Question 17591: 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=[15, 16], coefficients=[16, -17], 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=16, 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=[15, 16], coefficients=[16, -17], 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=16, 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=[15, 16];signed coefficients=[16, -17];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(16, 2)

finite prefix point0 index0 signed correction = 8

finite prefix point0 index0 function value := sum(15, 8)

finite prefix point0 index0 function value = 23

finite prefix point0 index0 reverse signed error := sub(23, 15)

finite prefix point0 index0 reverse signed error = 8

finite prefix point0 index0:original point=0;n=0;limit=15;signed coefficient=16

f_n(x)=23;f_n(x)-limit=8;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(16, 3)

finite prefix point0 index1 signed correction = 16/3

finite prefix point0 index1 function value := sum(15, 16/3)

finite prefix point0 index1 function value = 61/3

finite prefix point0 index1 reverse signed error := sub(61/3, 15)

finite prefix point0 index1 reverse signed error = 16/3

finite prefix point0 index1:original point=0;n=1;limit=15;signed coefficient=16

f_n(x)=61/3;f_n(x)-limit=16/3;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(16, 4)

finite prefix point0 index2 signed correction = 4

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

finite prefix point0 index2 function value = 19

finite prefix point0 index2 reverse signed error := sub(19, 15)

finite prefix point0 index2 reverse signed error = 4

finite prefix point0 index2:original point=0;n=2;limit=15;signed coefficient=16

f_n(x)=19;f_n(x)-limit=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(16, 5)

finite prefix point0 index3 signed correction = 16/5

finite prefix point0 index3 function value := sum(15, 16/5)

finite prefix point0 index3 function value = 91/5

finite prefix point0 index3 reverse signed error := sub(91/5, 15)

finite prefix point0 index3 reverse signed error = 16/5

finite prefix point0 index3:original point=0;n=3;limit=15;signed coefficient=16

f_n(x)=91/5;f_n(x)-limit=16/5;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(16, 6)

finite prefix point0 index4 signed correction = 8/3

finite prefix point0 index4 function value := sum(15, 8/3)

finite prefix point0 index4 function value = 53/3

finite prefix point0 index4 reverse signed error := sub(53/3, 15)

finite prefix point0 index4 reverse signed error = 8/3

finite prefix point0 index4:original point=0;n=4;limit=15;signed coefficient=16

f_n(x)=53/3;f_n(x)-limit=8/3;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(16, 7)

finite prefix point0 index5 signed correction = 16/7

finite prefix point0 index5 function value := sum(15, 16/7)

finite prefix point0 index5 function value = 121/7

finite prefix point0 index5 reverse signed error := sub(121/7, 15)

finite prefix point0 index5 reverse signed error = 16/7

finite prefix point0 index5:original point=0;n=5;limit=15;signed coefficient=16

f_n(x)=121/7;f_n(x)-limit=16/7;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(16, 8)

finite prefix point0 index6 signed correction = 2

finite prefix point0 index6 function value := sum(15, 2)

finite prefix point0 index6 function value = 17

finite prefix point0 index6 reverse signed error := sub(17, 15)

finite prefix point0 index6 reverse signed error = 2

finite prefix point0 index6:original point=0;n=6;limit=15;signed coefficient=16

f_n(x)=17;f_n(x)-limit=2;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(16, 9)

finite prefix point0 index7 signed correction = 16/9

finite prefix point0 index7 function value := sum(15, 16/9)

finite prefix point0 index7 function value = 151/9

finite prefix point0 index7 reverse signed error := sub(151/9, 15)

finite prefix point0 index7 reverse signed error = 16/9

finite prefix point0 index7:original point=0;n=7;limit=15;signed coefficient=16

f_n(x)=151/9;f_n(x)-limit=16/9;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(16, 10)

finite prefix point0 index8 signed correction = 8/5

finite prefix point0 index8 function value := sum(15, 8/5)

finite prefix point0 index8 function value = 83/5

finite prefix point0 index8 reverse signed error := sub(83/5, 15)

finite prefix point0 index8 reverse signed error = 8/5

finite prefix point0 index8:original point=0;n=8;limit=15;signed coefficient=16

f_n(x)=83/5;f_n(x)-limit=8/5;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(16, 11)

finite prefix point0 index9 signed correction = 16/11

finite prefix point0 index9 function value := sum(15, 16/11)

finite prefix point0 index9 function value = 181/11

finite prefix point0 index9 reverse signed error := sub(181/11, 15)

finite prefix point0 index9 reverse signed error = 16/11

finite prefix point0 index9:original point=0;n=9;limit=15;signed coefficient=16

f_n(x)=181/11;f_n(x)-limit=16/11;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(16, 12)

finite prefix point0 index10 signed correction = 4/3

finite prefix point0 index10 function value := sum(15, 4/3)

finite prefix point0 index10 function value = 49/3

finite prefix point0 index10 reverse signed error := sub(49/3, 15)

finite prefix point0 index10 reverse signed error = 4/3

finite prefix point0 index10:original point=0;n=10;limit=15;signed coefficient=16

f_n(x)=49/3;f_n(x)-limit=4/3;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(16, 13)

finite prefix point0 index11 signed correction = 16/13

finite prefix point0 index11 function value := sum(15, 16/13)

finite prefix point0 index11 function value = 211/13

finite prefix point0 index11 reverse signed error := sub(211/13, 15)

finite prefix point0 index11 reverse signed error = 16/13

finite prefix point0 index11:original point=0;n=11;limit=15;signed coefficient=16

f_n(x)=211/13;f_n(x)-limit=16/13;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(16, 14)

finite prefix point0 index12 signed correction = 8/7

finite prefix point0 index12 function value := sum(15, 8/7)

finite prefix point0 index12 function value = 113/7

finite prefix point0 index12 reverse signed error := sub(113/7, 15)

finite prefix point0 index12 reverse signed error = 8/7

finite prefix point0 index12:original point=0;n=12;limit=15;signed coefficient=16

f_n(x)=113/7;f_n(x)-limit=8/7;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(16, 15)

finite prefix point0 index13 signed correction = 16/15

finite prefix point0 index13 function value := sum(15, 16/15)

finite prefix point0 index13 function value = 241/15

finite prefix point0 index13 reverse signed error := sub(241/15, 15)

finite prefix point0 index13 reverse signed error = 16/15

finite prefix point0 index13:original point=0;n=13;limit=15;signed coefficient=16

f_n(x)=241/15;f_n(x)-limit=16/15;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(16, 16)

finite prefix point0 index14 signed correction = 1

finite prefix point0 index14 function value := sum(15, 1)

finite prefix point0 index14 function value = 16

finite prefix point0 index14 reverse signed error := sub(16, 15)

finite prefix point0 index14 reverse signed error = 1

finite prefix point0 index14:original point=0;n=14;limit=15;signed coefficient=16

f_n(x)=16;f_n(x)-limit=1;this error field remains SIGNED

finite prefix point0 index15 denominator := sum(15, 2)

finite prefix point0 index15 denominator = 17

finite prefix point0 index15 signed correction := div(16, 17)

finite prefix point0 index15 signed correction = 16/17

finite prefix point0 index15 function value := sum(15, 16/17)

finite prefix point0 index15 function value = 271/17

finite prefix point0 index15 reverse signed error := sub(271/17, 15)

finite prefix point0 index15 reverse signed error = 16/17

finite prefix point0 index15:original point=0;n=15;limit=15;signed coefficient=16

f_n(x)=271/17;f_n(x)-limit=16/17;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(-17, 2)

finite prefix point1 index0 signed correction = -17/2

finite prefix point1 index0 function value := sum(16, -17/2)

finite prefix point1 index0 function value = 15/2

finite prefix point1 index0 reverse signed error := sub(15/2, 16)

finite prefix point1 index0 reverse signed error = -17/2

finite prefix point1 index0:original point=1;n=0;limit=16;signed coefficient=-17

f_n(x)=15/2;f_n(x)-limit=-17/2;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(-17, 3)

finite prefix point1 index1 signed correction = -17/3

finite prefix point1 index1 function value := sum(16, -17/3)

finite prefix point1 index1 function value = 31/3

finite prefix point1 index1 reverse signed error := sub(31/3, 16)

finite prefix point1 index1 reverse signed error = -17/3

finite prefix point1 index1:original point=1;n=1;limit=16;signed coefficient=-17

f_n(x)=31/3;f_n(x)-limit=-17/3;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(-17, 4)

finite prefix point1 index2 signed correction = -17/4

finite prefix point1 index2 function value := sum(16, -17/4)

finite prefix point1 index2 function value = 47/4

finite prefix point1 index2 reverse signed error := sub(47/4, 16)

finite prefix point1 index2 reverse signed error = -17/4

finite prefix point1 index2:original point=1;n=2;limit=16;signed coefficient=-17

f_n(x)=47/4;f_n(x)-limit=-17/4;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(-17, 5)

finite prefix point1 index3 signed correction = -17/5

finite prefix point1 index3 function value := sum(16, -17/5)

finite prefix point1 index3 function value = 63/5

finite prefix point1 index3 reverse signed error := sub(63/5, 16)

finite prefix point1 index3 reverse signed error = -17/5

finite prefix point1 index3:original point=1;n=3;limit=16;signed coefficient=-17

f_n(x)=63/5;f_n(x)-limit=-17/5;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(-17, 6)

finite prefix point1 index4 signed correction = -17/6

finite prefix point1 index4 function value := sum(16, -17/6)

finite prefix point1 index4 function value = 79/6

finite prefix point1 index4 reverse signed error := sub(79/6, 16)

finite prefix point1 index4 reverse signed error = -17/6

finite prefix point1 index4:original point=1;n=4;limit=16;signed coefficient=-17

f_n(x)=79/6;f_n(x)-limit=-17/6;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(-17, 7)

finite prefix point1 index5 signed correction = -17/7

finite prefix point1 index5 function value := sum(16, -17/7)

finite prefix point1 index5 function value = 95/7

finite prefix point1 index5 reverse signed error := sub(95/7, 16)

finite prefix point1 index5 reverse signed error = -17/7

finite prefix point1 index5:original point=1;n=5;limit=16;signed coefficient=-17

f_n(x)=95/7;f_n(x)-limit=-17/7;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(-17, 8)

finite prefix point1 index6 signed correction = -17/8

finite prefix point1 index6 function value := sum(16, -17/8)

finite prefix point1 index6 function value = 111/8

finite prefix point1 index6 reverse signed error := sub(111/8, 16)

finite prefix point1 index6 reverse signed error = -17/8

finite prefix point1 index6:original point=1;n=6;limit=16;signed coefficient=-17

f_n(x)=111/8;f_n(x)-limit=-17/8;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(-17, 9)

finite prefix point1 index7 signed correction = -17/9

finite prefix point1 index7 function value := sum(16, -17/9)

finite prefix point1 index7 function value = 127/9

finite prefix point1 index7 reverse signed error := sub(127/9, 16)

finite prefix point1 index7 reverse signed error = -17/9

finite prefix point1 index7:original point=1;n=7;limit=16;signed coefficient=-17

f_n(x)=127/9;f_n(x)-limit=-17/9;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(-17, 10)

finite prefix point1 index8 signed correction = -17/10

finite prefix point1 index8 function value := sum(16, -17/10)

finite prefix point1 index8 function value = 143/10

finite prefix point1 index8 reverse signed error := sub(143/10, 16)

finite prefix point1 index8 reverse signed error = -17/10

finite prefix point1 index8:original point=1;n=8;limit=16;signed coefficient=-17

f_n(x)=143/10;f_n(x)-limit=-17/10;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(-17, 11)

finite prefix point1 index9 signed correction = -17/11

finite prefix point1 index9 function value := sum(16, -17/11)

finite prefix point1 index9 function value = 159/11

finite prefix point1 index9 reverse signed error := sub(159/11, 16)

finite prefix point1 index9 reverse signed error = -17/11

finite prefix point1 index9:original point=1;n=9;limit=16;signed coefficient=-17

f_n(x)=159/11;f_n(x)-limit=-17/11;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(-17, 12)

finite prefix point1 index10 signed correction = -17/12

finite prefix point1 index10 function value := sum(16, -17/12)

finite prefix point1 index10 function value = 175/12

finite prefix point1 index10 reverse signed error := sub(175/12, 16)

finite prefix point1 index10 reverse signed error = -17/12

finite prefix point1 index10:original point=1;n=10;limit=16;signed coefficient=-17

f_n(x)=175/12;f_n(x)-limit=-17/12;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(-17, 13)

finite prefix point1 index11 signed correction = -17/13

finite prefix point1 index11 function value := sum(16, -17/13)

finite prefix point1 index11 function value = 191/13

finite prefix point1 index11 reverse signed error := sub(191/13, 16)

finite prefix point1 index11 reverse signed error = -17/13

finite prefix point1 index11:original point=1;n=11;limit=16;signed coefficient=-17

f_n(x)=191/13;f_n(x)-limit=-17/13;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(-17, 14)

finite prefix point1 index12 signed correction = -17/14

finite prefix point1 index12 function value := sum(16, -17/14)

finite prefix point1 index12 function value = 207/14

finite prefix point1 index12 reverse signed error := sub(207/14, 16)

finite prefix point1 index12 reverse signed error = -17/14

finite prefix point1 index12:original point=1;n=12;limit=16;signed coefficient=-17

f_n(x)=207/14;f_n(x)-limit=-17/14;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(-17, 15)

finite prefix point1 index13 signed correction = -17/15

finite prefix point1 index13 function value := sum(16, -17/15)

finite prefix point1 index13 function value = 223/15

finite prefix point1 index13 reverse signed error := sub(223/15, 16)

finite prefix point1 index13 reverse signed error = -17/15

finite prefix point1 index13:original point=1;n=13;limit=16;signed coefficient=-17

f_n(x)=223/15;f_n(x)-limit=-17/15;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(-17, 16)

finite prefix point1 index14 signed correction = -17/16

finite prefix point1 index14 function value := sum(16, -17/16)

finite prefix point1 index14 function value = 239/16

finite prefix point1 index14 reverse signed error := sub(239/16, 16)

finite prefix point1 index14 reverse signed error = -17/16

finite prefix point1 index14:original point=1;n=14;limit=16;signed coefficient=-17

f_n(x)=239/16;f_n(x)-limit=-17/16;this error field remains SIGNED

finite prefix point1 index15 denominator := sum(15, 2)

finite prefix point1 index15 denominator = 17

finite prefix point1 index15 signed correction := div(-17, 17)

finite prefix point1 index15 signed correction = -1

finite prefix point1 index15 function value := sum(16, -1)

finite prefix point1 index15 function value = 15

finite prefix point1 index15 reverse signed error := sub(15, 16)

finite prefix point1 index15 reverse signed error = -1

finite prefix point1 index15:original point=1;n=15;limit=16;signed coefficient=-17

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

finite observed length=16;complete domain-order rows=[[0, [23, 61/3, 19, 91/5, 53/3, 121/7, 17, 151/9, 83/5, 181/11, 49/3, 211/13, 113/7, 241/15, 16, 271/17]], [1, [15/2, 31/3, 47/4, 63/5, 79/6, 95/7, 111/8, 127/9, 143/10, 159/11, 175/12, 191/13, 207/14, 223/15, 239/16, 15]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S160 · premise_and_result_audit — Recompute finite prefix point1 index15 reverse signed error with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute finite prefix point1 index15 function value with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute finite prefix point1 index15 signed correction with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute finite prefix point1 index15 denominator with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S155 · premise_and_result_audit — Recompute finite prefix point1 index14 reverse signed error with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute finite prefix point1 index14 function value with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute finite prefix point1 index14 signed correction with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute finite prefix point1 index14 denominator with exact rational arithmetic, not rounded decimals.
• 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 index13 reverse signed error with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute finite prefix point1 index13 function value with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute finite prefix point1 index13 signed correction with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute finite prefix point1 index13 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 index12 reverse signed error with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute finite prefix point1 index12 function value with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute finite prefix point1 index12 signed correction with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute finite prefix point1 index12 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 index11 reverse signed error with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute finite prefix point1 index11 function value with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute finite prefix point1 index11 signed correction with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute finite prefix point1 index11 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 index10 reverse signed error with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute finite prefix point1 index10 function value with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute finite prefix point1 index10 signed correction with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute finite prefix point1 index10 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 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute finite prefix point1 index9 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 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute finite prefix point1 index8 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 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute finite prefix point1 index7 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 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute finite prefix point1 index6 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 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute finite prefix point1 index5 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 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute finite prefix point1 index4 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 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute finite prefix point1 index3 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 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute finite prefix point1 index2 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 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute finite prefix point1 index1 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 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute finite prefix point1 index0 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 point0 index15 reverse signed error with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute finite prefix point0 index15 function value with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute finite prefix point0 index15 signed correction with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute finite prefix point0 index15 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

(16, [[0, [23, 61/3, 19, 91/5, 53/3, 121/7, 17, 151/9, 83/5, 181/11, 49/3, 211/13, 113/7, 241/15, 16, 271/17]], [1, [15/2, 31/3, 47/4, 63/5, 79/6, 95/7, 111/8, 127/9, 143/10, 159/11, 175/12, 191/13, 207/14, 223/15, 239/16, 15]]], 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.