Question 17590: A Long Prefix Still Does Not Replace the All-n Tail

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=15, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].

• G001 — Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=15, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-pointwise-convergence-of-functions

Step-by-step derivation

same COMPLETE finite domain Fin(2);original aligned limits=[14, 15];signed coefficients=[15, -16];positive shift=2

declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)

finite prefix point0 index0 denominator := sum(0, 2)

finite prefix point0 index0 denominator = 2

finite prefix point0 index0 signed correction := div(15, 2)

finite prefix point0 index0 signed correction = 15/2

finite prefix point0 index0 function value := sum(14, 15/2)

finite prefix point0 index0 function value = 43/2

finite prefix point0 index0 reverse signed error := sub(43/2, 14)

finite prefix point0 index0 reverse signed error = 15/2

finite prefix point0 index0:original point=0;n=0;limit=14;signed coefficient=15

f_n(x)=43/2;f_n(x)-limit=15/2;this error field remains SIGNED

finite prefix point0 index1 denominator := sum(1, 2)

finite prefix point0 index1 denominator = 3

finite prefix point0 index1 signed correction := div(15, 3)

finite prefix point0 index1 signed correction = 5

finite prefix point0 index1 function value := sum(14, 5)

finite prefix point0 index1 function value = 19

finite prefix point0 index1 reverse signed error := sub(19, 14)

finite prefix point0 index1 reverse signed error = 5

finite prefix point0 index1:original point=0;n=1;limit=14;signed coefficient=15

f_n(x)=19;f_n(x)-limit=5;this error field remains SIGNED

finite prefix point0 index2 denominator := sum(2, 2)

finite prefix point0 index2 denominator = 4

finite prefix point0 index2 signed correction := div(15, 4)

finite prefix point0 index2 signed correction = 15/4

finite prefix point0 index2 function value := sum(14, 15/4)

finite prefix point0 index2 function value = 71/4

finite prefix point0 index2 reverse signed error := sub(71/4, 14)

finite prefix point0 index2 reverse signed error = 15/4

finite prefix point0 index2:original point=0;n=2;limit=14;signed coefficient=15

f_n(x)=71/4;f_n(x)-limit=15/4;this error field remains SIGNED

finite prefix point0 index3 denominator := sum(3, 2)

finite prefix point0 index3 denominator = 5

finite prefix point0 index3 signed correction := div(15, 5)

finite prefix point0 index3 signed correction = 3

finite prefix point0 index3 function value := sum(14, 3)

finite prefix point0 index3 function value = 17

finite prefix point0 index3 reverse signed error := sub(17, 14)

finite prefix point0 index3 reverse signed error = 3

finite prefix point0 index3:original point=0;n=3;limit=14;signed coefficient=15

f_n(x)=17;f_n(x)-limit=3;this error field remains SIGNED

finite prefix point0 index4 denominator := sum(4, 2)

finite prefix point0 index4 denominator = 6

finite prefix point0 index4 signed correction := div(15, 6)

finite prefix point0 index4 signed correction = 5/2

finite prefix point0 index4 function value := sum(14, 5/2)

finite prefix point0 index4 function value = 33/2

finite prefix point0 index4 reverse signed error := sub(33/2, 14)

finite prefix point0 index4 reverse signed error = 5/2

finite prefix point0 index4:original point=0;n=4;limit=14;signed coefficient=15

f_n(x)=33/2;f_n(x)-limit=5/2;this error field remains SIGNED

finite prefix point0 index5 denominator := sum(5, 2)

finite prefix point0 index5 denominator = 7

finite prefix point0 index5 signed correction := div(15, 7)

finite prefix point0 index5 signed correction = 15/7

finite prefix point0 index5 function value := sum(14, 15/7)

finite prefix point0 index5 function value = 113/7

finite prefix point0 index5 reverse signed error := sub(113/7, 14)

finite prefix point0 index5 reverse signed error = 15/7

finite prefix point0 index5:original point=0;n=5;limit=14;signed coefficient=15

f_n(x)=113/7;f_n(x)-limit=15/7;this error field remains SIGNED

finite prefix point0 index6 denominator := sum(6, 2)

finite prefix point0 index6 denominator = 8

finite prefix point0 index6 signed correction := div(15, 8)

finite prefix point0 index6 signed correction = 15/8

finite prefix point0 index6 function value := sum(14, 15/8)

finite prefix point0 index6 function value = 127/8

finite prefix point0 index6 reverse signed error := sub(127/8, 14)

finite prefix point0 index6 reverse signed error = 15/8

finite prefix point0 index6:original point=0;n=6;limit=14;signed coefficient=15

f_n(x)=127/8;f_n(x)-limit=15/8;this error field remains SIGNED

finite prefix point0 index7 denominator := sum(7, 2)

finite prefix point0 index7 denominator = 9

finite prefix point0 index7 signed correction := div(15, 9)

finite prefix point0 index7 signed correction = 5/3

finite prefix point0 index7 function value := sum(14, 5/3)

finite prefix point0 index7 function value = 47/3

finite prefix point0 index7 reverse signed error := sub(47/3, 14)

finite prefix point0 index7 reverse signed error = 5/3

finite prefix point0 index7:original point=0;n=7;limit=14;signed coefficient=15

f_n(x)=47/3;f_n(x)-limit=5/3;this error field remains SIGNED

finite prefix point0 index8 denominator := sum(8, 2)

finite prefix point0 index8 denominator = 10

finite prefix point0 index8 signed correction := div(15, 10)

finite prefix point0 index8 signed correction = 3/2

finite prefix point0 index8 function value := sum(14, 3/2)

finite prefix point0 index8 function value = 31/2

finite prefix point0 index8 reverse signed error := sub(31/2, 14)

finite prefix point0 index8 reverse signed error = 3/2

finite prefix point0 index8:original point=0;n=8;limit=14;signed coefficient=15

f_n(x)=31/2;f_n(x)-limit=3/2;this error field remains SIGNED

finite prefix point0 index9 denominator := sum(9, 2)

finite prefix point0 index9 denominator = 11

finite prefix point0 index9 signed correction := div(15, 11)

finite prefix point0 index9 signed correction = 15/11

finite prefix point0 index9 function value := sum(14, 15/11)

finite prefix point0 index9 function value = 169/11

finite prefix point0 index9 reverse signed error := sub(169/11, 14)

finite prefix point0 index9 reverse signed error = 15/11

finite prefix point0 index9:original point=0;n=9;limit=14;signed coefficient=15

f_n(x)=169/11;f_n(x)-limit=15/11;this error field remains SIGNED

finite prefix point0 index10 denominator := sum(10, 2)

finite prefix point0 index10 denominator = 12

finite prefix point0 index10 signed correction := div(15, 12)

finite prefix point0 index10 signed correction = 5/4

finite prefix point0 index10 function value := sum(14, 5/4)

finite prefix point0 index10 function value = 61/4

finite prefix point0 index10 reverse signed error := sub(61/4, 14)

finite prefix point0 index10 reverse signed error = 5/4

finite prefix point0 index10:original point=0;n=10;limit=14;signed coefficient=15

f_n(x)=61/4;f_n(x)-limit=5/4;this error field remains SIGNED

finite prefix point0 index11 denominator := sum(11, 2)

finite prefix point0 index11 denominator = 13

finite prefix point0 index11 signed correction := div(15, 13)

finite prefix point0 index11 signed correction = 15/13

finite prefix point0 index11 function value := sum(14, 15/13)

finite prefix point0 index11 function value = 197/13

finite prefix point0 index11 reverse signed error := sub(197/13, 14)

finite prefix point0 index11 reverse signed error = 15/13

finite prefix point0 index11:original point=0;n=11;limit=14;signed coefficient=15

f_n(x)=197/13;f_n(x)-limit=15/13;this error field remains SIGNED

finite prefix point0 index12 denominator := sum(12, 2)

finite prefix point0 index12 denominator = 14

finite prefix point0 index12 signed correction := div(15, 14)

finite prefix point0 index12 signed correction = 15/14

finite prefix point0 index12 function value := sum(14, 15/14)

finite prefix point0 index12 function value = 211/14

finite prefix point0 index12 reverse signed error := sub(211/14, 14)

finite prefix point0 index12 reverse signed error = 15/14

finite prefix point0 index12:original point=0;n=12;limit=14;signed coefficient=15

f_n(x)=211/14;f_n(x)-limit=15/14;this error field remains SIGNED

finite prefix point0 index13 denominator := sum(13, 2)

finite prefix point0 index13 denominator = 15

finite prefix point0 index13 signed correction := div(15, 15)

finite prefix point0 index13 signed correction = 1

finite prefix point0 index13 function value := sum(14, 1)

finite prefix point0 index13 function value = 15

finite prefix point0 index13 reverse signed error := sub(15, 14)

finite prefix point0 index13 reverse signed error = 1

finite prefix point0 index13:original point=0;n=13;limit=14;signed coefficient=15

f_n(x)=15;f_n(x)-limit=1;this error field remains SIGNED

finite prefix point0 index14 denominator := sum(14, 2)

finite prefix point0 index14 denominator = 16

finite prefix point0 index14 signed correction := div(15, 16)

finite prefix point0 index14 signed correction = 15/16

finite prefix point0 index14 function value := sum(14, 15/16)

finite prefix point0 index14 function value = 239/16

finite prefix point0 index14 reverse signed error := sub(239/16, 14)

finite prefix point0 index14 reverse signed error = 15/16

finite prefix point0 index14:original point=0;n=14;limit=14;signed coefficient=15

f_n(x)=239/16;f_n(x)-limit=15/16;this error field remains SIGNED

finite prefix point1 index0 denominator := sum(0, 2)

finite prefix point1 index0 denominator = 2

finite prefix point1 index0 signed correction := div(-16, 2)

finite prefix point1 index0 signed correction = -8

finite prefix point1 index0 function value := sum(15, -8)

finite prefix point1 index0 function value = 7

finite prefix point1 index0 reverse signed error := sub(7, 15)

finite prefix point1 index0 reverse signed error = -8

finite prefix point1 index0:original point=1;n=0;limit=15;signed coefficient=-16

f_n(x)=7;f_n(x)-limit=-8;this error field remains SIGNED

finite prefix point1 index1 denominator := sum(1, 2)

finite prefix point1 index1 denominator = 3

finite prefix point1 index1 signed correction := div(-16, 3)

finite prefix point1 index1 signed correction = -16/3

finite prefix point1 index1 function value := sum(15, -16/3)

finite prefix point1 index1 function value = 29/3

finite prefix point1 index1 reverse signed error := sub(29/3, 15)

finite prefix point1 index1 reverse signed error = -16/3

finite prefix point1 index1:original point=1;n=1;limit=15;signed coefficient=-16

f_n(x)=29/3;f_n(x)-limit=-16/3;this error field remains SIGNED

finite prefix point1 index2 denominator := sum(2, 2)

finite prefix point1 index2 denominator = 4

finite prefix point1 index2 signed correction := div(-16, 4)

finite prefix point1 index2 signed correction = -4

finite prefix point1 index2 function value := sum(15, -4)

finite prefix point1 index2 function value = 11

finite prefix point1 index2 reverse signed error := sub(11, 15)

finite prefix point1 index2 reverse signed error = -4

finite prefix point1 index2:original point=1;n=2;limit=15;signed coefficient=-16

f_n(x)=11;f_n(x)-limit=-4;this error field remains SIGNED

finite prefix point1 index3 denominator := sum(3, 2)

finite prefix point1 index3 denominator = 5

finite prefix point1 index3 signed correction := div(-16, 5)

finite prefix point1 index3 signed correction = -16/5

finite prefix point1 index3 function value := sum(15, -16/5)

finite prefix point1 index3 function value = 59/5

finite prefix point1 index3 reverse signed error := sub(59/5, 15)

finite prefix point1 index3 reverse signed error = -16/5

finite prefix point1 index3:original point=1;n=3;limit=15;signed coefficient=-16

f_n(x)=59/5;f_n(x)-limit=-16/5;this error field remains SIGNED

finite prefix point1 index4 denominator := sum(4, 2)

finite prefix point1 index4 denominator = 6

finite prefix point1 index4 signed correction := div(-16, 6)

finite prefix point1 index4 signed correction = -8/3

finite prefix point1 index4 function value := sum(15, -8/3)

finite prefix point1 index4 function value = 37/3

finite prefix point1 index4 reverse signed error := sub(37/3, 15)

finite prefix point1 index4 reverse signed error = -8/3

finite prefix point1 index4:original point=1;n=4;limit=15;signed coefficient=-16

f_n(x)=37/3;f_n(x)-limit=-8/3;this error field remains SIGNED

finite prefix point1 index5 denominator := sum(5, 2)

finite prefix point1 index5 denominator = 7

finite prefix point1 index5 signed correction := div(-16, 7)

finite prefix point1 index5 signed correction = -16/7

finite prefix point1 index5 function value := sum(15, -16/7)

finite prefix point1 index5 function value = 89/7

finite prefix point1 index5 reverse signed error := sub(89/7, 15)

finite prefix point1 index5 reverse signed error = -16/7

finite prefix point1 index5:original point=1;n=5;limit=15;signed coefficient=-16

f_n(x)=89/7;f_n(x)-limit=-16/7;this error field remains SIGNED

finite prefix point1 index6 denominator := sum(6, 2)

finite prefix point1 index6 denominator = 8

finite prefix point1 index6 signed correction := div(-16, 8)

finite prefix point1 index6 signed correction = -2

finite prefix point1 index6 function value := sum(15, -2)

finite prefix point1 index6 function value = 13

finite prefix point1 index6 reverse signed error := sub(13, 15)

finite prefix point1 index6 reverse signed error = -2

finite prefix point1 index6:original point=1;n=6;limit=15;signed coefficient=-16

f_n(x)=13;f_n(x)-limit=-2;this error field remains SIGNED

finite prefix point1 index7 denominator := sum(7, 2)

finite prefix point1 index7 denominator = 9

finite prefix point1 index7 signed correction := div(-16, 9)

finite prefix point1 index7 signed correction = -16/9

finite prefix point1 index7 function value := sum(15, -16/9)

finite prefix point1 index7 function value = 119/9

finite prefix point1 index7 reverse signed error := sub(119/9, 15)

finite prefix point1 index7 reverse signed error = -16/9

finite prefix point1 index7:original point=1;n=7;limit=15;signed coefficient=-16

f_n(x)=119/9;f_n(x)-limit=-16/9;this error field remains SIGNED

finite prefix point1 index8 denominator := sum(8, 2)

finite prefix point1 index8 denominator = 10

finite prefix point1 index8 signed correction := div(-16, 10)

finite prefix point1 index8 signed correction = -8/5

finite prefix point1 index8 function value := sum(15, -8/5)

finite prefix point1 index8 function value = 67/5

finite prefix point1 index8 reverse signed error := sub(67/5, 15)

finite prefix point1 index8 reverse signed error = -8/5

finite prefix point1 index8:original point=1;n=8;limit=15;signed coefficient=-16

f_n(x)=67/5;f_n(x)-limit=-8/5;this error field remains SIGNED

finite prefix point1 index9 denominator := sum(9, 2)

finite prefix point1 index9 denominator = 11

finite prefix point1 index9 signed correction := div(-16, 11)

finite prefix point1 index9 signed correction = -16/11

finite prefix point1 index9 function value := sum(15, -16/11)

finite prefix point1 index9 function value = 149/11

finite prefix point1 index9 reverse signed error := sub(149/11, 15)

finite prefix point1 index9 reverse signed error = -16/11

finite prefix point1 index9:original point=1;n=9;limit=15;signed coefficient=-16

f_n(x)=149/11;f_n(x)-limit=-16/11;this error field remains SIGNED

finite prefix point1 index10 denominator := sum(10, 2)

finite prefix point1 index10 denominator = 12

finite prefix point1 index10 signed correction := div(-16, 12)

finite prefix point1 index10 signed correction = -4/3

finite prefix point1 index10 function value := sum(15, -4/3)

finite prefix point1 index10 function value = 41/3

finite prefix point1 index10 reverse signed error := sub(41/3, 15)

finite prefix point1 index10 reverse signed error = -4/3

finite prefix point1 index10:original point=1;n=10;limit=15;signed coefficient=-16

f_n(x)=41/3;f_n(x)-limit=-4/3;this error field remains SIGNED

finite prefix point1 index11 denominator := sum(11, 2)

finite prefix point1 index11 denominator = 13

finite prefix point1 index11 signed correction := div(-16, 13)

finite prefix point1 index11 signed correction = -16/13

finite prefix point1 index11 function value := sum(15, -16/13)

finite prefix point1 index11 function value = 179/13

finite prefix point1 index11 reverse signed error := sub(179/13, 15)

finite prefix point1 index11 reverse signed error = -16/13

finite prefix point1 index11:original point=1;n=11;limit=15;signed coefficient=-16

f_n(x)=179/13;f_n(x)-limit=-16/13;this error field remains SIGNED

finite prefix point1 index12 denominator := sum(12, 2)

finite prefix point1 index12 denominator = 14

finite prefix point1 index12 signed correction := div(-16, 14)

finite prefix point1 index12 signed correction = -8/7

finite prefix point1 index12 function value := sum(15, -8/7)

finite prefix point1 index12 function value = 97/7

finite prefix point1 index12 reverse signed error := sub(97/7, 15)

finite prefix point1 index12 reverse signed error = -8/7

finite prefix point1 index12:original point=1;n=12;limit=15;signed coefficient=-16

f_n(x)=97/7;f_n(x)-limit=-8/7;this error field remains SIGNED

finite prefix point1 index13 denominator := sum(13, 2)

finite prefix point1 index13 denominator = 15

finite prefix point1 index13 signed correction := div(-16, 15)

finite prefix point1 index13 signed correction = -16/15

finite prefix point1 index13 function value := sum(15, -16/15)

finite prefix point1 index13 function value = 209/15

finite prefix point1 index13 reverse signed error := sub(209/15, 15)

finite prefix point1 index13 reverse signed error = -16/15

finite prefix point1 index13:original point=1;n=13;limit=15;signed coefficient=-16

f_n(x)=209/15;f_n(x)-limit=-16/15;this error field remains SIGNED

finite prefix point1 index14 denominator := sum(14, 2)

finite prefix point1 index14 denominator = 16

finite prefix point1 index14 signed correction := div(-16, 16)

finite prefix point1 index14 signed correction = -1

finite prefix point1 index14 function value := sum(15, -1)

finite prefix point1 index14 function value = 14

finite prefix point1 index14 reverse signed error := sub(14, 15)

finite prefix point1 index14 reverse signed error = -1

finite prefix point1 index14:original point=1;n=14;limit=15;signed coefficient=-16

f_n(x)=14;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=15;complete domain-order rows=[[0, [43/2, 19, 71/4, 17, 33/2, 113/7, 127/8, 47/3, 31/2, 169/11, 61/4, 197/13, 211/14, 15, 239/16]], [1, [7, 29/3, 11, 59/5, 37/3, 89/7, 13, 119/9, 67/5, 149/11, 41/3, 179/13, 97/7, 209/15, 14]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S150 · premise_and_result_audit — Recompute finite prefix point1 index14 reverse signed error with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute finite prefix point1 index14 function value with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute finite prefix point1 index14 signed correction with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute finite prefix point1 index14 denominator with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Recompute finite prefix point1 index13 reverse signed error with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute finite prefix point1 index13 function value with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute finite prefix point1 index13 signed correction with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute finite prefix point1 index13 denominator with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Recompute finite prefix point1 index12 reverse signed error with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute finite prefix point1 index12 function value with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute finite prefix point1 index12 signed correction with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute finite prefix point1 index12 denominator with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Recompute finite prefix point1 index11 reverse signed error with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute finite prefix point1 index11 function value with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute finite prefix point1 index11 signed correction with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute finite prefix point1 index11 denominator with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S130 · premise_and_result_audit — Recompute finite prefix point1 index10 reverse signed error with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute finite prefix point1 index10 function value with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute finite prefix point1 index10 signed correction with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute finite prefix point1 index10 denominator with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S125 · premise_and_result_audit — Recompute finite prefix point1 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute finite prefix point1 index9 denominator with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S120 · premise_and_result_audit — Recompute finite prefix point1 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute finite prefix point1 index8 denominator with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S115 · premise_and_result_audit — Recompute finite prefix point1 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute finite prefix point1 index7 denominator with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S110 · premise_and_result_audit — Recompute finite prefix point1 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute finite prefix point1 index6 denominator with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S105 · premise_and_result_audit — Recompute finite prefix point1 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute finite prefix point1 index5 denominator with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Recompute finite prefix point1 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute finite prefix point1 index4 denominator with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Recompute finite prefix point1 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute finite prefix point1 index3 denominator with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Recompute finite prefix point1 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute finite prefix point1 index2 denominator with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Recompute finite prefix point1 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute finite prefix point1 index1 denominator with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Recompute finite prefix point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute finite prefix point1 index0 denominator with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Recompute finite prefix point0 index14 reverse signed error with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute finite prefix point0 index14 function value with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute finite prefix point0 index14 signed correction with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute finite prefix point0 index14 denominator with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Recompute finite prefix point0 index13 reverse signed error with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute finite prefix point0 index13 function value with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute finite prefix point0 index13 signed correction with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute finite prefix point0 index13 denominator with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Recompute finite prefix point0 index12 reverse signed error with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute finite prefix point0 index12 function value with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute finite prefix point0 index12 signed correction with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute finite prefix point0 index12 denominator with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Recompute finite prefix point0 index11 reverse signed error with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute finite prefix point0 index11 function value with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute finite prefix point0 index11 signed correction with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute finite prefix point0 index11 denominator with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Recompute finite prefix point0 index10 reverse signed error with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute finite prefix point0 index10 function value with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute finite prefix point0 index10 signed correction with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute finite prefix point0 index10 denominator with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute finite prefix point0 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute finite prefix point0 index9 function value with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute finite prefix point0 index9 signed correction with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute finite prefix point0 index9 denominator with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Recompute finite prefix point0 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute finite prefix point0 index8 function value with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute finite prefix point0 index8 signed correction with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute finite prefix point0 index8 denominator with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Recompute finite prefix point0 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute finite prefix point0 index7 function value with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute finite prefix point0 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute finite prefix point0 index7 denominator with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute finite prefix point0 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute finite prefix point0 index6 function value with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute finite prefix point0 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute finite prefix point0 index6 denominator with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute finite prefix point0 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute finite prefix point0 index5 function value with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute finite prefix point0 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute finite prefix point0 index5 denominator with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute finite prefix point0 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute finite prefix point0 index4 function value with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute finite prefix point0 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute finite prefix point0 index4 denominator with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute finite prefix point0 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute finite prefix point0 index3 function value with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute finite prefix point0 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute finite prefix point0 index3 denominator with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute finite prefix point0 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute finite prefix point0 index2 function value with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute finite prefix point0 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute finite prefix point0 index2 denominator with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute finite prefix point0 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute finite prefix point0 index1 function value with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute finite prefix point0 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute finite prefix point0 index1 denominator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute finite prefix point0 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute finite prefix point0 index0 function value with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute finite prefix point0 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute finite prefix point0 index0 denominator with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(15, [[0, [43/2, 19, 71/4, 17, 33/2, 113/7, 127/8, 47/3, 31/2, 169/11, 61/4, 197/13, 211/14, 15, 239/16]], [1, [7, 29/3, 11, 59/5, 37/3, 89/7, 13, 119/9, 67/5, 149/11, 41/3, 179/13, 97/7, 209/15, 14]]], No, finite_prefix_not_pointwise_convergence_certificate)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.