+ 2,
725
+ 1
726
+ ],
727
+ "min_accepted_improvement": null,
728
+ "update_bound": 4,
729
+ "updates": 0
730
+ },
731
+ {
732
+ "converged_round": 1,
733
+ "epsilon": 0.25,
734
+ "final_nash_gap": 0.5,
735
+ "matrix": [
736
+ 0,
737
+ 2,
738
+ 2,
739
+ 2
740
+ ],
741
+ "min_accepted_improvement": null,
742
+ "update_bound": 8,
743
+ "updates": 0
744
+ },
745
+ {
746
+ "converged_round": 1,
747
+ "epsilon": 0.5,
748
+ "final_nash_gap": 0.5,
749
+ "matrix": [
750
+ 0,
751
+ 2,
752
+ 2,
753
+ 2
754
+ ],
755
+ "min_accepted_improvement": null,
756
+ "update_bound": 4,
757
+ "updates": 0
758
+ },
759
+ {
760
+ "converged_round": 1,
761
+ "epsilon": 0.25,
762
+ "final_nash_gap": 0.25,
763
+ "matrix": [
764
+ 1,
765
+ 0,
766
+ 0,
767
+ 0
768
+ ],
769
+ "min_accepted_improvement": null,
770
+ "update_bound": 4,
771
+ "updates": 0
772
+ },
773
+ {
774
+ "converged_round": 1,
775
+ "epsilon": 0.5,
776
+ "final_nash_gap": 0.25,
777
+ "matrix": [
778
+ 1,
779
+ 0,
780
+ 0,
781
+ 0
782
+ ],
783
+ "min_accepted_improvement": null,
784
+ "update_bound": 2,
785
+ "updates": 0
786
+ },
787
+ {
788
+ "converged_round": 1,
789
+ "epsilon": 0.25,
790
+ "final_nash_gap": 0.0,
791
+ "matrix": [
792
+ 1,
793
+ 0,
794
+ 0,
795
+ 1
796
+ ],
797
+ "min_accepted_improvement": null,
798
+ "update_bound": 4,
799
+ "updates": 0
800
+ },
801
+ {
802
+ "converged_round": 1,
803
+ "epsilon": 0.5,
804
+ "final_nash_gap": 0.0,
805
+ "matrix": [
806
+ 1,
807
+ 0,
808
+ 0,
809
+ 1
810
+ ],
811
+ "min_accepted_improvement": null,
812
+ "update_bound": 2,
813
+ "updates": 0
814
+ },
815
+ {
816
+ "converged_round": 1,
817
+ "epsilon": 0.25,
818
+ "final_nash_gap": 0.25,
819
+ "matrix": [
820
+ 1,
821
+ 0,
822
+ 0,
823
+ 2
824
+ ],
825
+ "min_accepted_improvement": null,
826
+ "update_bound": 8,
827
+ "updates": 0
828
+ },
829
+ {
830
+ "converged_round": 1,
831
+ "epsilon": 0.5,
832
+ "final_nash_gap": 0.25,
833
+ "matrix": [
834
+ 1,
835
+ 0,
836
+ 0,
837
+ 2
838
+ ],
839
+ "min_accepted_improvement": null,
840
+ "update_bound": 4,
841
+ "updates": 0
842
+ },
843
+ {
844
+ "converged_round": 1,
845
+ "epsilon": 0.25,
846
+ "final_nash_gap": 0.5,
847
+ "matrix": [
848
+ 1,
849
+ 0,
850
+ 1,
851
+ 0
852
+ ],
853
+ "min_accepted_improvement": null,
854
+ "update_bound": 4,
855
+ "updates": 0
856
+ },
857
+ {
858
+ "converged_round": 1,
859
+ "epsilon": 0.5,
860
+ "final_nash_gap": 0.5,
861
+ "matrix": [
862
+ 1,
863
+ 0,
864
+ 1,
865
+ 0
866
+ ],
867
+ "min_accepted_improvement": null,
868
+ "update_bound": 2,
869
+ "updates": 0
870
+ },
871
+ {
872
+ "converged_round": 1,
873
+ "epsilon": 0.25,
874
+ "final_nash_gap": 0.25,
875
+ "matrix": [
876
+ 1,
877
+ 0,
878
+ 1,
879
+ 1
880
+ ],
881
+ "min_accepted_improvement": null,
882
+ "update_bound": 4,
883
+ "updates": 0
884
+ },
885
+ {
886
+ "converged_round": 1,
887
+ "epsilon": 0.5,
888
+ "final_nash_gap": 0.25,
889
+ "matrix": [
890
+ 1,
891
+ 0,
892
+ 1,
893
+ 1
894
+ ],
895
+ "min_accepted_improvement": null,
896
+ "update_bound": 2,
897
+ "updates": 0
898
+ },
899
+ {
900
+ "converged_round": 1,
901
+ "epsilon": 0.25,
902
+ "final_nash_gap": 0.5,
903
+ "matrix": [
904
+ 1,
905
+ 0,
906
+ 1,
907
+ 2
908
+ ],
909
+ "min_accepted_improvement": null,
910
+ "update_bound": 8,
911
+ "updates": 0
912
+ },
913
+ {
914
+ "converged_round": 1,
915
+ "epsilon": 0.5,
916
+ "final_nash_gap": 0.5,
917
+ "matrix": [
918
+ 1,
919
+ 0,
920
+ 1,
921
+ 2
922
+ ],
923
+ "min_accepted_improvement": null,
924
+ "update_bound": 4,
925
+ "updates": 0
926
+ },
927
+ {
928
+ "converged_round": 3,
929
+ "epsilon": 0.25,
930
+ "final_nash_gap": 0.4464790992674148,
931
+ "matrix": [
932
+ 1,
933
+ 0,
934
+ 2,
935
+ 0
936
+ ],
937
+ "min_accepted_improvement": 0.5894372978022444,
938
+ "update_bound": 8,
939
+ "updates": 1
940
+ },
941
+ {
942
+ "converged_round": 1,
943
+ "epsilon": 0.5,
944
+ "final_nash_gap": 0.75,
945
+ "matrix": [
946
+ 1,
947
+ 0,
948
+ 2,
949
+ 0
950
+ ],
951
+ "min_accepted_improvement": null,
952
+ "update_bound": 4,
953
+ "updates": 0
954
+ },
955
+ {
956
+ "converged_round": 1,
957
+ "epsilon": 0.25,
958
+ "final_nash_gap": 0.5,
959
+ "matrix": [
960
+ 1,
961
+ 0,
962
+ 2,
963
+ 1
964
+ ],
965
+ "min_accepted_improvement": null,
966
+ "update_bound": 8,
967
+ "updates": 0
968
+ },
969
+ {
970
+ "converged_round": 1,
971
+ "epsilon": 0.5,
972
+ "final_nash_gap": 0.5,
973
+ "matrix": [
974
+ 1,
975
+ 0,
976
+ 2,
977
+ 1
978
+ ],
979
+ "min_accepted_improvement": null,
980
+ "update_bound": 4,
981
+ "updates": 0
982
+ },
983
+ {
984
+ "converged_round": 2,
985
+ "epsilon": 0.25,
986
+ "final_nash_gap": 0.2736382857095345,
987
+ "matrix": [
988
+ 1,
989
+ 0,
990
+ 2,
991
+ 2
992
+ ],
993
+ "min_accepted_improvement": 0.4763617142904655,
994
+ "update_bound": 8,
995
+ "updates": 1
996
+ },
997
+ {
998
+ "converged_round": 1,
999
+ "epsilon": 0.5,
1000
+ "final_nash_gap": 0.75,
1001
+ "matrix": [
1002
+ 1,
1003
+ 0,
1004
+ 2,
1005
+ 2
1006
+ ],
1007
+ "min_accepted_improvement": null,
1008
+ "update_bound": 4,
1009
+ "updates": 0
1010
+ },
1011
+ {
1012
+ "converged_round": 1,
1013
+ "epsilon": 0.25,
1014
+ "final_nash_gap": 0.5,
1015
+ "matrix": [
1016
+ 1,
1017
+ 1,
1018
+ 0,
1019
+ 0
1020
+ ],
1021
+ "min_accepted_improvement": null,
1022
+ "update_bound": 4,
1023
+ "updates": 0
1024
+ },
1025
+ {
1026
+ "converged_round": 1,
1027
+ "epsilon": 0.5,
1028
+ "final_nash_gap": 0.5,
1029
+ "matrix": [
1030
+ 1,
1031
+ 1,
1032
+ 0,
1033
+ 0
1034
+ ],
1035
+ "min_accepted_improvement": null,
1036
+ "update_bound": 2,
1037
+ "updates": 0
1038
+ },
1039
+ {
1040
+ "converged_round": 1,
1041
+ "epsilon": 0.25,
1042
+ "final_nash_gap": 0.25,
1043
+ "matrix": [
1044
+ 1,
1045
+ 1,
1046
+ 0,
1047
+ 1
1048
+ ],
1049
+ "min_accepted_improvement": null,
1050
+ "update_bound": 4,
1051
+ "updates": 0
1052
+ },
1053
+ {
1054
+ "converged_round": 1,
1055
+ "epsilon": 0.5,
1056
+ "final_nash_gap": 0.25,
1057
+ "matrix": [
1058
+ 1,
1059
+ 1,
1060
+ 0,
1061
+ 1
1062
+ ],
1063
+ "min_accepted_improvement": null,
1064
+ "update_bound": 2,
1065
+ "updates": 0
1066
+ },
1067
+ {
1068
+ "converged_round": 1,
1069
+ "epsilon": 0.25,
1070
+ "final_nash_gap": 0.5,
1071
+ "matrix": [
1072
+ 1,
1073
+ 1,
1074
+ 0,
1075
+ 2
1076
+ ],
1077
+ "min_accepted_improvement": null,
1078
+ "update_bound": 8,
1079
+ "updates": 0
1080
+ },
1081
+ {
1082
+ "converged_round": 1,
1083
+ "epsilon": 0.5,
1084
+ "final_nash_gap": 0.5,
1085
+ "matrix": [
1086
+ 1,
1087
+ 1,
1088
+ 0,
1089
+ 2
1090
+ ],
1091
+ "min_accepted_improvement": null,
1092
+ "update_bound": 4,
1093
+ "updates": 0
1094
+ },
1095
+ {
1096
+ "converged_round": 1,
1097
+ "epsilon": 0.25,
1098
+ "final_nash_gap": 0.25,
1099
+ "matrix": [
1100
+ 1,
1101
+ 1,
1102
+ 1,
1103
+ 0
1104
+ ],
1105
+ "min_accepted_improvement": null,
1106
+ "update_bound": 4,
1107
+ "updates": 0
1108
+ },
1109
+ {
1110
+ "converged_round": 1,
1111
+ "epsilon": 0.5,
1112
+ "final_nash_gap": 0.25,
1113
+ "matrix": [
1114
+ 1,
1115
+ 1,
1116
+ 1,
1117
+ 0
1118
+ ],
1119
+ "min_accepted_improvement": null,
1120
+ "update_bound": 2,
1121
+ "updates": 0
1122
+ },
1123
+ {
1124
+ "converged_round": 1,
1125
+ "epsilon": 0.25,
1126
+ "final_nash_gap": 0.0,
1127
+ "matrix": [
1128
+ 1,
1129
+ 1,
1130
+ 1,
1131
+ 1
1132
+ ],
1133
+ "min_accepted_improvement": null,
1134
+ "update_bound": 0,
1135
+ "updates": 0
1136
+ },
1137
+ {
1138
+ "converged_round": 1,
1139
+ "epsilon": 0.5,
1140
+ "final_nash_gap": 0.0,
1141
+ "matrix": [
1142
+ 1,
1143
+ 1,
1144
+ 1,
1145
+ 1
1146
+ ],
1147
+ "min_accepted_improvement": null,
1148
+ "update_bound": 0,
1149
+ "updates": 0
1150
+ },
1151
+ {
1152
+ "converged_round": 1,
1153
+ "epsilon": 0.25,
1154
+ "final_nash_gap": 0.25,
1155
+ "matrix": [
1156
+ 1,
1157
+ 1,
1158
+ 1,
1159
+ 2
1160
+ ],
1161
+ "min_accepted_improvement": null,
1162
+ "update_bound": 4,
1163
+ "updates": 0
1164
+ },
1165
+ {
1166
+ "converged_round": 1,
1167
+ "epsilon": 0.5,
1168
+ "final_nash_gap": 0.25,
1169
+ "matrix": [
1170
+ 1,
1171
+ 1,
1172
+ 1,
1173
+ 2
1174
+ ],
1175
+ "min_accepted_improvement": null,
1176
+ "update_bound": 2,
1177
+ "updates": 0
1178
+ },
1179
+ {
1180
+ "converged_round": 1,
1181
+ "epsilon": 0.25,
1182
+ "final_nash_gap": 0.5,
1183
+ "matrix": [
1184
+ 1,
1185
+ 1,
1186
+ 2,
1187
+ 0
1188
+ ],
1189
+ "min_accepted_improvement": null,
1190
+ "update_bound": 8,
1191
+ "updates": 0
1192
+ },
1193
+ {
1194
+ "converged_round": 1,
1195
+ "epsilon": 0.5,
1196
+ "final_nash_gap": 0.5,
1197
+ "matrix": [
1198
+ 1,
1199
+ 1,
1200
+ 2,
1201
+ 0
1202
+ ],
1203
+ "min_accepted_improvement": null,
1204
+ "update_bound": 4,
1205
+ "updates": 0
1206
+ },
1207
+ {
1208
+ "converged_round": 1,
1209
+ "epsilon": 0.25,
1210
+ "final_nash_gap": 0.25,
1211
+ "matrix": [
1212
+ 1,
1213
+ 1,
1214
+ 2,
1215
+ 1
1216
+ ],
1217
+ "min_accepted_improvement": null,
1218
+ "update_bound": 4,
1219
+ "updates": 0
1220
+ },
1221
+ {
1222
+ "converged_round": 1,
1223
+ "epsilon": 0.5,
1224
+ "final_nash_gap": 0.25,
1225
+ "matrix": [
1226
+ 1,
1227
+ 1,
1228
+ 2,
1229
+ 1
1230
+ ],
1231
+ "min_accepted_improvement": null,
1232
+ "update_bound": 2,
1233
+ "updates": 0
1234
+ },
1235
+ {
1236
+ "converged_round": 1,
1237
+ "epsilon": 0.25,
1238
+ "final_nash_gap": 0.5,
1239
+ "matrix": [
1240
+ 1,
1241
+ 1,
1242
+ 2,
1243
+ 2
1244
+ ],
1245
+ "min_accepted_improvement": null,
1246
+ "update_bound": 4,
1247
+ "updates": 0
1248
+ },
1249
+ {
1250
+ "converged_round": 1,
1251
+ "epsilon": 0.5,
1252
+ "final_nash_gap": 0.5,
1253
+ "matrix": [
1254
+ 1,
1255
+ 1,
1256
+ 2,
1257
+ 2
1258
+ ],
1259
+ "min_accepted_improvement": null,
1260
+ "update_bound": 2,
1261
+ "updates": 0
1262
+ },
1263
+ {
1264
+ "converged_round": 2,
1265
+ "epsilon": 0.25,
1266
+ "final_nash_gap": 0.4087872380968218,
1267
+ "matrix": [
1268
+ 1,
1269
+ 2,
1270
+ 0,
1271
+ 0
1272
+ ],
1273
+ "min_accepted_improvement": 0.4763617142904655,
1274
+ "update_bound": 8,
1275
+ "updates": 1
1276
+ },
1277
+ {
1278
+ "converged_round": 1,
1279
+ "epsilon": 0.5,
1280
+ "final_nash_gap": 0.75,
1281
+ "matrix": [
1282
+ 1,
1283
+ 2,
1284
+ 0,
1285
+ 0
1286
+ ],
1287
+ "min_accepted_improvement": null,
1288
+ "update_bound": 4,
1289
+ "updates": 0
1290
+ },
1291
+ {
1292
+ "converged_round": 1,
1293
+ "epsilon": 0.25,
1294
+ "final_nash_gap": 0.5,
1295
+ "matrix": [
1296
+ 1,
1297
+ 2,
1298
+ 0,
1299
+ 1
1300
+ ],
1301
+ "min_accepted_improvement": null,
1302
+ "update_bound": 8,
1303
+ "updates": 0
1304
+ },
1305
+ {
1306
+ "converged_round": 1,
1307
+ "epsilon": 0.5,
1308
+ "final_nash_gap": 0.5,
1309
+ "matrix": [
1310
+ 1,
1311
+ 2,
1312
+ 0,
1313
+ 1
1314
+ ],
1315
+ "min_accepted_improvement": null,
1316
+ "update_bound": 4,
1317
+ "updates": 0
1318
+ },
1319
+ {
1320
+ "converged_round": 3,
1321
+ "epsilon": 0.25,
1322
+ "final_nash_gap": 0.1605627021977556,
1323
+ "matrix": [
1324
+ 1,
1325
+ 2,
1326
+ 0,
1327
+ 2
1328
+ ],
1329
+ "min_accepted_improvement": 0.5894372978022444,
1330
+ "update_bound": 8,
1331
+ "updates": 1
1332
+ },
1333
+ {
1334
+ "converged_round": 1,
1335
+ "epsilon": 0.5,
1336
+ "final_nash_gap": 0.75,
1337
+ "matrix": [
1338
+ 1,
1339
+ 2,
1340
+ 0,
1341
+ 2
1342
+ ],
1343
+ "min_accepted_improvement": null,
1344
+ "update_bound": 4,
1345
+ "updates": 0
1346
+ },
1347
+ {
1348
+ "converged_round": 1,
1349
+ "epsilon": 0.25,
1350
+ "final_nash_gap": 0.5,
1351
+ "matrix": [
1352
+ 1,
1353
+ 2,
1354
+ 1,
1355
+ 0
1356
+ ],
1357
+ "min_accepted_improvement": null,
1358
+ "update_bound": 8,
1359
+ "updates": 0
1360
+ },
1361
+ {
1362
+ "converged_round": 1,
1363
+ "epsilon": 0.5,
1364
+ "final_nash_gap": 0.5,
1365
+ "matrix": [
1366
+ 1,
1367
+ 2,
1368
+ 1,
1369
+ 0
1370
+ ],
1371
+ "min_accepted_improvement": null,
1372
+ "update_bound": 4,
1373
+ "updates": 0
1374
+ },
1375
+ {
1376
+ "converged_round": 1,
1377
+ "epsilon": 0.25,
1378
+ "final_nash_gap": 0.25,
1379
+ "matrix": [
1380
+ 1,
1381
+ 2,
1382
+ 1,
1383
+ 1
1384
+ ],
1385
+ "min_accepted_improvement": null,
1386
+ "update_bound": 4,
1387
+ "updates": 0
1388
+ },
1389
+ {
1390
+ "converged_round": 1,
1391
+ "epsilon": 0.5,
1392
+ "final_nash_gap": 0.25,
1393
+ "matrix": [
1394
+ 1,
1395
+ 2,
1396
+ 1,
1397
+ 1
1398
+ ],
1399
+ "min_accepted_improvement": null,
1400
+ "update_bound": 2,
1401
+ "updates": 0
1402
+ },
1403
+ {
1404
+ "converged_round": 1,
1405
+ "epsilon": 0.25,
1406
+ "final_nash_gap": 0.5,
1407
+ "matrix": [
1408
+ 1,
1409
+ 2,
1410
+ 1,
1411
+ 2
1412
+ ],
1413
+ "min_accepted_improvement": null,
1414
+ "update_bound": 4,
1415
+ "updates": 0
1416
+ },
1417
+ {
1418
+ "converged_round": 1,
1419
+ "epsilon": 0.5,
1420
+ "final_nash_gap": 0.5,
1421
+ "matrix": [
1422
+ 1,
1423
+ 2,
1424
+ 1,
1425
+ 2
1426
+ ],
1427
+ "min_accepted_improvement": null,
1428
+ "update_bound": 2,
1429
+ "updates": 0
1430
+ },
1431
+ {
1432
+ "converged_round": 1,
1433
+ "epsilon": 0.25,
1434
+ "final_nash_gap": 0.25,
1435
+ "matrix": [
1436
+ 1,
1437
+ 2,
1438
+ 2,
1439
+ 0
1440
+ ],
1441
+ "min_accepted_improvement": null,
1442
+ "update_bound": 8,
1443
+ "updates": 0
1444
+ },
1445
+ {
1446
+ "converged_round": 1,
1447
+ "epsilon": 0.5,
1448
+ "final_nash_gap": 0.25,
1449
+ "matrix": [
1450
+ 1,
1451
+ 2,
1452
+ 2,
1453
+ 0
1454
+ ],
1455
+ "min_accepted_improvement": null,
1456
+ "update_bound": 4,
1457
+ "updates": 0
1458
+ },
1459
+ {
1460
+ "converged_round": 1,
1461
+ "epsilon": 0.25,
1462
+ "final_nash_gap": 0.0,
1463
+ "matrix": [
1464
+ 1,
1465
+ 2,
1466
+ 2,
1467
+ 1
1468
+ ],
1469
+ "min_accepted_improvement": null,
1470
+ "update_bound": 4,
1471
+ "updates": 0
1472
+ },
1473
+ {
1474
+ "converged_round": 1,
1475
+ "epsilon": 0.5,
1476
+ "final_nash_gap": 0.0,
1477
+ "matrix": [
1478
+ 1,
1479
+ 2,
1480
+ 2,
1481
+ 1
1482
+ ],
1483
+ "min_accepted_improvement": null,
1484
+ "update_bound": 2,
1485
+ "updates": 0
1486
+ },
1487
+ {
1488
+ "converged_round": 1,
1489
+ "epsilon": 0.25,
1490
+ "final_nash_gap": 0.25,
1491
+ "matrix": [
1492
+ 1,
1493
+ 2,
1494
+ 2,
1495
+ 2
1496
+ ],
1497
+ "min_accepted_improvement": null,
1498
+ "update_bound": 4,
1499
+ "updates": 0
1500
+ },
1501
+ {
1502
+ "converged_round": 1,
1503
+ "epsilon": 0.5,
1504
+ "final_nash_gap": 0.25,
1505
+ "matrix": [
1506
+ 1,
1507
+ 2,
1508
+ 2,
1509
+ 2
1510
+ ],
1511
+ "min_accepted_improvement": null,
1512
+ "update_bound": 2,
1513
+ "updates": 0
1514
+ },
1515
+ {
1516
+ "converged_round": 1,
1517
+ "epsilon": 0.25,
1518
+ "final_nash_gap": 0.5,
1519
+ "matrix": [
1520
+ 2,
1521
+ 0,
1522
+ 0,
1523
+ 0
1524
+ ],
1525
+ "min_accepted_improvement": null,
1526
+ "update_bound": 8,
1527
+ "updates": 0
1528
+ },
1529
+ {
1530
+ "converged_round": 1,
1531
+ "epsilon": 0.5,
1532
+ "final_nash_gap": 0.5,
1533
+ "matrix": [
1534
+ 2,
1535
+ 0,
1536
+ 0,
1537
+ 0
1538
+ ],
1539
+ "min_accepted_improvement": null,
1540
+ "update_bound": 4,
1541
+ "updates": 0
1542
+ },
1543
+ {
1544
+ "converged_round": 1,
1545
+ "epsilon": 0.25,
1546
+ "final_nash_gap": 0.25,
1547
+ "matrix": [
1548
+ 2,
1549
+ 0,
1550
+ 0,
1551
+ 1
1552
+ ],
1553
+ "min_accepted_improvement": null,
1554
+ "update_bound": 8,
1555
+ "updates": 0
1556
+ },
1557
+ {
1558
+ "converged_round": 1,
1559
+ "epsilon": 0.5,
1560
+ "final_nash_gap": 0.25,
1561
+ "matrix": [
1562
+ 2,
1563
+ 0,
1564
+ 0,
1565
+ 1
1566
+ ],
1567
+ "min_accepted_improvement": null,
1568
+ "update_bound": 4,
1569
+ "updates": 0
1570
+ },
1571
+ {
1572
+ "converged_round": 1,
1573
+ "epsilon": 0.25,
1574
+ "final_nash_gap": 0.0,
1575
+ "matrix": [
1576
+ 2,
1577
+ 0,
1578
+ 0,
1579
+ 2
1580
+ ],
1581
+ "min_accepted_improvement": null,
1582
+ "update_bound": 8,
1583
+ "updates": 0
1584
+ },
1585
+ {
1586
+ "converged_round": 1,
1587
+ "epsilon": 0.5,
1588
+ "final_nash_gap": 0.0,
1589
+ "matrix": [
1590
+ 2,
1591
+ 0,
1592
+ 0,
1593
+ 2
1594
+ ],
1595
+ "min_accepted_improvement": null,
1596
+ "update_bound": 4,
1597
+ "updates": 0
1598
+ },
1599
+ {
1600
+ "converged_round": 3,
1601
+ "epsilon": 0.25,
1602
+ "final_nash_gap": 0.4464790992674148,
1603
+ "matrix": [
1604
+ 2,
1605
+ 0,
1606
+ 1,
1607
+ 0
1608
+ ],
1609
+ "min_accepted_improvement": 0.5894372978022444,
1610
+ "update_bound": 8,
1611
+ "updates": 1
1612
+ },
1613
+ {
1614
+ "converged_round": 1,
1615
+ "epsilon": 0.5,
1616
+ "final_nash_gap": 0.75,
1617
+ "matrix": [
1618
+ 2,
1619
+ 0,
1620
+ 1,
1621
+ 0
1622
+ ],
1623
+ "min_accepted_improvement": null,
1624
+ "update_bound": 4,
1625
+ "updates": 0
1626
+ },
1627
+ {
1628
+ "converged_round": 1,
1629
+ "epsilon": 0.25,
1630
+ "final_nash_gap": 0.5,
1631
+ "matrix": [
1632
+ 2,
1633
+ 0,
1634
+ 1,
1635
+ 1
1636
+ ],
1637
+ "min_accepted_improvement": null,
1638
+ "update_bound": 8,
1639
+ "updates": 0
1640
+ },
1641
+ {
1642
+ "converged_round": 1,
1643
+ "epsilon": 0.5,
1644
+ "final_nash_gap": 0.5,
1645
+ "matrix": [
1646
+ 2,
1647
+ 0,
1648
+ 1,
1649
+ 1
1650
+ ],
1651
+ "min_accepted_improvement": null,
1652
+ "update_bound": 4,
1653
+ "updates": 0
1654
+ },
1655
+ {
1656
+ "converged_round": 1,
1657
+ "epsilon": 0.25,
1658
+ "final_nash_gap": 0.25,
1659
+ "matrix": [
1660
+ 2,
1661
+ 0,
1662
+ 1,
1663
+ 2
1664
+ ],
1665
+ "min_accepted_improvement": null,
1666
+ "update_bound": 8,
1667
+ "updates": 0
1668
+ },
1669
+ {
1670
+ "converged_round": 1,
1671
+ "epsilon": 0.5,
1672
+ "final_nash_gap": 0.25,
1673
+ "matrix": [
1674
+ 2,
1675
+ 0,
1676
+ 1,
1677
+ 2
1678
+ ],
1679
+ "min_accepted_improvement": null,
1680
+ "update_bound": 4,
1681
+ "updates": 0
1682
+ },
1683
+ {
1684
+ "converged_round": 3,
1685
+ "epsilon": 0.25,
1686
+ "final_nash_gap": 0.11161443841433938,
1687
+ "matrix": [
1688
+ 2,
1689
+ 0,
1690
+ 2,
1691
+ 0
1692
+ ],
1693
+ "min_accepted_improvement": 0.8883855615856606,
1694
+ "update_bound": 8,
1695
+ "updates": 1
1696
+ },
1697
+ {
1698
+ "converged_round": 1,
1699
+ "epsilon": 0.5,
1700
+ "final_nash_gap": 1.0,
1701
+ "matrix": [
1702
+ 2,
1703
+ 0,
1704
+ 2,
1705
+ 0
1706
+ ],
1707
+ "min_accepted_improvement": null,
1708
+ "update_bound": 4,
1709
+ "updates": 0
1710
+ },
1711
+ {
1712
+ "converged_round": 3,
1713
+ "epsilon": 0.25,
1714
+ "final_nash_gap": 0.1605627021977556,
1715
+ "matrix": [
1716
+ 2,
1717
+ 0,
1718
+ 2,
1719
+ 1
1720
+ ],
1721
+ "min_accepted_improvement": 0.5894372978022444,
1722
+ "update_bound": 8,
1723
+ "updates": 1
1724
+ },
1725
+ {
1726
+ "converged_round": 1,
1727
+ "epsilon": 0.5,
1728
+ "final_nash_gap": 0.75,
1729
+ "matrix": [
1730
+ 2,
1731
+ 0,
1732
+ 2,
1733
+ 1
1734
+ ],
1735
+ "min_accepted_improvement": null,
1736
+ "update_bound": 4,
1737
+ "updates": 0
1738
+ },
1739
+ {
1740
+ "converged_round": 1,
1741
+ "epsilon": 0.25,
1742
+ "final_nash_gap": 0.5,
1743
+ "matrix": [
1744
+ 2,
1745
+ 0,
1746
+ 2,
1747
+ 2
1748
+ ],
1749
+ "min_accepted_improvement": null,
1750
+ "update_bound": 8,
1751
+ "updates": 0
1752
+ },
1753
+ {
1754
+ "converged_round": 1,
1755
+ "epsilon": 0.5,
1756
+ "final_nash_gap": 0.5,
1757
+ "matrix": [
1758
+ 2,
1759
+ 0,
1760
+ 2,
1761
+ 2
1762
+ ],
1763
+ "min_accepted_improvement": null,
1764
+ "update_bound": 4,
1765
+ "updates": 0
1766
+ },
1767
+ {
1768
+ "converged_round": 2,
1769
+ "epsilon": 0.25,
1770
+ "final_nash_gap": 0.4087872380968218,
1771
+ "matrix": [
1772
+ 2,
1773
+ 1,
1774
+ 0,
1775
+ 0
1776
+ ],
1777
+ "min_accepted_improvement": 0.4763617142904655,
1778
+ "update_bound": 8,
1779
+ "updates": 1
1780
+ },
1781
+ {
1782
+ "converged_round": 1,
1783
+ "epsilon": 0.5,
1784
+ "final_nash_gap": 0.75,
1785
+ "matrix": [
1786
+ 2,
1787
+ 1,
1788
+ 0,
1789
+ 0
1790
+ ],
1791
+ "min_accepted_improvement": null,
1792
+ "update_bound": 4,
1793
+ "updates": 0
1794
+ },
1795
+ {
1796
+ "converged_round": 1,
1797
+ "epsilon": 0.25,
1798
+ "final_nash_gap": 0.5,
1799
+ "matrix": [
1800
+ 2,
1801
+ 1,
1802
+ 0,
1803
+ 1
1804
+ ],
1805
+ "min_accepted_improvement": null,
1806
+ "update_bound": 8,
1807
+ "updates": 0
1808
+ },
1809
+ {
1810
+ "converged_round": 1,
1811
+ "epsilon": 0.5,
1812
+ "final_nash_gap": 0.5,
1813
+ "matrix": [
1814
+ 2,
1815
+ 1,
1816
+ 0,
1817
+ 1
1818
+ ],
1819
+ "min_accepted_improvement": null,
1820
+ "update_bound": 4,
1821
+ "updates": 0
1822
+ },
1823
+ {
1824
+ "converged_round": 1,
1825
+ "epsilon": 0.25,
1826
+ "final_nash_gap": 0.25,
1827
+ "matrix": [
1828
+ 2,
1829
+ 1,
1830
+ 0,
1831
+ 2
1832
+ ],
1833
+ "min_accepted_improvement": null,
1834
+ "update_bound": 8,
1835
+ "updates": 0
1836
+ },
1837
+ {
1838
+ "converged_round": 1,
1839
+ "epsilon": 0.5,
1840
+ "final_nash_gap": 0.25,
1841
+ "matrix": [
1842
+ 2,
1843
+ 1,
1844
+ 0,
1845
+ 2
1846
+ ],
1847
+ "min_accepted_improvement": null,
1848
+ "update_bound": 4,
1849
+ "updates": 0
1850
+ },
1851
+ {
1852
+ "converged_round": 1,
1853
+ "epsilon": 0.25,
1854
+ "final_nash_gap": 0.5,
1855
+ "matrix": [
1856
+ 2,
1857
+ 1,
1858
+ 1,
1859
+ 0
1860
+ ],
1861
+ "min_accepted_improvement": null,
1862
+ "update_bound": 8,
1863
+ "updates": 0
1864
+ },
1865
+ {
1866
+ "converged_round": 1,
1867
+ "epsilon": 0.5,
1868
+ "final_nash_gap": 0.5,
1869
+ "matrix": [
1870
+ 2,
1871
+ 1,
1872
+ 1,
1873
+ 0
1874
+ ],
1875
+ "min_accepted_improvement": null,
1876
+ "update_bound": 4,
1877
+ "updates": 0
1878
+ },
1879
+ {
1880
+ "converged_round": 1,
1881
+ "epsilon": 0.25,
1882
+ "final_nash_gap": 0.25,
1883
+ "matrix": [
1884
+ 2,
1885
+ 1,
1886
+ 1,
1887
+ 1
1888
+ ],
1889
+ "min_accepted_improvement": null,
1890
+ "update_bound": 4,
1891
+ "updates": 0
1892
+ },
1893
+ {
1894
+ "converged_round": 1,
1895
+ "epsilon": 0.5,
1896
+ "final_nash_gap": 0.25,
1897
+ "matrix": [
1898
+ 2,
1899
+ 1,
1900
+ 1,
1901
+ 1
1902
+ ],
1903
+ "min_accepted_improvement": null,
1904
+ "update_bound": 2,
1905
+ "updates": 0
1906
+ },
1907
+ {
1908
+ "converged_round": 1,
1909
+ "epsilon": 0.25,
1910
+ "final_nash_gap": 0.0,
1911
+ "matrix": [
1912
+ 2,
1913
+ 1,
1914
+ 1,
1915
+ 2
1916
+ ],
1917
+ "min_accepted_improvement": null,
1918
+ "update_bound": 4,
1919
+ "updates": 0
1920
+ },
1921
+ {
1922
+ "converged_round": 1,
1923
+ "epsilon": 0.5,
1924
+ "final_nash_gap": 0.0,
1925
+ "matrix": [
1926
+ 2,
1927
+ 1,
1928
+ 1,
1929
+ 2
1930
+ ],
1931
+ "min_accepted_improvement": null,
1932
+ "update_bound": 2,
1933
+ "updates": 0
1934
+ },
1935
+ {
1936
+ "converged_round": 3,
1937
+ "epsilon": 0.25,
1938
+ "final_nash_gap": 0.1605627021977556,
1939
+ "matrix": [
1940
+ 2,
1941
+ 1,
1942
+ 2,
1943
+ 0
1944
+ ],
1945
+ "min_accepted_improvement": 0.5894372978022444,
1946
+ "update_bound": 8,
1947
+ "updates": 1
1948
+ },
1949
+ {
1950
+ "converged_round": 1,
1951
+ "epsilon": 0.5,
1952
+ "final_nash_gap": 0.75,
1953
+ "matrix": [
1954
+ 2,
1955
+ 1,
1956
+ 2,
1957
+ 0
1958
+ ],
1959
+ "min_accepted_improvement": null,
1960
+ "update_bound": 4,
1961
+ "updates": 0
1962
+ },
1963
+ {
1964
+ "converged_round": 1,
1965
+ "epsilon": 0.25,
1966
+ "final_nash_gap": 0.5,
1967
+ "matrix": [
1968
+ 2,
1969
+ 1,
1970
+ 2,
1971
+ 1
1972
+ ],
1973
+ "min_accepted_improvement": null,
1974
+ "update_bound": 4,
1975
+ "updates": 0
1976
+ },
1977
+ {
1978
+ "converged_round": 1,
1979
+ "epsilon": 0.5,
1980
+ "final_nash_gap": 0.5,
1981
+ "matrix": [
1982
+ 2,
1983
+ 1,
1984
+ 2,
1985
+ 1
1986
+ ],
1987
+ "min_accepted_improvement": null,
1988
+ "update_bound": 2,
1989
+ "updates": 0
1990
+ },
1991
+ {
1992
+ "converged_round": 1,
1993
+ "epsilon": 0.25,
1994
+ "final_nash_gap": 0.25,
1995
+ "matrix": [
1996
+ 2,
1997
+ 1,
1998
+ 2,
1999
+ 2
2000
+ ],
2001
+ "min_accepted_improvement": null,
2002
+ "update_bound": 4,
2003
+ "updates": 0
2004
+ },
2005
+ {
2006
+ "converged_round": 1,
2007
+ "epsilon": 0.5,
2008
+ "final_nash_gap": 0.25,
2009
+ "matrix": [
2010
+ 2,
2011
+ 1,
2012
+ 2,
2013
+ 2
2014
+ ],
2015
+ "min_accepted_improvement": null,
2016
+ "update_bound": 2,
2017
+ "updates": 0
2018
+ },
2019
+ {
2020
+ "converged_round": 2,
2021
+ "epsilon": 0.25,
2022
+ "final_nash_gap": 0.23840584404423537,
2023
+ "matrix": [
2024
+ 2,
2025
+ 2,
2026
+ 0,
2027
+ 0
2028
+ ],
2029
+ "min_accepted_improvement": 0.7615941559557646,
2030
+ "update_bound": 8,
2031
+ "updates": 1
2032
+ },
2033
+ {
2034
+ "converged_round": 1,
2035
+ "epsilon": 0.5,
2036
+ "final_nash_gap": 1.0,
2037
+ "matrix": [
2038
+ 2,
2039
+ 2,
2040
+ 0,
2041
+ 0
2042
+ ],
2043
+ "min_accepted_improvement": null,
2044
+ "update_bound": 4,
2045
+ "updates": 0
2046
+ },
2047
+ {
2048
+ "converged_round": 2,
2049
+ "epsilon": 0.25,
2050
+ "final_nash_gap": 0.2736382857095345,
2051
+ "matrix": [
2052
+ 2,
2053
+ 2,
2054
+ 0,
2055
+ 1
2056
+ ],
2057
+ "min_accepted_improvement": 0.4763617142904655,
2058
+ "update_bound": 8,
2059
+ "updates": 1
2060
+ },
2061
+ {
2062
+ "converged_round": 1,
2063
+ "epsilon": 0.5,
2064
+ "final_nash_gap": 0.75,
2065
+ "matrix": [
2066
+ 2,
2067
+ 2,
2068
+ 0,
2069
+ 1
2070
+ ],
2071
+ "min_accepted_improvement": null,
2072
+ "update_bound": 4,
2073
+ "updates": 0
2074
+ },
2075
+ {
2076
+ "converged_round": 1,
2077
+ "epsilon": 0.25,
2078
+ "final_nash_gap": 0.5,
2079
+ "matrix": [
2080
+ 2,
2081
+ 2,
2082
+ 0,
2083
+ 2
2084
+ ],
2085
+ "min_accepted_improvement": null,
2086
+ "update_bound": 8,
2087
+ "updates": 0
2088
+ },
2089
+ {
2090
+ "converged_round": 1,
2091
+ "epsilon": 0.5,
2092
+ "final_nash_gap": 0.5,
2093
+ "matrix": [
2094
+ 2,
2095
+ 2,
2096
+ 0,
2097
+ 2
2098
+ ],
2099
+ "min_accepted_improvement": null,
2100
+ "update_bound": 4,
2101
+ "updates": 0
2102
+ },
2103
+ {
2104
+ "converged_round": 2,
2105
+ "epsilon": 0.25,
2106
+ "final_nash_gap": 0.2736382857095345,
2107
+ "matrix": [
2108
+ 2,
2109
+ 2,
2110
+ 1,
2111
+ 0
2112
+ ],
2113
+ "min_accepted_improvement": 0.4763617142904655,
2114
+ "update_bound": 8,
2115
+ "updates": 1
2116
+ },
2117
+ {
2118
+ "converged_round": 1,
2119
+ "epsilon": 0.5,
2120
+ "final_nash_gap": 0.75,
2121
+ "matrix": [
2122
+ 2,
2123
+ 2,
2124
+ 1,
2125
+ 0
2126
+ ],
2127
+ "min_accepted_improvement": null,
2128
+ "update_bound": 4,
2129
+ "updates": 0
2130
+ },
2131
+ {
2132
+ "converged_round": 1,
2133
+ "epsilon": 0.25,
2134
+ "final_nash_gap": 0.5,
2135
+ "matrix": [
2136
+ 2,
2137
+ 2,
2138
+ 1,
2139
+ 1
2140
+ ],
2141
+ "min_accepted_improvement": null,
2142
+ "update_bound": 4,
2143
+ "updates": 0
2144
+ },
2145
+ {
2146
+ "converged_round": 1,
2147
+ "epsilon": 0.5,
2148
+ "final_nash_gap": 0.5,
2149
+ "matrix": [
2150
+ 2,
2151
+ 2,
2152
+ 1,
2153
+ 1
2154
+ ],
2155
+ "min_accepted_improvement": null,
2156
+ "update_bound": 2,
2157
+ "updates": 0
2158
+ },
2159
+ {
2160
+ "converged_round": 1,
2161
+ "epsilon": 0.25,
2162
+ "final_nash_gap": 0.25,
2163
+ "matrix": [
2164
+ 2,
2165
+ 2,
2166
+ 1,
2167
+ 2
2168
+ ],
2169
+ "min_accepted_improvement": null,
2170
+ "update_bound": 4,
2171
+ "updates": 0
2172
+ },
2173
+ {
2174
+ "converged_round": 1,
2175
+ "epsilon": 0.5,
2176
+ "final_nash_gap": 0.25,
2177
+ "matrix": [
2178
+ 2,
2179
+ 2,
2180
+ 1,
2181
+ 2
2182
+ ],
2183
+ "min_accepted_improvement": null,
2184
+ "update_bound": 2,
2185
+ "updates": 0
2186
+ },
2187
+ {
2188
+ "converged_round": 1,
2189
+ "epsilon": 0.25,
2190
+ "final_nash_gap": 0.5,
2191
+ "matrix": [
2192
+ 2,
2193
+ 2,
2194
+ 2,
2195
+ 0
2196
+ ],
2197
+ "min_accepted_improvement": null,
2198
+ "update_bound": 8,
2199
+ "updates": 0
2200
+ },
2201
+ {
2202
+ "converged_round": 1,
2203
+ "epsilon": 0.5,
2204
+ "final_nash_gap": 0.5,
2205
+ "matrix": [
2206
+ 2,
2207
+ 2,
2208
+ 2,
2209
+ 0
2210
+ ],
2211
+ "min_accepted_improvement": null,
2212
+ "update_bound": 4,
2213
+ "updates": 0
2214
+ },
2215
+ {
2216
+ "converged_round": 1,
2217
+ "epsilon": 0.25,
2218
+ "final_nash_gap": 0.25,
2219
+ "matrix": [
2220
+ 2,
2221
+ 2,
2222
+ 2,
2223
+ 1
2224
+ ],
2225
+ "min_accepted_improvement": null,
2226
+ "update_bound": 4,
2227
+ "updates": 0
2228
+ },
2229
+ {
2230
+ "converged_round": 1,
2231
+ "epsilon": 0.5,
2232
+ "final_nash_gap": 0.25,
2233
+ "matrix": [
2234
+ 2,
2235
+ 2,
2236
+ 2,
2237
+ 1
2238
+ ],
2239
+ "min_accepted_improvement": null,
2240
+ "update_bound": 2,
2241
+ "updates": 0
2242
+ },
2243
+ {
2244
+ "converged_round": 1,
2245
+ "epsilon": 0.25,
2246
+ "final_nash_gap": 0.0,
2247
+ "matrix": [
2248
+ 2,
2249
+ 2,
2250
+ 2,
2251
+ 2
2252
+ ],
2253
+ "min_accepted_improvement": null,
2254
+ "update_bound": 0,
2255
+ "updates": 0
2256
+ },
2257
+ {
2258
+ "converged_round": 1,
2259
+ "epsilon": 0.5,
2260
+ "final_nash_gap": 0.0,
2261
+ "matrix": [
2262
+ 2,
2263
+ 2,
2264
+ 2,
2265
+ 2
2266
+ ],
2267
+ "min_accepted_improvement": null,
2268
+ "update_bound": 0,
2269
+ "updates": 0
2270
+ }
2271
+ ],
2272
+ "claim": 3,
2273
+ "factor_two_note": "Theorem 3.4 concludes 2epsilon-Nash; Theorem 1.1 is the epsilon-reparameterized statement.",
2274
+ "gates": {
2275
+ "all_exhaustive_games_converge": true,
2276
+ "all_final_gaps_at_most_two_epsilon": true,
2277
+ "all_update_counts_telescope": true,
2278
+ "every_accepted_update_meets_lazy_threshold": true,
2279
+ "negative_control_cycles_without_alternation": true
2280
+ },
2281
+ "limitations": "Exhaustive over all 81 two-player 2x2 identical-interest payoff tables with entries in {0,1,2}, not all finite potential games.",
2282
+ "negative_controls": {
2283
+ "simultaneous_greedy_response": {
2284
+ "cycles": true,
2285
+ "states": [
2286
+ [
2287
+ 0,
2288
+ 0
2289
+ ],
2290
+ [
2291
+ 1,
2292
+ 1
2293
+ ],
2294
+ [
2295
+ 0,
2296
+ 0
2297
+ ],
2298
+ [
2299
+ 1,
2300
+ 1
2301
+ ],
2302
+ [
2303
+ 0,
2304
+ 0
2305
+ ],
2306
+ [
2307
+ 1,
2308
+ 1
2309
+ ],
2310
+ [
2311
+ 0,
2312
+ 0
2313
+ ],
2314
+ [
2315
+ 1,
2316
+ 1
2317
+ ],
2318
+ [
2319
+ 0,
2320
+ 0
2321
+ ],
2322
+ [
2323
+ 1,
2324
+ 1
2325
+ ],
2326
+ [
2327
+ 0,
2328
+ 0
2329
+ ],
2330
+ [
2331
+ 1,
2332
+ 1
2333
+ ],
2334
+ [
2335
+ 0,
2336
+ 0
2337
+ ],
2338
+ [
2339
+ 1,
2340
+ 1
2341
+ ],
2342
+ [
2343
+ 0,
2344
+ 0
2345
+ ],
2346
+ [
2347
+ 1,
2348
+ 1
2349
+ ],
2350
+ [
2351
+ 0,
2352
+ 0
2353
+ ],
2354
+ [
2355
+ 1,
2356
+ 1
2357
+ ],
2358
+ [
2359
+ 0,
2360
+ 0
2361
+ ],
2362
+ [
2363
+ 1,
2364
+ 1
2365
+ ]
2366
+ ]
2367
+ }
2368
+ },
2369
+ "result_sha256": "6e39ee99d795a9c8681722119e0b38ae3e92d50a477137826d65177eb12b6846",
2370
+ "runtime_seconds": 0.00728992186486721,
2371
+ "source_pdf_sha256": "9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218",
2372
+ "status": "VERIFIED"
2373
+ }
evidence/raw/claim_4.json ADDED
@@ -0,0 +1,25 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "claim": 4,
3
+ "gates": {
4
+ "all_small_step_per_step_inequalities_pass": true,
5
+ "all_small_step_potentials_non_decreasing": true,
6
+ "all_telescoping_bounds_pass": true,
7
+ "large_step_negative_control_has_violations": true
8
+ },
9
+ "limitations": "Finite exhaustive/random numerical audit using exact spectral L. It checks two canonical regularizers and is not a proof of the universal proposition.",
10
+ "negative_controls": {
11
+ "eta_over_L_violation_count": 38
12
+ },
13
+ "result_sha256": "fc4444eb94677d8191fc85b663c274f4fe62bb3ba0d8a61e7687c46edba32ed0",
14
+ "runtime_seconds": 4.859731135889888,
15
+ "seed": 17353,
16
+ "source_pdf_sha256": "9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218",
17
+ "status": "VERIFIED",
18
+ "summary": {
19
+ "large_step_cases": 896,
20
+ "large_step_violation_count": 38,
21
+ "maximum_path_to_bound_ratio": 0.34596427985662237,
22
+ "minimum_slack": -2.842193761588269e-14,
23
+ "positive_cases": 672
24
+ }
25
+ }
evidence/raw/claim_5.json ADDED
@@ -0,0 +1,518 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "claim": 5,
3
+ "gates": {
4
+ "all_errors_are_O_one_over_t": true,
5
+ "all_movements_are_O_one_over_t": true,
6
+ "all_post_switch_strategies_highly_suboptimal": true,
7
+ "reset_history_causes_order_one_movement": true
8
+ },
9
+ "limitations": "Finite logarithmic t-grid. Entropy and Euclidean formulas are closed form; the log-barrier optimum is solved to a fixed 160 bisection steps.",
10
+ "negative_controls": {
11
+ "history_resets": [
12
+ {
13
+ "regularizer": "entropy",
14
+ "reset_movement": 1.4621171572600098
15
+ },
16
+ {
17
+ "regularizer": "euclidean",
18
+ "reset_movement": 2.0
19
+ },
20
+ {
21
+ "regularizer": "log_barrier",
22
+ "reset_movement": 1.2360660701529758
23
+ }
24
+ ]
25
+ },
26
+ "records": [
27
+ {
28
+ "error": 0.00033535013046637197,
29
+ "error_times_t": 0.0026828010437309757,
30
+ "movement_times_t": 0.009211217022947185,
31
+ "post_switch_best_response_gap": 0.9990889488055994,
32
+ "post_switch_movement": 0.001151402127868398,
33
+ "regularizer": "entropy",
34
+ "t": 8
35
+ },
36
+ {
37
+ "error": 1.1253516207787584e-07,
38
+ "error_times_t": 1.8005625932460134e-06,
39
+ "movement_times_t": 6.187746076579416e-06,
40
+ "post_switch_best_response_gap": 0.999999694097773,
41
+ "post_switch_movement": 3.867341297862135e-07,
42
+ "regularizer": "entropy",
43
+ "t": 16
44
+ },
45
+ {
46
+ "error": 1.2656542480726785e-14,
47
+ "error_times_t": 4.050093593832571e-13,
48
+ "movement_times_t": 1.3926712581842444e-12,
49
+ "post_switch_best_response_gap": 0.9999999999999656,
50
+ "post_switch_movement": 4.352097681825764e-14,
51
+ "regularizer": "entropy",
52
+ "t": 32
53
+ },
54
+ {
55
+ "error": 0.0,
56
+ "error_times_t": 0.0,
57
+ "movement_times_t": 1.7637114300892435e-26,
58
+ "post_switch_best_response_gap": 1.0,
59
+ "post_switch_movement": 2.755799109514443e-28,
60
+ "regularizer": "entropy",
61
+ "t": 64
62
+ },
63
+ {
64
+ "error": 0.0,
65
+ "error_times_t": 0.0,
66
+ "movement_times_t": 5.657319198724482e-54,
67
+ "post_switch_best_response_gap": 1.0,
68
+ "post_switch_movement": 4.419780624003502e-56,
69
+ "regularizer": "entropy",
70
+ "t": 128
71
+ },
72
+ {
73
+ "error": 0.0,
74
+ "error_times_t": 0.0,
75
+ "movement_times_t": 2.9103618933977983e-109,
76
+ "post_switch_best_response_gap": 1.0,
77
+ "post_switch_movement": 1.136860114608515e-111,
78
+ "regularizer": "entropy",
79
+ "t": 256
80
+ },
81
+ {
82
+ "error": 0.0,
83
+ "error_times_t": 0.0,
84
+ "movement_times_t": 3.851142811243827e-220,
85
+ "post_switch_best_response_gap": 1.0,
86
+ "post_switch_movement": 7.5217633032106e-223,
87
+ "regularizer": "entropy",
88
+ "t": 512
89
+ },
90
+ {
91
+ "error": 0.0,
92
+ "error_times_t": 0.0,
93
+ "movement_times_t": 0.0,
94
+ "post_switch_best_response_gap": 1.0,
95
+ "post_switch_movement": 0.0,
96
+ "regularizer": "entropy",
97
+ "t": 1024
98
+ },
99
+ {
100
+ "error": 0.0,
101
+ "error_times_t": 0.0,
102
+ "movement_times_t": 0.0,
103
+ "post_switch_best_response_gap": 1.0,
104
+ "post_switch_movement": 0.0,
105
+ "regularizer": "entropy",
106
+ "t": 2048
107
+ },
108
+ {
109
+ "error": 0.0,
110
+ "error_times_t": 0.0,
111
+ "movement_times_t": 0.0,
112
+ "post_switch_best_response_gap": 1.0,
113
+ "post_switch_movement": 0.0,
114
+ "regularizer": "entropy",
115
+ "t": 4096
116
+ },
117
+ {
118
+ "error": 0.0,
119
+ "error_times_t": 0.0,
120
+ "movement_times_t": 0.0,
121
+ "post_switch_best_response_gap": 1.0,
122
+ "post_switch_movement": 0.0,
123
+ "regularizer": "entropy",
124
+ "t": 8192
125
+ },
126
+ {
127
+ "error": 0.0,
128
+ "error_times_t": 0.0,
129
+ "movement_times_t": 0.0,
130
+ "post_switch_best_response_gap": 1.0,
131
+ "post_switch_movement": 0.0,
132
+ "regularizer": "entropy",
133
+ "t": 16384
134
+ },
135
+ {
136
+ "error": 0.0,
137
+ "error_times_t": 0.0,
138
+ "movement_times_t": 0.0,
139
+ "post_switch_best_response_gap": 1.0,
140
+ "post_switch_movement": 0.0,
141
+ "regularizer": "entropy",
142
+ "t": 32768
143
+ },
144
+ {
145
+ "error": 0.0,
146
+ "error_times_t": 0.0,
147
+ "movement_times_t": 0.0,
148
+ "post_switch_best_response_gap": 1.0,
149
+ "post_switch_movement": 0.0,
150
+ "regularizer": "entropy",
151
+ "t": 65536
152
+ },
153
+ {
154
+ "error": 0.0,
155
+ "error_times_t": 0.0,
156
+ "movement_times_t": 0.0,
157
+ "post_switch_best_response_gap": 1.0,
158
+ "post_switch_movement": 0.0,
159
+ "regularizer": "entropy",
160
+ "t": 131072
161
+ },
162
+ {
163
+ "error": 0.0,
164
+ "error_times_t": 0.0,
165
+ "movement_times_t": 0.0,
166
+ "post_switch_best_response_gap": 1.0,
167
+ "post_switch_movement": 0.0,
168
+ "regularizer": "entropy",
169
+ "t": 262144
170
+ },
171
+ {
172
+ "error": 0.0,
173
+ "error_times_t": 0.0,
174
+ "movement_times_t": 0.0,
175
+ "post_switch_best_response_gap": 1.0,
176
+ "post_switch_movement": 0.0,
177
+ "regularizer": "entropy",
178
+ "t": 524288
179
+ },
180
+ {
181
+ "error": 0.0,
182
+ "error_times_t": 0.0,
183
+ "movement_times_t": 0.0,
184
+ "post_switch_best_response_gap": 1.0,
185
+ "post_switch_movement": 0.0,
186
+ "regularizer": "entropy",
187
+ "t": 1048576
188
+ },
189
+ {
190
+ "error": 0.0,
191
+ "error_times_t": 0.0,
192
+ "movement_times_t": 0.0,
193
+ "post_switch_best_response_gap": 1.0,
194
+ "post_switch_movement": 0.0,
195
+ "regularizer": "euclidean",
196
+ "t": 8
197
+ },
198
+ {
199
+ "error": 0.0,
200
+ "error_times_t": 0.0,
201
+ "movement_times_t": 0.0,
202
+ "post_switch_best_response_gap": 1.0,
203
+ "post_switch_movement": 0.0,
204
+ "regularizer": "euclidean",
205
+ "t": 16
206
+ },
207
+ {
208
+ "error": 0.0,
209
+ "error_times_t": 0.0,
210
+ "movement_times_t": 0.0,
211
+ "post_switch_best_response_gap": 1.0,
212
+ "post_switch_movement": 0.0,
213
+ "regularizer": "euclidean",
214
+ "t": 32
215
+ },
216
+ {
217
+ "error": 0.0,
218
+ "error_times_t": 0.0,
219
+ "movement_times_t": 0.0,
220
+ "post_switch_best_response_gap": 1.0,
221
+ "post_switch_movement": 0.0,
222
+ "regularizer": "euclidean",
223
+ "t": 64
224
+ },
225
+ {
226
+ "error": 0.0,
227
+ "error_times_t": 0.0,
228
+ "movement_times_t": 0.0,
229
+ "post_switch_best_response_gap": 1.0,
230
+ "post_switch_movement": 0.0,
231
+ "regularizer": "euclidean",
232
+ "t": 128
233
+ },
234
+ {
235
+ "error": 0.0,
236
+ "error_times_t": 0.0,
237
+ "movement_times_t": 0.0,
238
+ "post_switch_best_response_gap": 1.0,
239
+ "post_switch_movement": 0.0,
240
+ "regularizer": "euclidean",
241
+ "t": 256
242
+ },
243
+ {
244
+ "error": 0.0,
245
+ "error_times_t": 0.0,
246
+ "movement_times_t": 0.0,
247
+ "post_switch_best_response_gap": 1.0,
248
+ "post_switch_movement": 0.0,
249
+ "regularizer": "euclidean",
250
+ "t": 512
251
+ },
252
+ {
253
+ "error": 0.0,
254
+ "error_times_t": 0.0,
255
+ "movement_times_t": 0.0,
256
+ "post_switch_best_response_gap": 1.0,
257
+ "post_switch_movement": 0.0,
258
+ "regularizer": "euclidean",
259
+ "t": 1024
260
+ },
261
+ {
262
+ "error": 0.0,
263
+ "error_times_t": 0.0,
264
+ "movement_times_t": 0.0,
265
+ "post_switch_best_response_gap": 1.0,
266
+ "post_switch_movement": 0.0,
267
+ "regularizer": "euclidean",
268
+ "t": 2048
269
+ },
270
+ {
271
+ "error": 0.0,
272
+ "error_times_t": 0.0,
273
+ "movement_times_t": 0.0,
274
+ "post_switch_best_response_gap": 1.0,
275
+ "post_switch_movement": 0.0,
276
+ "regularizer": "euclidean",
277
+ "t": 4096
278
+ },
279
+ {
280
+ "error": 0.0,
281
+ "error_times_t": 0.0,
282
+ "movement_times_t": 0.0,
283
+ "post_switch_best_response_gap": 1.0,
284
+ "post_switch_movement": 0.0,
285
+ "regularizer": "euclidean",
286
+ "t": 8192
287
+ },
288
+ {
289
+ "error": 0.0,
290
+ "error_times_t": 0.0,
291
+ "movement_times_t": 0.0,
292
+ "post_switch_best_response_gap": 1.0,
293
+ "post_switch_movement": 0.0,
294
+ "regularizer": "euclidean",
295
+ "t": 16384
296
+ },
297
+ {
298
+ "error": 0.0,
299
+ "error_times_t": 0.0,
300
+ "movement_times_t": 0.0,
301
+ "post_switch_best_response_gap": 1.0,
302
+ "post_switch_movement": 0.0,
303
+ "regularizer": "euclidean",
304
+ "t": 32768
305
+ },
306
+ {
307
+ "error": 0.0,
308
+ "error_times_t": 0.0,
309
+ "movement_times_t": 0.0,
310
+ "post_switch_best_response_gap": 1.0,
311
+ "post_switch_movement": 0.0,
312
+ "regularizer": "euclidean",
313
+ "t": 65536
314
+ },
315
+ {
316
+ "error": 0.0,
317
+ "error_times_t": 0.0,
318
+ "movement_times_t": 0.0,
319
+ "post_switch_best_response_gap": 1.0,
320
+ "post_switch_movement": 0.0,
321
+ "regularizer": "euclidean",
322
+ "t": 131072
323
+ },
324
+ {
325
+ "error": 0.0,
326
+ "error_times_t": 0.0,
327
+ "movement_times_t": 0.0,
328
+ "post_switch_best_response_gap": 1.0,
329
+ "post_switch_movement": 0.0,
330
+ "regularizer": "euclidean",
331
+ "t": 262144
332
+ },
333
+ {
334
+ "error": 0.0,
335
+ "error_times_t": 0.0,
336
+ "movement_times_t": 0.0,
337
+ "post_switch_best_response_gap": 1.0,
338
+ "post_switch_movement": 0.0,
339
+ "regularizer": "euclidean",
340
+ "t": 524288
341
+ },
342
+ {
343
+ "error": 0.0,
344
+ "error_times_t": 0.0,
345
+ "movement_times_t": 0.0,
346
+ "post_switch_best_response_gap": 1.0,
347
+ "post_switch_movement": 0.0,
348
+ "regularizer": "euclidean",
349
+ "t": 1048576
350
+ },
351
+ {
352
+ "error": 0.10961179679779254,
353
+ "error_times_t": 0.8768943743823403,
354
+ "movement_times_t": 0.2117999492004401,
355
+ "post_switch_best_response_gap": 0.87715070637718,
356
+ "post_switch_movement": 0.026474993650055012,
357
+ "regularizer": "log_barrier",
358
+ "t": 8
359
+ },
360
+ {
361
+ "error": 0.05860889073134068,
362
+ "error_times_t": 0.9377422517014509,
363
+ "movement_times_t": 0.11625314948076948,
364
+ "post_switch_best_response_gap": 0.9377581983473853,
365
+ "post_switch_movement": 0.0072658218425480925,
366
+ "regularizer": "log_barrier",
367
+ "t": 16
368
+ },
369
+ {
370
+ "error": 0.030274389316206296,
371
+ "error_times_t": 0.9687804581186015,
372
+ "movement_times_t": 0.060427074453755836,
373
+ "post_switch_best_response_gap": 0.9687814376454538,
374
+ "post_switch_movement": 0.0018883460766798699,
375
+ "regularizer": "log_barrier",
376
+ "t": 32
377
+ },
378
+ {
379
+ "error": 0.015380918950558709,
380
+ "error_times_t": 0.9843788128357573,
381
+ "movement_times_t": 0.03074659042732719,
382
+ "post_switch_best_response_gap": 0.9843788733117278,
383
+ "post_switch_movement": 0.00048041547542698737,
384
+ "regularizer": "log_barrier",
385
+ "t": 64
386
+ },
387
+ {
388
+ "error": 0.007751468568585551,
389
+ "error_times_t": 0.9921879767789505,
390
+ "movement_times_t": 0.015501030140001149,
391
+ "post_switch_best_response_gap": 0.9921879805324301,
392
+ "post_switch_movement": 0.00012110179796875897,
393
+ "regularizer": "log_barrier",
394
+ "t": 128
395
+ },
396
+ {
397
+ "error": 0.0038909914437610382,
398
+ "error_times_t": 0.9960938096028258,
399
+ "movement_times_t": 0.007781744479871122,
400
+ "post_switch_best_response_gap": 0.9960938098365517,
401
+ "post_switch_movement": 3.039743937449657e-05,
402
+ "regularizer": "log_barrier",
403
+ "t": 256
404
+ },
405
+ {
406
+ "error": 0.0019493103172862902,
407
+ "error_times_t": 0.9980468824505806,
408
+ "movement_times_t": 0.0038985908324775664,
409
+ "post_switch_best_response_gap": 0.9980468824651039,
410
+ "post_switch_movement": 7.614435219682747e-06,
411
+ "regularizer": "log_barrier",
412
+ "t": 512
413
+ },
414
+ {
415
+ "error": 0.0009756088265930885,
416
+ "error_times_t": 0.9990234384313226,
417
+ "movement_times_t": 0.0019512139278958784,
418
+ "post_switch_best_response_gap": 0.9990234384322312,
419
+ "post_switch_movement": 1.9054823514608188e-06,
420
+ "regularizer": "log_barrier",
421
+ "t": 1024
422
+ },
423
+ {
424
+ "error": 0.00048804283147774186,
425
+ "error_times_t": 0.9995117188664153,
426
+ "movement_times_t": 0.0009760851971805096,
427
+ "post_switch_best_response_gap": 0.9995117188664722,
428
+ "post_switch_movement": 4.766041001857957e-07,
429
+ "regularizer": "log_barrier",
430
+ "t": 2048
431
+ },
432
+ {
433
+ "error": 0.00024408102035877732,
434
+ "error_times_t": 0.9997558593895519,
435
+ "movement_times_t": 0.0004881619825027883,
436
+ "post_switch_best_response_gap": 0.9997558593895555,
437
+ "post_switch_movement": 1.191801715094698e-07,
438
+ "regularizer": "log_barrier",
439
+ "t": 4096
440
+ },
441
+ {
442
+ "error": 0.0001220554113390282,
443
+ "error_times_t": 0.999877929689319,
444
+ "movement_times_t": 0.0002441108154016547,
445
+ "post_switch_best_response_gap": 0.9998779296893192,
446
+ "post_switch_movement": 2.97986835207098e-08,
447
+ "regularizer": "log_barrier",
448
+ "t": 8192
449
+ },
450
+ {
451
+ "error": 6.103143095970154e-05,
452
+ "error_times_t": 0.99993896484375,
453
+ "movement_times_t": 0.00012206286191940308,
454
+ "post_switch_best_response_gap": 0.9999389648439774,
455
+ "post_switch_movement": 7.450125849572942e-09,
456
+ "regularizer": "log_barrier",
457
+ "t": 16384
458
+ },
459
+ {
460
+ "error": 3.0516646802425385e-05,
461
+ "error_times_t": 0.999969482421875,
462
+ "movement_times_t": 6.103329360485077e-05,
463
+ "post_switch_best_response_gap": 0.9999694824219034,
464
+ "post_switch_movement": 1.8625883058120962e-09,
465
+ "regularizer": "log_barrier",
466
+ "t": 32768
467
+ },
468
+ {
469
+ "error": 1.5258556231856346e-05,
470
+ "error_times_t": 0.9999847412109375,
471
+ "movement_times_t": 3.0517112463712692e-05,
472
+ "post_switch_best_response_gap": 0.999984741210941,
473
+ "post_switch_movement": 4.6565418188038166e-10,
474
+ "regularizer": "log_barrier",
475
+ "t": 65536
476
+ },
477
+ {
478
+ "error": 7.6293363235890865e-06,
479
+ "error_times_t": 0.9999923706054688,
480
+ "movement_times_t": 1.5258672647178173e-05,
481
+ "post_switch_best_response_gap": 0.9999923706054692,
482
+ "post_switch_movement": 1.1641443364851511e-10,
483
+ "regularizer": "log_barrier",
484
+ "t": 131072
485
+ },
486
+ {
487
+ "error": 3.8146827137097716e-06,
488
+ "error_times_t": 0.9999961853027344,
489
+ "movement_times_t": 7.62939453125e-06,
490
+ "post_switch_best_response_gap": 0.9999961853027344,
491
+ "post_switch_movement": 2.9103830456733704e-11,
492
+ "regularizer": "log_barrier",
493
+ "t": 262144
494
+ },
495
+ {
496
+ "error": 1.907344994833693e-06,
497
+ "error_times_t": 0.9999980926513672,
498
+ "movement_times_t": 3.814697265625e-06,
499
+ "post_switch_best_response_gap": 0.9999980926513672,
500
+ "post_switch_movement": 7.275957614183426e-12,
501
+ "regularizer": "log_barrier",
502
+ "t": 524288
503
+ },
504
+ {
505
+ "error": 9.536734069115482e-07,
506
+ "error_times_t": 0.9999990463256836,
507
+ "movement_times_t": 1.9073486328125e-06,
508
+ "post_switch_best_response_gap": 0.9999990463256836,
509
+ "post_switch_movement": 1.8189894035458565e-12,
510
+ "regularizer": "log_barrier",
511
+ "t": 1048576
512
+ }
513
+ ],
514
+ "result_sha256": "b9a8f75ff333deb870670567c7585a7a40bf3b84ff51f655960d64c68d803918",
515
+ "runtime_seconds": 0.004537451080977917,
516
+ "source_pdf_sha256": "9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218",
517
+ "status": "VERIFIED"
518
+ }
evidence/raw/claim_6.json ADDED
@@ -0,0 +1,755 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "claim": 6,
3
+ "cross_claim_evidence": "Claim 1 direct trajectories audit Property 4.3 end-to-end on the paper's A construction.",
4
+ "figure_2b_gamma_100": [
5
+ [
6
+ 3,
7
+ 0,
8
+ 0,
9
+ 0,
10
+ -3
11
+ ],
12
+ [
13
+ 0,
14
+ 7,
15
+ 0,
16
+ 6,
17
+ -113
18
+ ],
19
+ [
20
+ 0,
21
+ 8,
22
+ 9,
23
+ 0,
24
+ -117
25
+ ],
26
+ [
27
+ 4,
28
+ 0,
29
+ 0,
30
+ 5,
31
+ -109
32
+ ],
33
+ [
34
+ -7,
35
+ -115,
36
+ -109,
37
+ -111,
38
+ -200
39
+ ]
40
+ ],
41
+ "gates": {
42
+ "all_structural_checks_pass": true,
43
+ "figure_2b_exact": true,
44
+ "lemma_4_1_has_no_violations": true,
45
+ "lemma_4_1_has_non_vacuous_cells": true,
46
+ "negative_controls_activate": true,
47
+ "period_count_source_correction_verified": true
48
+ },
49
+ "lemma_4_1": {
50
+ "cells": 72960,
51
+ "non_vacuous_cells": 50502,
52
+ "shrunk_range_factor": 0.24994473008507856,
53
+ "shrunk_range_would_violate": true,
54
+ "tightest_probability_to_bound_ratio": 0.4998894601701571,
55
+ "violations": 0
56
+ },
57
+ "limitations": "Property 4.3 is directly exercised on the small exact trajectories in Claim 1; the broad m-grid checks its combinatorial prerequisites.",
58
+ "negative_controls": {
59
+ "matrix_mutation_caught": true,
60
+ "range_shrink_caught": true
61
+ },
62
+ "result_sha256": "ab4d7056f7456bb024fc1a6e40dde6c905f39d848c2f8aa6d1a35a4ef9d0c1dd",
63
+ "runtime_seconds": 0.3144537899643183,
64
+ "source_correction": "Labels k=3,...,2m-1 give 2m-3 periods, not 2m-1 periods.",
65
+ "source_pdf_sha256": "9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218",
66
+ "status": "VERIFIED",
67
+ "structural": [
68
+ {
69
+ "even_row_shift": true,
70
+ "extra_row_column_negative": true,
71
+ "m": 5,
72
+ "matrix_sha256": "64c34943af681e8fb23b913f016955f05c8f3b48fd8266a41901edd00a66ab42",
73
+ "max_entry": 9,
74
+ "odd_column_shift": true,
75
+ "period_count": 7,
76
+ "period_labels": [
77
+ 3,
78
+ 9
79
+ ],
80
+ "positive_entries_exact": true
81
+ },
82
+ {
83
+ "even_row_shift": true,
84
+ "extra_row_column_negative": true,
85
+ "m": 7,
86
+ "matrix_sha256": "bfde8d870b91a7b586ed1f800a6190d84ad3c7f5fe55f1087853f544d0a85175",
87
+ "max_entry": 13,
88
+ "odd_column_shift": true,
89
+ "period_count": 11,
90
+ "period_labels": [
91
+ 3,
92
+ 13
93
+ ],
94
+ "positive_entries_exact": true
95
+ },
96
+ {
97
+ "even_row_shift": true,
98
+ "extra_row_column_negative": true,
99
+ "m": 9,
100
+ "matrix_sha256": "00d60a5c2f07b36ba4180ee88420d23e9943374e5ba9b4c5567c6e205a2a5d54",
101
+ "max_entry": 17,
102
+ "odd_column_shift": true,
103
+ "period_count": 15,
104
+ "period_labels": [
105
+ 3,
106
+ 17
107
+ ],
108
+ "positive_entries_exact": true
109
+ },
110
+ {
111
+ "even_row_shift": true,
112
+ "extra_row_column_negative": true,
113
+ "m": 11,
114
+ "matrix_sha256": "4fe2312a7b45ee23c3d7c00e1d203f544123531e9313489d47a697a915e95288",
115
+ "max_entry": 21,
116
+ "odd_column_shift": true,
117
+ "period_count": 19,
118
+ "period_labels": [
119
+ 3,
120
+ 21
121
+ ],
122
+ "positive_entries_exact": true
123
+ },
124
+ {
125
+ "even_row_shift": true,
126
+ "extra_row_column_negative": true,
127
+ "m": 13,
128
+ "matrix_sha256": "026fcae2763b627effea9123dc4888a9f99f5d6e2852348b73c81290607e1e6c",
129
+ "max_entry": 25,
130
+ "odd_column_shift": true,
131
+ "period_count": 23,
132
+ "period_labels": [
133
+ 3,
134
+ 25
135
+ ],
136
+ "positive_entries_exact": true
137
+ },
138
+ {
139
+ "even_row_shift": true,
140
+ "extra_row_column_negative": true,
141
+ "m": 15,
142
+ "matrix_sha256": "4a2d7e36087e90e1c0b51cebd46828c55c42b00c4eb9d714ca51b8fdd0f252c2",
143
+ "max_entry": 29,
144
+ "odd_column_shift": true,
145
+ "period_count": 27,
146
+ "period_labels": [
147
+ 3,
148
+ 29
149
+ ],
150
+ "positive_entries_exact": true
151
+ },
152
+ {
153
+ "even_row_shift": true,
154
+ "extra_row_column_negative": true,
155
+ "m": 17,
156
+ "matrix_sha256": "1fbe76ca9b80b7ca8bc572e1dbe002c54777455e200739d0399b68f94cfbdfab",
157
+ "max_entry": 33,
158
+ "odd_column_shift": true,
159
+ "period_count": 31,
160
+ "period_labels": [
161
+ 3,
162
+ 33
163
+ ],
164
+ "positive_entries_exact": true
165
+ },
166
+ {
167
+ "even_row_shift": true,
168
+ "extra_row_column_negative": true,
169
+ "m": 19,
170
+ "matrix_sha256": "4037b114be6f471221d1ffb569fe5e46c2dcb3e81c63c5ed8af13148fdd1198a",
171
+ "max_entry": 37,
172
+ "odd_column_shift": true,
173
+ "period_count": 35,
174
+ "period_labels": [
175
+ 3,
176
+ 37
177
+ ],
178
+ "positive_entries_exact": true
179
+ },
180
+ {
181
+ "even_row_shift": true,
182
+ "extra_row_column_negative": true,
183
+ "m": 21,
184
+ "matrix_sha256": "c98534adf4573aa23a7bb854204b980c132c3a1d8acdda7b2555edfe673bee7f",
185
+ "max_entry": 41,
186
+ "odd_column_shift": true,
187
+ "period_count": 39,
188
+ "period_labels": [
189
+ 3,
190
+ 41
191
+ ],
192
+ "positive_entries_exact": true
193
+ },
194
+ {
195
+ "even_row_shift": true,
196
+ "extra_row_column_negative": true,
197
+ "m": 23,
198
+ "matrix_sha256": "fb1b9015aec30f727588448aecb70af76e23a9a3ff435d2829314b8691683b9f",
199
+ "max_entry": 45,
200
+ "odd_column_shift": true,
201
+ "period_count": 43,
202
+ "period_labels": [
203
+ 3,
204
+ 45
205
+ ],
206
+ "positive_entries_exact": true
207
+ },
208
+ {
209
+ "even_row_shift": true,
210
+ "extra_row_column_negative": true,
211
+ "m": 25,
212
+ "matrix_sha256": "a4ab1b74ecd50ca71ba0ab0c887b8ccb7864dd02b633f416c67d82ebfd0ec800",
213
+ "max_entry": 49,
214
+ "odd_column_shift": true,
215
+ "period_count": 47,
216
+ "period_labels": [
217
+ 3,
218
+ 49
219
+ ],
220
+ "positive_entries_exact": true
221
+ },
222
+ {
223
+ "even_row_shift": true,
224
+ "extra_row_column_negative": true,
225
+ "m": 27,
226
+ "matrix_sha256": "153aac373a33dbbc1f89a01a241086491610951cf31c68d60159a0f38a2a6b35",
227
+ "max_entry": 53,
228
+ "odd_column_shift": true,
229
+ "period_count": 51,
230
+ "period_labels": [
231
+ 3,
232
+ 53
233
+ ],
234
+ "positive_entries_exact": true
235
+ },
236
+ {
237
+ "even_row_shift": true,
238
+ "extra_row_column_negative": true,
239
+ "m": 29,
240
+ "matrix_sha256": "c02476a8e7ed9f1e87cca89457e99d0e590394fa3b3e81ef6597a82013671dab",
241
+ "max_entry": 57,
242
+ "odd_column_shift": true,
243
+ "period_count": 55,
244
+ "period_labels": [
245
+ 3,
246
+ 57
247
+ ],
248
+ "positive_entries_exact": true
249
+ },
250
+ {
251
+ "even_row_shift": true,
252
+ "extra_row_column_negative": true,
253
+ "m": 31,
254
+ "matrix_sha256": "ee79dd31121b747bb69c4419e7c6fb47d254ca8f8b42028b62494836f61f4a21",
255
+ "max_entry": 61,
256
+ "odd_column_shift": true,
257
+ "period_count": 59,
258
+ "period_labels": [
259
+ 3,
260
+ 61
261
+ ],
262
+ "positive_entries_exact": true
263
+ },
264
+ {
265
+ "even_row_shift": true,
266
+ "extra_row_column_negative": true,
267
+ "m": 33,
268
+ "matrix_sha256": "36b3b59495cca1a8baa512f40370f3ef8256307d075c15ee65e3e0f00689c8fe",
269
+ "max_entry": 65,
270
+ "odd_column_shift": true,
271
+ "period_count": 63,
272
+ "period_labels": [
273
+ 3,
274
+ 65
275
+ ],
276
+ "positive_entries_exact": true
277
+ },
278
+ {
279
+ "even_row_shift": true,
280
+ "extra_row_column_negative": true,
281
+ "m": 35,
282
+ "matrix_sha256": "b1f3146d3ba5b3629b83ac2fc49f0e4bc4a922042f1ea060337f883cd4254324",
283
+ "max_entry": 69,
284
+ "odd_column_shift": true,
285
+ "period_count": 67,
286
+ "period_labels": [
287
+ 3,
288
+ 69
289
+ ],
290
+ "positive_entries_exact": true
291
+ },
292
+ {
293
+ "even_row_shift": true,
294
+ "extra_row_column_negative": true,
295
+ "m": 37,
296
+ "matrix_sha256": "64e4d7c875dcaf99c4c3d61868f43ca53196c5f2f765e926d8e1546dfd25411b",
297
+ "max_entry": 73,
298
+ "odd_column_shift": true,
299
+ "period_count": 71,
300
+ "period_labels": [
301
+ 3,
302
+ 73
303
+ ],
304
+ "positive_entries_exact": true
305
+ },
306
+ {
307
+ "even_row_shift": true,
308
+ "extra_row_column_negative": true,
309
+ "m": 39,
310
+ "matrix_sha256": "25dd22866e6751fad15333ad404dfe32ad39d22b8607450560202043a29623a4",
311
+ "max_entry": 77,
312
+ "odd_column_shift": true,
313
+ "period_count": 75,
314
+ "period_labels": [
315
+ 3,
316
+ 77
317
+ ],
318
+ "positive_entries_exact": true
319
+ },
320
+ {
321
+ "even_row_shift": true,
322
+ "extra_row_column_negative": true,
323
+ "m": 41,
324
+ "matrix_sha256": "d3d7f688f7b6ce057f38ad4f2abc71c35c4910e66199963ced4b598f9025d7e7",
325
+ "max_entry": 81,
326
+ "odd_column_shift": true,
327
+ "period_count": 79,
328
+ "period_labels": [
329
+ 3,
330
+ 81
331
+ ],
332
+ "positive_entries_exact": true
333
+ },
334
+ {
335
+ "even_row_shift": true,
336
+ "extra_row_column_negative": true,
337
+ "m": 43,
338
+ "matrix_sha256": "af1dbb736abd6fab7f696882de9f03ec92851f3a7836cfc62cac6e9b67ae6048",
339
+ "max_entry": 85,
340
+ "odd_column_shift": true,
341
+ "period_count": 83,
342
+ "period_labels": [
343
+ 3,
344
+ 85
345
+ ],
346
+ "positive_entries_exact": true
347
+ },
348
+ {
349
+ "even_row_shift": true,
350
+ "extra_row_column_negative": true,
351
+ "m": 45,
352
+ "matrix_sha256": "c8aa670a6477c90719515b08ffedf6658b2b4d9072b18c352312e6807dfa9a70",
353
+ "max_entry": 89,
354
+ "odd_column_shift": true,
355
+ "period_count": 87,
356
+ "period_labels": [
357
+ 3,
358
+ 89
359
+ ],
360
+ "positive_entries_exact": true
361
+ },
362
+ {
363
+ "even_row_shift": true,
364
+ "extra_row_column_negative": true,
365
+ "m": 47,
366
+ "matrix_sha256": "b437e623b2e4c9f2d837c5a3d80b084243f17f1a76c1d0f389c489fd22e0111e",
367
+ "max_entry": 93,
368
+ "odd_column_shift": true,
369
+ "period_count": 91,
370
+ "period_labels": [
371
+ 3,
372
+ 93
373
+ ],
374
+ "positive_entries_exact": true
375
+ },
376
+ {
377
+ "even_row_shift": true,
378
+ "extra_row_column_negative": true,
379
+ "m": 49,
380
+ "matrix_sha256": "7971fb6bd070b0bfe4f1d47837cf2de929174fea1cfcf07f4694be6f4ed15c35",
381
+ "max_entry": 97,
382
+ "odd_column_shift": true,
383
+ "period_count": 95,
384
+ "period_labels": [
385
+ 3,
386
+ 97
387
+ ],
388
+ "positive_entries_exact": true
389
+ },
390
+ {
391
+ "even_row_shift": true,
392
+ "extra_row_column_negative": true,
393
+ "m": 51,
394
+ "matrix_sha256": "b9647ebdd4e81ae1e7e0fd70ffe8a52ff77a640e74c1a324669fe4741db1cfd4",
395
+ "max_entry": 101,
396
+ "odd_column_shift": true,
397
+ "period_count": 99,
398
+ "period_labels": [
399
+ 3,
400
+ 101
401
+ ],
402
+ "positive_entries_exact": true
403
+ },
404
+ {
405
+ "even_row_shift": true,
406
+ "extra_row_column_negative": true,
407
+ "m": 53,
408
+ "matrix_sha256": "95e62497ef2f7efad1f2382a462b954435c33b58c583b0592fd638a9c9ee7cb3",
409
+ "max_entry": 105,
410
+ "odd_column_shift": true,
411
+ "period_count": 103,
412
+ "period_labels": [
413
+ 3,
414
+ 105
415
+ ],
416
+ "positive_entries_exact": true
417
+ },
418
+ {
419
+ "even_row_shift": true,
420
+ "extra_row_column_negative": true,
421
+ "m": 55,
422
+ "matrix_sha256": "ddea9e882bec41c0815f0898581d29636f6f285a2b5d2308def7c33036d27f45",
423
+ "max_entry": 109,
424
+ "odd_column_shift": true,
425
+ "period_count": 107,
426
+ "period_labels": [
427
+ 3,
428
+ 109
429
+ ],
430
+ "positive_entries_exact": true
431
+ },
432
+ {
433
+ "even_row_shift": true,
434
+ "extra_row_column_negative": true,
435
+ "m": 57,
436
+ "matrix_sha256": "166c96f69ea4df1dc89209fc570cebb82f7c9127c8129f4f245483a81e0668b6",
437
+ "max_entry": 113,
438
+ "odd_column_shift": true,
439
+ "period_count": 111,
440
+ "period_labels": [
441
+ 3,
442
+ 113
443
+ ],
444
+ "positive_entries_exact": true
445
+ },
446
+ {
447
+ "even_row_shift": true,
448
+ "extra_row_column_negative": true,
449
+ "m": 59,
450
+ "matrix_sha256": "8984e07cf3369f71de8f8edaa74f51ba0e46c2a248140990aef6841fe85246a0",
451
+ "max_entry": 117,
452
+ "odd_column_shift": true,
453
+ "period_count": 115,
454
+ "period_labels": [
455
+ 3,
456
+ 117
457
+ ],
458
+ "positive_entries_exact": true
459
+ },
460
+ {
461
+ "even_row_shift": true,
462
+ "extra_row_column_negative": true,
463
+ "m": 61,
464
+ "matrix_sha256": "2a380066779682895da08e478ac222f3ef3bad6b2991b00cd8a5ffef79d5ad6e",
465
+ "max_entry": 121,
466
+ "odd_column_shift": true,
467
+ "period_count": 119,
468
+ "period_labels": [
469
+ 3,
470
+ 121
471
+ ],
472
+ "positive_entries_exact": true
473
+ },
474
+ {
475
+ "even_row_shift": true,
476
+ "extra_row_column_negative": true,
477
+ "m": 63,
478
+ "matrix_sha256": "2019d2eda7624eb6be6ca3b174a28aca372ddc07e56f698a1c3d6b07ccf8aa67",
479
+ "max_entry": 125,
480
+ "odd_column_shift": true,
481
+ "period_count": 123,
482
+ "period_labels": [
483
+ 3,
484
+ 125
485
+ ],
486
+ "positive_entries_exact": true
487
+ },
488
+ {
489
+ "even_row_shift": true,
490
+ "extra_row_column_negative": true,
491
+ "m": 65,
492
+ "matrix_sha256": "8fce60ab4db255fac187736afbe8040e0f103c05d6156f27c696c689d62509fe",
493
+ "max_entry": 129,
494
+ "odd_column_shift": true,
495
+ "period_count": 127,
496
+ "period_labels": [
497
+ 3,
498
+ 129
499
+ ],
500
+ "positive_entries_exact": true
501
+ },
502
+ {
503
+ "even_row_shift": true,
504
+ "extra_row_column_negative": true,
505
+ "m": 67,
506
+ "matrix_sha256": "be19289c66ed9d3017cc57f861b0604a00f88d1e6c1d251e6592ec59b647a77b",
507
+ "max_entry": 133,
508
+ "odd_column_shift": true,
509
+ "period_count": 131,
510
+ "period_labels": [
511
+ 3,
512
+ 133
513
+ ],
514
+ "positive_entries_exact": true
515
+ },
516
+ {
517
+ "even_row_shift": true,
518
+ "extra_row_column_negative": true,
519
+ "m": 69,
520
+ "matrix_sha256": "9ca307b2fa339a9b595f4a03a66946f5bfb9e87f402cbaf18eb0d0d7297ef8af",
521
+ "max_entry": 137,
522
+ "odd_column_shift": true,
523
+ "period_count": 135,
524
+ "period_labels": [
525
+ 3,
526
+ 137
527
+ ],
528
+ "positive_entries_exact": true
529
+ },
530
+ {
531
+ "even_row_shift": true,
532
+ "extra_row_column_negative": true,
533
+ "m": 71,
534
+ "matrix_sha256": "14e1c3b50dd11acd9f94c92a5d846235aca9261e66d1038485c5eaf4a803c227",
535
+ "max_entry": 141,
536
+ "odd_column_shift": true,
537
+ "period_count": 139,
538
+ "period_labels": [
539
+ 3,
540
+ 141
541
+ ],
542
+ "positive_entries_exact": true
543
+ },
544
+ {
545
+ "even_row_shift": true,
546
+ "extra_row_column_negative": true,
547
+ "m": 73,
548
+ "matrix_sha256": "d08d72e3577356897c52389796648ec1ce9491c44ca0a0af03cde38c213736d6",
549
+ "max_entry": 145,
550
+ "odd_column_shift": true,
551
+ "period_count": 143,
552
+ "period_labels": [
553
+ 3,
554
+ 145
555
+ ],
556
+ "positive_entries_exact": true
557
+ },
558
+ {
559
+ "even_row_shift": true,
560
+ "extra_row_column_negative": true,
561
+ "m": 75,
562
+ "matrix_sha256": "6fcabfa0476529c65e3a70abd537c2f5c9ee30bcc35e9a209cff90c0dedc64c1",
563
+ "max_entry": 149,
564
+ "odd_column_shift": true,
565
+ "period_count": 147,
566
+ "period_labels": [
567
+ 3,
568
+ 149
569
+ ],
570
+ "positive_entries_exact": true
571
+ },
572
+ {
573
+ "even_row_shift": true,
574
+ "extra_row_column_negative": true,
575
+ "m": 77,
576
+ "matrix_sha256": "f6e6c748a56873fc5301ea239da89cff8f9011c1b927375699bc15f556894d3d",
577
+ "max_entry": 153,
578
+ "odd_column_shift": true,
579
+ "period_count": 151,
580
+ "period_labels": [
581
+ 3,
582
+ 153
583
+ ],
584
+ "positive_entries_exact": true
585
+ },
586
+ {
587
+ "even_row_shift": true,
588
+ "extra_row_column_negative": true,
589
+ "m": 79,
590
+ "matrix_sha256": "5e2ac4f1af95651b1e97233380a9702cc588f10b5f1dc89699e169adc69611f2",
591
+ "max_entry": 157,
592
+ "odd_column_shift": true,
593
+ "period_count": 155,
594
+ "period_labels": [
595
+ 3,
596
+ 157
597
+ ],
598
+ "positive_entries_exact": true
599
+ },
600
+ {
601
+ "even_row_shift": true,
602
+ "extra_row_column_negative": true,
603
+ "m": 81,
604
+ "matrix_sha256": "b4f59efa1caa8eb5a5f5a5b7f8345c6127e1f49fa9768722da762281388cde27",
605
+ "max_entry": 161,
606
+ "odd_column_shift": true,
607
+ "period_count": 159,
608
+ "period_labels": [
609
+ 3,
610
+ 161
611
+ ],
612
+ "positive_entries_exact": true
613
+ },
614
+ {
615
+ "even_row_shift": true,
616
+ "extra_row_column_negative": true,
617
+ "m": 83,
618
+ "matrix_sha256": "ee9efda7c8eb692756ec376cafac56945d396597ce63eb18a3af0a56f70ea001",
619
+ "max_entry": 165,
620
+ "odd_column_shift": true,
621
+ "period_count": 163,
622
+ "period_labels": [
623
+ 3,
624
+ 165
625
+ ],
626
+ "positive_entries_exact": true
627
+ },
628
+ {
629
+ "even_row_shift": true,
630
+ "extra_row_column_negative": true,
631
+ "m": 85,
632
+ "matrix_sha256": "d272d28a65d6c2dd4c4d745d7f17a694274e95c3d34fbfe64afc4ad1fecc29ff",
633
+ "max_entry": 169,
634
+ "odd_column_shift": true,
635
+ "period_count": 167,
636
+ "period_labels": [
637
+ 3,
638
+ 169
639
+ ],
640
+ "positive_entries_exact": true
641
+ },
642
+ {
643
+ "even_row_shift": true,
644
+ "extra_row_column_negative": true,
645
+ "m": 87,
646
+ "matrix_sha256": "25094aa7e4639be2709d2d15154cfc10ce79976a398d8f9c9b254dda0e38c7c9",
647
+ "max_entry": 173,
648
+ "odd_column_shift": true,
649
+ "period_count": 171,
650
+ "period_labels": [
651
+ 3,
652
+ 173
653
+ ],
654
+ "positive_entries_exact": true
655
+ },
656
+ {
657
+ "even_row_shift": true,
658
+ "extra_row_column_negative": true,
659
+ "m": 89,
660
+ "matrix_sha256": "160a2c8e2329867b32a992fb4b87f65eaa2a80b6d3f5f3d7576a02ffd2f34be8",
661
+ "max_entry": 177,
662
+ "odd_column_shift": true,
663
+ "period_count": 175,
664
+ "period_labels": [
665
+ 3,
666
+ 177
667
+ ],
668
+ "positive_entries_exact": true
669
+ },
670
+ {
671
+ "even_row_shift": true,
672
+ "extra_row_column_negative": true,
673
+ "m": 91,
674
+ "matrix_sha256": "45a2b3eeb463666a0d46dfda72a592bcb25e1ee77b546ebb82c8f0c23523ca79",
675
+ "max_entry": 181,
676
+ "odd_column_shift": true,
677
+ "period_count": 179,
678
+ "period_labels": [
679
+ 3,
680
+ 181
681
+ ],
682
+ "positive_entries_exact": true
683
+ },
684
+ {
685
+ "even_row_shift": true,
686
+ "extra_row_column_negative": true,
687
+ "m": 93,
688
+ "matrix_sha256": "abef506bbf098d952272798aa3d30c266d1a368c0f3b58dcc4a94603f5a6e99c",
689
+ "max_entry": 185,
690
+ "odd_column_shift": true,
691
+ "period_count": 183,
692
+ "period_labels": [
693
+ 3,
694
+ 185
695
+ ],
696
+ "positive_entries_exact": true
697
+ },
698
+ {
699
+ "even_row_shift": true,
700
+ "extra_row_column_negative": true,
701
+ "m": 95,
702
+ "matrix_sha256": "ed83641c82f17bb7db70cb80ba512e205aa4395164721fc4713e711d111424db",
703
+ "max_entry": 189,
704
+ "odd_column_shift": true,
705
+ "period_count": 187,
706
+ "period_labels": [
707
+ 3,
708
+ 189
709
+ ],
710
+ "positive_entries_exact": true
711
+ },
712
+ {
713
+ "even_row_shift": true,
714
+ "extra_row_column_negative": true,
715
+ "m": 97,
716
+ "matrix_sha256": "c0b5094ceda2ae95ac03607fd90e9effa0afcf2ecb7672e76f6a6beeb849136d",
717
+ "max_entry": 193,
718
+ "odd_column_shift": true,
719
+ "period_count": 191,
720
+ "period_labels": [
721
+ 3,
722
+ 193
723
+ ],
724
+ "positive_entries_exact": true
725
+ },
726
+ {
727
+ "even_row_shift": true,
728
+ "extra_row_column_negative": true,
729
+ "m": 99,
730
+ "matrix_sha256": "672d7c55bd6b7b456298a268fa87335f78810e2a54ee665b92a2a1d0e1ef75d9",
731
+ "max_entry": 197,
732
+ "odd_column_shift": true,
733
+ "period_count": 195,
734
+ "period_labels": [
735
+ 3,
736
+ 197
737
+ ],
738
+ "positive_entries_exact": true
739
+ },
740
+ {
741
+ "even_row_shift": true,
742
+ "extra_row_column_negative": true,
743
+ "m": 101,
744
+ "matrix_sha256": "0bc17e2483a065cf32444566bb57f6a02b88bfe077b694c722b0bd75dccceb02",
745
+ "max_entry": 201,
746
+ "odd_column_shift": true,
747
+ "period_count": 199,
748
+ "period_labels": [
749
+ 3,
750
+ 201
751
+ ],
752
+ "positive_entries_exact": true
753
+ }
754
+ ]
755
+ }
evidence/raw/job_manifest.json ADDED
@@ -0,0 +1,28 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "commands": [
3
+ [
4
+ "/root/.cache/uv/environments-v2/hf-job-ca539c160c0ca8be/bin/python",
5
+ "repro/run_claims.py",
6
+ "--out",
7
+ "/tmp/ftrl-repro-a60xjgc_/repo/evidence/raw"
8
+ ],
9
+ [
10
+ "/root/.cache/uv/environments-v2/hf-job-ca539c160c0ca8be/bin/python",
11
+ "repro/verify_evidence.py",
12
+ "--evidence",
13
+ "/tmp/ftrl-repro-a60xjgc_/repo/evidence/raw",
14
+ "--write",
15
+ "/tmp/ftrl-repro-a60xjgc_/repo/evidence/verification/cumulative.json"
16
+ ]
17
+ ],
18
+ "cpu_count": 64,
19
+ "flavor": "cpu-upgrade",
20
+ "job_id": "6a6cde22a00abefd4b289f1a",
21
+ "repository": "https://github.com/MachineLearning-Nerd/icml26-repro-l6KZJO7w48-doubly-exponential-lower-bounds-for-follow-the-regularized-leader-in-potenti.git",
22
+ "repository_sha": "1f50506f5c18e2ade654515e6d8a2e533c0afc2f",
23
+ "run_stdout_tail": "claim 1: VERIFIED (225.032s)\nclaim 2: VERIFIED (11.334s)\nclaim 3: VERIFIED (0.007s)\nclaim 4: VERIFIED (4.860s)\nclaim 5: VERIFIED (0.005s)\nclaim 6: VERIFIED (0.314s)\n{\"backend\": \"huggingface-jobs\", \"claims\": [{\"claim\": 1, \"result_sha256\": \"55c94bd08adb809b85617bd7a2e2bb973ed1509391ac7200b1f1e47ed58a9eb8\", \"runtime_seconds\": 225.03165634302422, \"status\": \"VERIFIED\"}, {\"claim\": 2, \"result_sha256\": \"e8f2d32160fb43829919a63d9cdaf5602e45fec015aec9f6aa4fb8391178f8e8\", \"runtime_seconds\": 11.333668414037675, \"status\": \"VERIFIED\"}, {\"claim\": 3, \"result_sha256\": \"6e39ee99d795a9c8681722119e0b38ae3e92d50a477137826d65177eb12b6846\", \"runtime_seconds\": 0.00728992186486721, \"status\": \"VERIFIED\"}, {\"claim\": 4, \"result_sha256\": \"fc4444eb94677d8191fc85b663c274f4fe62bb3ba0d8a61e7687c46edba32ed0\", \"runtime_seconds\": 4.859731135889888, \"status\": \"VERIFIED\"}, {\"claim\": 5, \"result_sha256\": \"b9a8f75ff333deb870670567c7585a7a40bf3b84ff51f655960d64c68d803918\", \"runtime_seconds\": 0.004537451080977917, \"status\": \"VERIFIED\"}, {\"claim\": 6, \"result_sha256\": \"ab4d7056f7456bb024fc1a6e40dde6c905f39d848c2f8aa6d1a35a4ef9d0c1dd\", \"runtime_seconds\": 0.3144537899643183, \"status\": \"VERIFIED\"}], \"cpu_count\": 64, \"manifest_sha256\": \"59a7c14fdbe448f7b2ad597f4d2a3ea712e1c339cd167be40b7fc99de6dba737\", \"numpy\": \"2.3.2\", \"paper\": \"2601.23248v1\", \"platform\": \"Linux-6.12.90-120.164.amzn2023.x86_64-x86_64-with-glibc2.36\", \"python\": \"3.12.12\", \"required_flavor\": \"cpu-upgrade\", \"schema_version\": 1, \"seed\": 17353, \"source_pdf_sha256\": \"9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218\", \"total_runtime_seconds\": 241.5836918039713}\n",
24
+ "runtime_seconds": 242.8970369210001,
25
+ "schema_version": 1,
26
+ "space_id": "DineshAI/l6KZJO7w48",
27
+ "verify_stdout": "{\"claims_verified\": [{\"checks\": [\"lower-bound exponent recalculated\", \"right-censored transitions are consecutive\", \"2,000,000-round preterminal Nash gaps checked\"], \"claim\": 1}, {\"checks\": [\"all snakes independently checked for chords\", \"factorial recurrence recomputed\"], \"claim\": 2}, {\"checks\": [\"162 exhaustive cases present\", \"2epsilon and update bounds checked\"], \"claim\": 3}, {\"checks\": [\"per-step slack threshold checked\", \"large-step violations required\"], \"claim\": 4}, {\"checks\": [\"all 54 history points checked\", \"reset-history controls checked\"], \"claim\": 5}, {\"checks\": [\"Figure 2b exact\", \"period count source correction checked\", \"Lemma 4.1 non-vacuous\"], \"claim\": 6}], \"runtime_seconds\": 0.01477994304150343, \"schema_version\": 1, \"sha256\": \"b8cc0e1128542bd5357fcf1b94ec4a1f2783313331c97273cac6ffc2b14d11ef\", \"source_pdf_sha256\": \"9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218\", \"verdict\": \"PASS\"}\n"
28
+ }
evidence/raw/run_manifest.json ADDED
@@ -0,0 +1,52 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "backend": "huggingface-jobs",
3
+ "claims": [
4
+ {
5
+ "claim": 1,
6
+ "result_sha256": "55c94bd08adb809b85617bd7a2e2bb973ed1509391ac7200b1f1e47ed58a9eb8",
7
+ "runtime_seconds": 225.03165634302422,
8
+ "status": "VERIFIED"
9
+ },
10
+ {
11
+ "claim": 2,
12
+ "result_sha256": "e8f2d32160fb43829919a63d9cdaf5602e45fec015aec9f6aa4fb8391178f8e8",
13
+ "runtime_seconds": 11.333668414037675,
14
+ "status": "VERIFIED"
15
+ },
16
+ {
17
+ "claim": 3,
18
+ "result_sha256": "6e39ee99d795a9c8681722119e0b38ae3e92d50a477137826d65177eb12b6846",
19
+ "runtime_seconds": 0.00728992186486721,
20
+ "status": "VERIFIED"
21
+ },
22
+ {
23
+ "claim": 4,
24
+ "result_sha256": "fc4444eb94677d8191fc85b663c274f4fe62bb3ba0d8a61e7687c46edba32ed0",
25
+ "runtime_seconds": 4.859731135889888,
26
+ "status": "VERIFIED"
27
+ },
28
+ {
29
+ "claim": 5,
30
+ "result_sha256": "b9a8f75ff333deb870670567c7585a7a40bf3b84ff51f655960d64c68d803918",
31
+ "runtime_seconds": 0.004537451080977917,
32
+ "status": "VERIFIED"
33
+ },
34
+ {
35
+ "claim": 6,
36
+ "result_sha256": "ab4d7056f7456bb024fc1a6e40dde6c905f39d848c2f8aa6d1a35a4ef9d0c1dd",
37
+ "runtime_seconds": 0.3144537899643183,
38
+ "status": "VERIFIED"
39
+ }
40
+ ],
41
+ "cpu_count": 64,
42
+ "manifest_sha256": "59a7c14fdbe448f7b2ad597f4d2a3ea712e1c339cd167be40b7fc99de6dba737",
43
+ "numpy": "2.3.2",
44
+ "paper": "2601.23248v1",
45
+ "platform": "Linux-6.12.90-120.164.amzn2023.x86_64-x86_64-with-glibc2.36",
46
+ "python": "3.12.12",
47
+ "required_flavor": "cpu-upgrade",
48
+ "schema_version": 1,
49
+ "seed": 17353,
50
+ "source_pdf_sha256": "9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218",
51
+ "total_runtime_seconds": 241.5836918039713
52
+ }
evidence/verification/cumulative.json ADDED
@@ -0,0 +1,53 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "claims_verified": [
3
+ {
4
+ "checks": [
5
+ "lower-bound exponent recalculated",
6
+ "right-censored transitions are consecutive",
7
+ "2,000,000-round preterminal Nash gaps checked"
8
+ ],
9
+ "claim": 1
10
+ },
11
+ {
12
+ "checks": [
13
+ "all snakes independently checked for chords",
14
+ "factorial recurrence recomputed"
15
+ ],
16
+ "claim": 2
17
+ },
18
+ {
19
+ "checks": [
20
+ "162 exhaustive cases present",
21
+ "2epsilon and update bounds checked"
22
+ ],
23
+ "claim": 3
24
+ },
25
+ {
26
+ "checks": [
27
+ "per-step slack threshold checked",
28
+ "large-step violations required"
29
+ ],
30
+ "claim": 4
31
+ },
32
+ {
33
+ "checks": [
34
+ "all 54 history points checked",
35
+ "reset-history controls checked"
36
+ ],
37
+ "claim": 5
38
+ },
39
+ {
40
+ "checks": [
41
+ "Figure 2b exact",
42
+ "period count source correction checked",
43
+ "Lemma 4.1 non-vacuous"
44
+ ],
45
+ "claim": 6
46
+ }
47
+ ],
48
+ "runtime_seconds": 0.01477994304150343,
49
+ "schema_version": 1,
50
+ "sha256": "b8cc0e1128542bd5357fcf1b94ec4a1f2783313331c97273cac6ffc2b14d11ef",
51
+ "source_pdf_sha256": "9c43d01c6c7b13230dd89d9744aaf7802c197bdc478ba0045883c0b96edd8218",
52
+ "verdict": "PASS"
53
+ }