algo/v0/complétion_minorée_m15_vers_m16.md

Complétion par frères au palier 2^16 (m=15 → m=16)

Introduction

Ce document poursuit la stratégie de preuve par registre K : il traite la transition m=15 → m=16 en séparant les parents en cas « one » (un seul enfant non couvert) et « both » (deux enfants non couverts), puis en indiquant la complétion systématique des cas « one » par clauses de descente minorées (lemme de frère).

Données de départ (issues du registre exact)

• Palier m=15 : |R_15| = 1345 (résidus impairs non couverts modulo 2^15 = 32768).
• Palier m=16 : |R_16| = 2446 (résidus impairs non couverts modulo 2^16 = 65536).

Chaque parent r ∈ R_15 a deux enfants au palier suivant : r et r + 2^15.

Décomposition parents → enfants (m=15 → m=16)

• Parents « zero » (0 enfant non couvert) : 0
• Parents « one » (1 enfant non couvert) : 244
• Parents « both » (2 enfants non couverts): 1101

Vérification de cohérence :

• |R_16| = 2*|both| + |one| = 2*1101 + 244 = 2446

Conséquence après complétion par clauses minorées

Le lemme de frère implique que chaque enfant « one » est fermable au même palier par une clause de descente minorée, dès que l’horizon k de la clause exacte qui ferme le frère satisfait 3^k < 2^16 (condition automatiquement vraie pour k ≤ 10).

Après ajout des clauses minorées correspondantes :

• résidu restant au palier 2^16 : |R_16^comp| = 2*|both| = 2202

Coefficient de survie sur la transition :

• sans complétion : q_15 = |R_16|/(2|R_15|) = 2446/(2*1345) = 0.9092936802973978
• avec complétion : q_15^comp = |R_16^comp|/(2|R_15|) = 2202/(2*1345) = 0.8185873605947955

Répartition modulo 32 (parents)

Parents « one » par classe modulo 32 :

• 7 : 57
• 15 : 41
• 27 : 57
• 31 : 89

Parents « both » par classe modulo 32 :

• 7 : 213
• 15 : 168
• 27 : 213
• 31 : 507

Listes exhaustives

Parents « one » au palier 2^15 (244 résidus modulo 32768)

127, 303, 415, 583, 623, 831, 839, 943, 1095, 1151, 1247, 1275, 1327, 1567, 1647, 1727 1775, 1887, 2119, 2271, 2299, 2303, 2331, 2351, 2471, 2591, 2719, 2727, 2799, 2831, 2983, 3135 3163, 3295, 3303, 3455, 3611, 3743, 4007, 4031, 4079, 4159, 4187, 4199, 4287, 4479, 4655, 5023 5103, 5183, 5231, 5311, 5599, 5631, 5787, 6047, 6127, 6175, 6255, 6503, 6651, 6759, 6783, 6907 7071, 7163, 7199, 7487, 7495, 7783, 8063, 8187, 8431, 8539, 8795, 9051, 9087, 9371, 9375, 9383 9711, 9959, 10075, 10267, 10287, 10607, 10655, 10735, 10863, 11079, 11231, 11311, 11551, 11567, 11679, 11807 11967, 12255, 12415, 12511, 12543, 12571, 12827, 12967, 13119, 13383, 13563, 13567, 13695, 13851, 14023, 14063 14143, 14271, 14399, 14407, 14439, 14895, 15007, 15271, 15295, 15343, 15431, 15451, 15591, 15839, 15911, 15919 16027, 16287, 16295, 16575, 16615, 16743, 16863, 17147, 17319, 17519, 17599, 17727, 17735, 17767, 17799, 18463 18623, 18751, 19035, 19047, 19199, 19451, 19623, 20071, 20091, 20199, 20271, 20351, 20475, 20507, 20527, 20783 20927, 21095, 21103, 21215, 21223, 21339, 21471, 21499, 21659, 21727, 21807, 21979, 21999, 22047, 22207, 22363 22655, 22683, 22751, 22811, 22943, 23103, 23231, 23359, 23367, 23387, 23399, 23615, 23803, 23835, 23867, 23935 24303, 24391, 24559, 24639, 24647, 24679, 24831, 25115, 25247, 25255, 25503, 25575, 25583, 25691, 25703, 25831 26139, 26267, 26279, 26527, 26535, 26559, 27163, 27183, 27291, 27759, 27839, 27975, 28127, 28703, 28999, 29031 29435, 29467, 29863, 30015, 30311, 30459, 30591, 30715, 30747, 30767, 30887, 31323, 31643, 31711, 31771, 31899 32239, 32347, 32487, 32603

Enfants « one » au palier 2^16 (244 résidus modulo 65536)

415, 831, 1151, 1247, 1327, 1567, 1647, 1727, 1887, 2119, 2299, 2303, 2719, 2799, 3135, 3295 3455, 3611, 4007, 4031, 4187, 4287, 5023, 5103, 5183, 5599, 6175, 6255, 6651, 6907, 7071, 7163 7487, 7783, 8063, 8539, 8795, 9051, 9371, 9375, 9711, 10267, 10287, 10607, 10863, 11079, 11679, 12255 12415, 12511, 12571, 12827, 12967, 13383, 13563, 13567, 14023, 14063, 14143, 14399, 14439, 14895, 15271, 15295 15431, 15451, 15591, 15911, 16027, 16287, 16743, 16863, 17319, 17519, 17599, 17735, 17799, 18751, 19047, 19623 20199, 20475, 21095, 21215, 21471, 21727, 21807, 22047, 22207, 22363, 22683, 22943, 23103, 23359, 23399, 23615 23835, 23935, 24391, 24647, 24831, 25247, 25503, 25583, 25703, 26139, 26279, 26535, 26559, 27291, 28127, 28703 28999, 29435, 30015, 30311, 30591, 30747, 30767, 30887, 31323, 31643, 31899, 32239, 32895, 33071, 33351, 33391 33607, 33711, 33863, 34043, 34543, 35039, 35099, 35119, 35239, 35359, 35495, 35599, 35751, 35931, 36071, 36511 36847, 36927, 36967, 37247, 37423, 37999, 38079, 38399, 38555, 38815, 38895, 39271, 39527, 39551, 39967, 40263 40955, 41199, 41855, 42151, 42727, 42843, 43423, 43503, 43999, 44079, 44319, 44335, 44575, 44735, 45311, 45887 46463, 46619, 47039, 47175, 47775, 48111, 48607, 48687, 49063, 49343, 49383, 49915, 50495, 50535, 51231, 51391 51803, 51967, 52219, 52839, 52859, 53039, 53119, 53275, 53295, 53551, 53695, 53871, 53991, 54107, 54267, 54427 54747, 54767, 55423, 55519, 55579, 55999, 56135, 56155, 56571, 56635, 57071, 57327, 57407, 57447, 57883, 58023 58343, 58459, 58599, 59035, 59295, 59931, 59951, 60527, 60607, 60743, 61799, 62235, 62631, 63227, 63483, 64479 64539, 65115, 65255, 65371

Parents « both » au palier 2^15 (1101 résidus modulo 32768)

27, 31, 47, 63, 71, 91, 103, 111, 159, 167, 223, 239, 251, 283, 319, 327 359, 447, 479, 495, 511, 559, 603, 639, 667, 671, 703, 743, 751, 763, 767, 795 859, 871, 895, 927, 959, 991, 1007, 1023, 1051, 1055, 1115, 1127, 1179, 1183, 1255, 1263 1279, 1307, 1343, 1383, 1407, 1439, 1471, 1503, 1519, 1535, 1583, 1639, 1663, 1691, 1695, 1767 1791, 1819, 1883, 1895, 1919, 1951, 1959, 2043, 2047, 2111, 2139, 2151, 2159, 2175, 2207, 2215 2287, 2367, 2375, 2407, 2463, 2495, 2527, 2543, 2559, 2651, 2671, 2687, 2715, 2751, 2791, 2811 2843, 2879, 2887, 2919, 2943, 3007, 3055, 3071, 3099, 3103, 3175, 3183, 3199, 3227, 3231, 3263 3311, 3323, 3327, 3355, 3375, 3391, 3399, 3431, 3487, 3519, 3567, 3583, 3615, 3631, 3711, 3739 3775, 3815, 3823, 3839, 3867, 3931, 3943, 3999, 4095, 4127, 4167, 4207, 4255, 4263, 4319, 4335 4347, 4351, 4379, 4399, 4415, 4423, 4511, 4575, 4591, 4607, 4635, 4699, 4719, 4735, 4763, 4767 4775, 4799, 4839, 4847, 4863, 4891, 4935, 4967, 4991, 5055, 5119, 5147, 5151, 5211, 5223, 5247 5279, 5287, 5343, 5351, 5359, 5375, 5403, 5423, 5447, 5479, 5503, 5535, 5567, 5659, 5663, 5679 5735, 5759, 5791, 5823, 5863, 5887, 5915, 5991, 6015, 6055, 6079, 6139, 6143, 6171, 6191, 6207 6235, 6247, 6303, 6311, 6367, 6383, 6395, 6399, 6427, 6463, 6471, 6591, 6623, 6639, 6655, 6703 6767, 6811, 6823, 6847, 6887, 6895, 6911, 6939, 7039, 7103, 7135, 7151, 7167, 7195, 7231, 7271 7279, 7295, 7323, 7327, 7335, 7359, 7407, 7423, 7451, 7471, 7527, 7551, 7583, 7615, 7647, 7663 7679, 7707, 7711, 7727, 7807, 7835, 7839, 7871, 7935, 7963, 8027, 8039, 8095, 8127, 8175, 8191 8219, 8223, 8239, 8255, 8283, 8295, 8303, 8319, 8351, 8359, 8415, 8443, 8475, 8511, 8519, 8551 8607, 8639, 8671, 8687, 8703, 8751, 8831, 8859, 8895, 8935, 8943, 8955, 8959, 8987, 9023, 9031 9063, 9119, 9151, 9183, 9199, 9215, 9243, 9247, 9287, 9319, 9343, 9455, 9467, 9471, 9499, 9535 9543, 9575, 9599, 9631, 9663, 9695, 9727, 9775, 9831, 9855, 9883, 9887, 9967, 9983, 10011, 10087 10111, 10143, 10235, 10239, 10303, 10311, 10331, 10343, 10351, 10367, 10399, 10407, 10479, 10491, 10495, 10559 10567, 10599, 10687, 10719, 10751, 10843, 10879, 10907, 10911, 10919, 10943, 10983, 10991, 11003, 11007, 11035 11071, 11111, 11135, 11175, 11199, 11247, 11263, 11291, 11295, 11355, 11367, 11391, 11419, 11423, 11431, 11495 11503, 11515, 11519, 11547, 11583, 11591, 11623, 11711, 11759, 11775, 11803, 11823, 11903, 11931, 11935, 12007 12015, 12031, 12059, 12123, 12135, 12191, 12199, 12223, 12287, 12315, 12335, 12359, 12379, 12399, 12447, 12455 12479, 12527, 12539, 12591, 12607, 12615, 12703, 12735, 12767, 12783, 12799, 12847, 12891, 12911, 12927, 12955 12959, 12991, 13031, 13039, 13055, 13127, 13159, 13183, 13247, 13279, 13311, 13339, 13343, 13403, 13415, 13423 13439, 13471, 13479, 13503, 13535, 13543, 13551, 13595, 13615, 13639, 13671, 13727, 13759, 13791, 13823, 13855 13871, 13951, 13983, 14015, 14055, 14079, 14107, 14183, 14247, 14335, 14363, 14367, 14383, 14427, 14447, 14463 14495, 14503, 14559, 14575, 14587, 14591, 14619, 14655, 14663, 14695, 14783, 14815, 14831, 14847, 14951, 14959 14975, 15003, 15015, 15039, 15079, 15087, 15099, 15103, 15131, 15167, 15207, 15231, 15327, 15355, 15359, 15387 15391, 15423, 15463, 15471, 15487, 15519, 15527, 15551, 15599, 15611, 15615, 15643, 15663, 15679, 15719, 15743 15775, 15807, 15871, 15899, 15903, 15975, 15999, 16031, 16063, 16111, 16127, 16155, 16231, 16255, 16319, 16367 16379, 16383, 16411, 16415, 16431, 16447, 16455, 16475, 16487, 16495, 16511, 16543, 16551, 16607, 16623, 16635 16667, 16703, 16711, 16831, 16879, 16895, 16943, 16987, 17023, 17051, 17055, 17087, 17127, 17135, 17151, 17179 17215, 17243, 17255, 17279, 17311, 17343, 17375, 17391, 17407, 17435, 17439, 17479, 17499, 17511, 17535, 17563 17567, 17639, 17647, 17659, 17663, 17691, 17791, 17823, 17855, 17887, 17903, 17919, 17967, 18023, 18047, 18075 18079, 18151, 18159, 18175, 18203, 18267, 18279, 18303, 18335, 18343, 18427, 18431, 18495, 18503, 18523, 18535 18543, 18559, 18591, 18599, 18671, 18683, 18759, 18791, 18847, 18879, 18911, 18927, 18943, 19055, 19071, 19099 19103, 19135, 19175, 19183, 19195, 19227, 19263, 19271, 19303, 19327, 19367, 19391, 19439, 19455, 19483, 19487 19547, 19559, 19567, 19583, 19611, 19615, 19647, 19687, 19695, 19707, 19711, 19739, 19759, 19775, 19783, 19815 19871, 19903, 19951, 19967, 19999, 20015, 20095, 20123, 20127, 20159, 20207, 20223, 20251, 20315, 20327, 20383 20391, 20479, 20511, 20551, 20571, 20591, 20639, 20647, 20671, 20703, 20719, 20731, 20735, 20763, 20799, 20807 20895, 20959, 20975, 20991, 21019, 21083, 21119, 21147, 21151, 21159, 21183, 21231, 21247, 21275, 21319, 21351 21375, 21439, 21503, 21531, 21535, 21595, 21607, 21615, 21631, 21663, 21671, 21695, 21735, 21743, 21759, 21787 21831, 21863, 21887, 21919, 21951, 22015, 22043, 22063, 22119, 22143, 22175, 22247, 22271, 22299, 22375, 22399 22439, 22463, 22523, 22527, 22555, 22559, 22575, 22591, 22619, 22631, 22639, 22687, 22695, 22767, 22779, 22783 22847, 22855, 22975, 23007, 23023, 23039, 23087, 23143, 23151, 23195, 23207, 23271, 23279, 23295, 23323, 23423 23487, 23519, 23535, 23547, 23551, 23579, 23583, 23655, 23663, 23679, 23707, 23711, 23719, 23743, 23791, 23807 23855, 23911, 23967, 23999, 24031, 24047, 24063, 24091, 24095, 24111, 24167, 24191, 24219, 24223, 24255, 24319 24347, 24411, 24423, 24447, 24479, 24511, 24571, 24575, 24603, 24607, 24623, 24667, 24687, 24703, 24735, 24743 24799, 24827, 24859, 24895, 24903, 24935, 24991, 25023, 25055, 25071, 25087, 25135, 25215, 25243, 25279, 25319 25327, 25339, 25343, 25371, 25407, 25415, 25435, 25447, 25471, 25535, 25567, 25599, 25627, 25631, 25671, 25727 25755, 25839, 25851, 25855, 25883, 25919, 25927, 25959, 25983, 26015, 26047, 26079, 26095, 26111, 26159, 26215 26239, 26271, 26351, 26367, 26395, 26459, 26471, 26495, 26619, 26623, 26687, 26695, 26715, 26727, 26735, 26751 26783, 26791, 26863, 26875, 26879, 26943, 26951, 26983, 27039, 27071, 27103, 27119, 27135, 27227, 27263, 27295 27303, 27327, 27367, 27375, 27387, 27391, 27419, 27455, 27463, 27495, 27519, 27559, 27583, 27631, 27647, 27675 27679, 27739, 27751, 27775, 27803, 27807, 27815, 27879, 27887, 27899, 27903, 27931, 27967, 28007, 28063, 28095 28143, 28159, 28187, 28207, 28287, 28315, 28319, 28391, 28399, 28415, 28443, 28507, 28519, 28575, 28583, 28607 28671, 28699, 28719, 28743, 28763, 28783, 28831, 28839, 28863, 28911, 28923, 28927, 28975, 28991, 29087, 29119 29151, 29167, 29183, 29211, 29231, 29275, 29295, 29311, 29339, 29343, 29351, 29375, 29415, 29423, 29439, 29511 29543, 29567, 29631, 29663, 29695, 29723, 29727, 29787, 29799, 29807, 29823, 29855, 29887, 29919, 29927, 29935 29951, 29979, 29999, 30023, 30055, 30111, 30143, 30175, 30207, 30235, 30239, 30255, 30335, 30367, 30399, 30439 30463, 30491, 30567, 30631, 30655, 30719, 30751, 30811, 30831, 30847, 30879, 30943, 30959, 30971, 30975, 31003 31039, 31047, 31079, 31167, 31199, 31215, 31231, 31279, 31335, 31343, 31359, 31387, 31399, 31423, 31463, 31471 31483, 31487, 31515, 31551, 31591, 31615, 31679, 31739, 31743, 31775, 31807, 31847, 31855, 31871, 31903, 31911 31935, 31983, 31995, 31999, 32027, 32047, 32063, 32103, 32127, 32159, 32191, 32223, 32255, 32283, 32287, 32303 32359, 32383, 32415, 32447, 32495, 32511, 32539, 32615, 32639, 32703, 32751, 32763, 32767