exact_figure_data.json

Paper IX: Arithmetic Packets, Dyadic Towers, and Local Polynomial Deformation

exact_figure_data.json

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1 {
2   "scope": "Exact finite illustration data; theorem-derived curves are not numerical evidence for universal proofs.",
3   "coefficient_definition": "C_N,j(n)=(Nn)![t^(Nn)](j! sum_q t^(Nq)/(Nq+j)!)^(-1)",
4   "modular_precision_bits": 256,
5   "modular_rows": 255,
6   "rational_crosscheck_rows": 12,
7   "assertion_counts": {
8     "complete_residue_permutation": 4,
9     "finite_period_repeat": 296,
10     "finite_scaled_isometry": 56,
11     "first_row_benchmark": 1,
12     "independent_rational_inversion": 52,
13     "mixed_counterexample_valuation": 1,
14     "mixed_counterexample_value": 1,
15     "odd_source_denominator": 1024,
16     "plotted_original_metric": 4,
17     "rational_modular_crosscheck": 52,
18     "sampled_address_two_filtration": 340,
19     "sampled_address_two_power_order_exact": 30,
20     "sampled_factorial_bound": 864,
21     "sampled_first_difference_exact": 288,
22     "sampled_homogeneous_equality": 432,
23     "sampled_mixed_factorial_bound": 1360,
24     "sampled_mixed_parity_rule": 680,
25     "stencil_exact_valuation": 1,
26     "stencil_offsets": 1,
27     "upper_half_counterexample": 3
28   },
29   "total_assertions": 5490,
30   "rational_first_rows": {
31     "0": [
32       "1",
33       "-1",
34       "69",
35       "-33661",
36       "60376809"
37     ],
38     "1": [
39       "1",
40       "-1/5",
41       "121/45",
42       "-16593/65",
43       "68164277/765"
44     ],
45     "2": [
46       "1",
47       "-1/15",
48       "13/45",
49       "-4009/455",
50       "151129/153"
51     ],
52     "3": [
53       "1",
54       "-1/35",
55       "59/1155",
56       "-407/637",
57       "15330979/522291"
58     ]
59   },
60   "mixed_stencil": {
61     "steps": [
62       1,
63       2,
64       4
65     ],
66     "offset_weights": [
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90       ],
91       [
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93         -1
94       ],
95       [
96         7,
97         1
98       ]
99     ],
100     "value": "-60924841253875487872",
101     "valuation": 7
102   },
103   "mixed_order_two": [
104     {
105       "steps": [
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107         1
108       ],
109       "lower": 3,
110       "actual": 3,
111       "value": "72"
112     },
113     {
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118       "lower": 4,
119       "actual": 6,
120       "value": "-33728"
121     },
122     {
123       "steps": [
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127       "lower": 5,
128       "actual": 5,
129       "value": "60376672"
130     },
131     {
132       "steps": [
133         1,
134         4
135       ],
136       "lower": 5,
137       "actual": 6,
138       "value": "-288354427328"
139     },
140     {
141       "steps": [
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143         4
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145       "lower": 6,
146       "actual": 6,
147       "value": "3019098102225472"
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149     {
150       "steps": [
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152         4
153       ],
154       "lower": 5,
155       "actual": 10,
156       "value": "-60921822444127689728"
157     }
158   ],
159   "residue_permutations": [
160     {
161       "j": 0,
162       "modulus": 256,
163       "B": 1,
164       "period": 128,
165       "Euler_residues": [
166         1,
167         255,
168         69,
169         131,
170         233,
171         39,
172         109,
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292         157,
293         27
294       ],
295       "scaled_residues": [
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297         127,
298         34,
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300         116,
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304         40,
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419         25,
420         76,
421         43,
422         78,
423         13
424       ]
425     },
426     {
427       "j": 1,
428       "modulus": 256,
429       "B": 1,
430       "period": 128,
431       "Euler_residues": [
432         1,
433         51,
434         253,
435         111,
436         25,
437         11,
438         213,
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440         177,
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558         165,
559         215
560       ],
561       "scaled_residues": [
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563         25,
564         126,
565         55,
566         12,
567         5,
568         106,
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683         1,
684         102,
685         95,
686         52,
687         109,
688         82,
689         107
690       ]
691     },
692     {
693       "j": 2,
694       "modulus": 256,
695       "B": 5,
696       "period": 8,
697       "Euler_residues": [
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699         34,
700         194,
701         226,
702         130,
703         162,
704         66,
705         98
706       ],
707       "scaled_residues": [
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709         1,
710         6,
711         7,
712         4,
713         5,
714         2,
715         3
716       ]
717     },
718     {
719       "j": 3,
720       "modulus": 256,
721       "B": 3,
722       "period": 32,
723       "Euler_residues": [
724         6,
725         190,
726         118,
727         46,
728         230,
729         158,
730         86,
731         14,
732         198,
733         126,
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736         166,
737         94,
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750         182,
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752         38,
753         222,
754         150,
755         78
756       ],
757       "scaled_residues": [
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759         23,
760         14,
761         5,
762         28,
763         19,
764         10,
765         1,
766         24,
767         15,
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771         11,
772         2,
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774         16,
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776         30,
777         21,
778         12,
779         3,
780         26,
781         17,
782         8,
783         31,
784         22,
785         13,
786         4,
787         27,
788         18,
789         9
790       ]
791     }
792   ],
793   "upper_half_counterexamples": [
794     {
795       "N": 16,
796       "j": 9,
797       "first_valuation": 5,
798       "step_two_valuation": 7,
799       "naive_prediction": 6,
800       "C0": "1",
801       "C1": "-1/2042975",
802       "C2": "319118897/2215923048625"
803     },
804     {
805       "N": 32,
806       "j": 18,
807       "first_valuation": 6,
808       "step_two_valuation": 6,
809       "naive_prediction": 7,
810       "C0": "1",
811       "C1": "-1/18053528883775",
812       "C2": "187347588441574511/33319533660275387449299375"
813     },
814     {
815       "N": 32,
816       "j": 19,
817       "first_valuation": 5,
818       "step_two_valuation": 7,
819       "naive_prediction": 6,
820       "C0": "1",
821       "C1": "-1/48459472266975",
822       "C2": "47098222049108039/60351555833951953390194375"
823     }
824   ],
825   "theorem_derived_unit_step_exponents": {
826     "orders": [
827       1,
828       2,
829       3,
830       4,
831       5,
832       6,
833       7,
834       8,
835       9,
836       10,
837       11,
838       12,
839       13,
840       14,
841       15,
842       16,
843       17,
844       18,
845       19,
846       20,
847       21,
848       22,
849       23,
850       24,
851       25,
852       26,
853       27,
854       28,
855       29,
856       30,
857       31,
858       32
859     ],
860     "Euler_j01": [
861       1,
862       3,
863       4,
864       7,
865       8,
866       10,
867       11,
868       15,
869       16,
870       18,
871       19,
872       22,
873       23,
874       25,
875       26,
876       31,
877       32,
878       34,
879       35,
880       38,
881       39,
882       41,
883       42,
884       46,
885       47,
886       49,
887       50,
888       53,
889       54,
890       56,
891       57,
892       63
893     ],
894     "Euler_j2": [
895       5,
896       7,
897       9,
898       14,
899       16,
900       20,
901       22,
902       28,
903       30,
904       34,
905       36,
906       41,
907       43,
908       47,
909       49,
910       56,
911       58,
912       62,
913       64,
914       69,
915       71,
916       75,
917       77,
918       83,
919       85,
920       89,
921       91,
922       96,
923       98,
924       102,
925       104,
926       112
927     ],
928     "Euler_j3": [
929       3,
930       5,
931       6,
932       9,
933       10,
934       12,
935       13,
936       17,
937       18,
938       20,
939       21,
940       24,
941       25,
942       27,
943       28,
944       33,
945       34,
946       36,
947       37,
948       40,
949       41,
950       43,
951       44,
952       48,
953       49,
954       51,
955       52,
956       55,
957       56,
958       58,
959       59,
960       65
961     ]
962   },
963   "known_exactness": "j0/j1 all homogeneous orders; j2 powers of two; no complete higher-order j3 formula claimed"
964 }

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