Question 17588: 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=[12, 13], coefficients=[13, -14], 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=13, 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=[12, 13], coefficients=[13, -14], 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=13, 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=[12, 13];signed coefficients=[13, -14];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(13, 2)

finite prefix point0 index0 signed correction = 13/2

finite prefix point0 index0 function value := sum(12, 13/2)

finite prefix point0 index0 function value = 37/2

finite prefix point0 index0 reverse signed error := sub(37/2, 12)

finite prefix point0 index0 reverse signed error = 13/2

finite prefix point0 index0:original point=0;n=0;limit=12;signed coefficient=13

f_n(x)=37/2;f_n(x)-limit=13/2;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(13, 3)

finite prefix point0 index1 signed correction = 13/3

finite prefix point0 index1 function value := sum(12, 13/3)

finite prefix point0 index1 function value = 49/3

finite prefix point0 index1 reverse signed error := sub(49/3, 12)

finite prefix point0 index1 reverse signed error = 13/3

finite prefix point0 index1:original point=0;n=1;limit=12;signed coefficient=13

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

finite prefix point0 index2 signed correction = 13/4

finite prefix point0 index2 function value := sum(12, 13/4)

finite prefix point0 index2 function value = 61/4

finite prefix point0 index2 reverse signed error := sub(61/4, 12)

finite prefix point0 index2 reverse signed error = 13/4

finite prefix point0 index2:original point=0;n=2;limit=12;signed coefficient=13

f_n(x)=61/4;f_n(x)-limit=13/4;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(13, 5)

finite prefix point0 index3 signed correction = 13/5

finite prefix point0 index3 function value := sum(12, 13/5)

finite prefix point0 index3 function value = 73/5

finite prefix point0 index3 reverse signed error := sub(73/5, 12)

finite prefix point0 index3 reverse signed error = 13/5

finite prefix point0 index3:original point=0;n=3;limit=12;signed coefficient=13

f_n(x)=73/5;f_n(x)-limit=13/5;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(13, 6)

finite prefix point0 index4 signed correction = 13/6

finite prefix point0 index4 function value := sum(12, 13/6)

finite prefix point0 index4 function value = 85/6

finite prefix point0 index4 reverse signed error := sub(85/6, 12)

finite prefix point0 index4 reverse signed error = 13/6

finite prefix point0 index4:original point=0;n=4;limit=12;signed coefficient=13

f_n(x)=85/6;f_n(x)-limit=13/6;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(13, 7)

finite prefix point0 index5 signed correction = 13/7

finite prefix point0 index5 function value := sum(12, 13/7)

finite prefix point0 index5 function value = 97/7

finite prefix point0 index5 reverse signed error := sub(97/7, 12)

finite prefix point0 index5 reverse signed error = 13/7

finite prefix point0 index5:original point=0;n=5;limit=12;signed coefficient=13

f_n(x)=97/7;f_n(x)-limit=13/7;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(13, 8)

finite prefix point0 index6 signed correction = 13/8

finite prefix point0 index6 function value := sum(12, 13/8)

finite prefix point0 index6 function value = 109/8

finite prefix point0 index6 reverse signed error := sub(109/8, 12)

finite prefix point0 index6 reverse signed error = 13/8

finite prefix point0 index6:original point=0;n=6;limit=12;signed coefficient=13

f_n(x)=109/8;f_n(x)-limit=13/8;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(13, 9)

finite prefix point0 index7 signed correction = 13/9

finite prefix point0 index7 function value := sum(12, 13/9)

finite prefix point0 index7 function value = 121/9

finite prefix point0 index7 reverse signed error := sub(121/9, 12)

finite prefix point0 index7 reverse signed error = 13/9

finite prefix point0 index7:original point=0;n=7;limit=12;signed coefficient=13

f_n(x)=121/9;f_n(x)-limit=13/9;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(13, 10)

finite prefix point0 index8 signed correction = 13/10

finite prefix point0 index8 function value := sum(12, 13/10)

finite prefix point0 index8 function value = 133/10

finite prefix point0 index8 reverse signed error := sub(133/10, 12)

finite prefix point0 index8 reverse signed error = 13/10

finite prefix point0 index8:original point=0;n=8;limit=12;signed coefficient=13

f_n(x)=133/10;f_n(x)-limit=13/10;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(13, 11)

finite prefix point0 index9 signed correction = 13/11

finite prefix point0 index9 function value := sum(12, 13/11)

finite prefix point0 index9 function value = 145/11

finite prefix point0 index9 reverse signed error := sub(145/11, 12)

finite prefix point0 index9 reverse signed error = 13/11

finite prefix point0 index9:original point=0;n=9;limit=12;signed coefficient=13

f_n(x)=145/11;f_n(x)-limit=13/11;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(13, 12)

finite prefix point0 index10 signed correction = 13/12

finite prefix point0 index10 function value := sum(12, 13/12)

finite prefix point0 index10 function value = 157/12

finite prefix point0 index10 reverse signed error := sub(157/12, 12)

finite prefix point0 index10 reverse signed error = 13/12

finite prefix point0 index10:original point=0;n=10;limit=12;signed coefficient=13

f_n(x)=157/12;f_n(x)-limit=13/12;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(13, 13)

finite prefix point0 index11 signed correction = 1

finite prefix point0 index11 function value := sum(12, 1)

finite prefix point0 index11 function value = 13

finite prefix point0 index11 reverse signed error := sub(13, 12)

finite prefix point0 index11 reverse signed error = 1

finite prefix point0 index11:original point=0;n=11;limit=12;signed coefficient=13

f_n(x)=13;f_n(x)-limit=1;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(13, 14)

finite prefix point0 index12 signed correction = 13/14

finite prefix point0 index12 function value := sum(12, 13/14)

finite prefix point0 index12 function value = 181/14

finite prefix point0 index12 reverse signed error := sub(181/14, 12)

finite prefix point0 index12 reverse signed error = 13/14

finite prefix point0 index12:original point=0;n=12;limit=12;signed coefficient=13

f_n(x)=181/14;f_n(x)-limit=13/14;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(-14, 2)

finite prefix point1 index0 signed correction = -7

finite prefix point1 index0 function value := sum(13, -7)

finite prefix point1 index0 function value = 6

finite prefix point1 index0 reverse signed error := sub(6, 13)

finite prefix point1 index0 reverse signed error = -7

finite prefix point1 index0:original point=1;n=0;limit=13;signed coefficient=-14

f_n(x)=6;f_n(x)-limit=-7;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(-14, 3)

finite prefix point1 index1 signed correction = -14/3

finite prefix point1 index1 function value := sum(13, -14/3)

finite prefix point1 index1 function value = 25/3

finite prefix point1 index1 reverse signed error := sub(25/3, 13)

finite prefix point1 index1 reverse signed error = -14/3

finite prefix point1 index1:original point=1;n=1;limit=13;signed coefficient=-14

f_n(x)=25/3;f_n(x)-limit=-14/3;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(-14, 4)

finite prefix point1 index2 signed correction = -7/2

finite prefix point1 index2 function value := sum(13, -7/2)

finite prefix point1 index2 function value = 19/2

finite prefix point1 index2 reverse signed error := sub(19/2, 13)

finite prefix point1 index2 reverse signed error = -7/2

finite prefix point1 index2:original point=1;n=2;limit=13;signed coefficient=-14

f_n(x)=19/2;f_n(x)-limit=-7/2;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(-14, 5)

finite prefix point1 index3 signed correction = -14/5

finite prefix point1 index3 function value := sum(13, -14/5)

finite prefix point1 index3 function value = 51/5

finite prefix point1 index3 reverse signed error := sub(51/5, 13)

finite prefix point1 index3 reverse signed error = -14/5

finite prefix point1 index3:original point=1;n=3;limit=13;signed coefficient=-14

f_n(x)=51/5;f_n(x)-limit=-14/5;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(-14, 6)

finite prefix point1 index4 signed correction = -7/3

finite prefix point1 index4 function value := sum(13, -7/3)

finite prefix point1 index4 function value = 32/3

finite prefix point1 index4 reverse signed error := sub(32/3, 13)

finite prefix point1 index4 reverse signed error = -7/3

finite prefix point1 index4:original point=1;n=4;limit=13;signed coefficient=-14

f_n(x)=32/3;f_n(x)-limit=-7/3;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(-14, 7)

finite prefix point1 index5 signed correction = -2

finite prefix point1 index5 function value := sum(13, -2)

finite prefix point1 index5 function value = 11

finite prefix point1 index5 reverse signed error := sub(11, 13)

finite prefix point1 index5 reverse signed error = -2

finite prefix point1 index5:original point=1;n=5;limit=13;signed coefficient=-14

f_n(x)=11;f_n(x)-limit=-2;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(-14, 8)

finite prefix point1 index6 signed correction = -7/4

finite prefix point1 index6 function value := sum(13, -7/4)

finite prefix point1 index6 function value = 45/4

finite prefix point1 index6 reverse signed error := sub(45/4, 13)

finite prefix point1 index6 reverse signed error = -7/4

finite prefix point1 index6:original point=1;n=6;limit=13;signed coefficient=-14

f_n(x)=45/4;f_n(x)-limit=-7/4;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(-14, 9)

finite prefix point1 index7 signed correction = -14/9

finite prefix point1 index7 function value := sum(13, -14/9)

finite prefix point1 index7 function value = 103/9

finite prefix point1 index7 reverse signed error := sub(103/9, 13)

finite prefix point1 index7 reverse signed error = -14/9

finite prefix point1 index7:original point=1;n=7;limit=13;signed coefficient=-14

f_n(x)=103/9;f_n(x)-limit=-14/9;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(-14, 10)

finite prefix point1 index8 signed correction = -7/5

finite prefix point1 index8 function value := sum(13, -7/5)

finite prefix point1 index8 function value = 58/5

finite prefix point1 index8 reverse signed error := sub(58/5, 13)

finite prefix point1 index8 reverse signed error = -7/5

finite prefix point1 index8:original point=1;n=8;limit=13;signed coefficient=-14

f_n(x)=58/5;f_n(x)-limit=-7/5;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(-14, 11)

finite prefix point1 index9 signed correction = -14/11

finite prefix point1 index9 function value := sum(13, -14/11)

finite prefix point1 index9 function value = 129/11

finite prefix point1 index9 reverse signed error := sub(129/11, 13)

finite prefix point1 index9 reverse signed error = -14/11

finite prefix point1 index9:original point=1;n=9;limit=13;signed coefficient=-14

f_n(x)=129/11;f_n(x)-limit=-14/11;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(-14, 12)

finite prefix point1 index10 signed correction = -7/6

finite prefix point1 index10 function value := sum(13, -7/6)

finite prefix point1 index10 function value = 71/6

finite prefix point1 index10 reverse signed error := sub(71/6, 13)

finite prefix point1 index10 reverse signed error = -7/6

finite prefix point1 index10:original point=1;n=10;limit=13;signed coefficient=-14

f_n(x)=71/6;f_n(x)-limit=-7/6;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(-14, 13)

finite prefix point1 index11 signed correction = -14/13

finite prefix point1 index11 function value := sum(13, -14/13)

finite prefix point1 index11 function value = 155/13

finite prefix point1 index11 reverse signed error := sub(155/13, 13)

finite prefix point1 index11 reverse signed error = -14/13

finite prefix point1 index11:original point=1;n=11;limit=13;signed coefficient=-14

f_n(x)=155/13;f_n(x)-limit=-14/13;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(-14, 14)

finite prefix point1 index12 signed correction = -1

finite prefix point1 index12 function value := sum(13, -1)

finite prefix point1 index12 function value = 12

finite prefix point1 index12 reverse signed error := sub(12, 13)

finite prefix point1 index12 reverse signed error = -1

finite prefix point1 index12:original point=1;n=12;limit=13;signed coefficient=-14

f_n(x)=12;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=13;complete domain-order rows=[[0, [37/2, 49/3, 61/4, 73/5, 85/6, 97/7, 109/8, 121/9, 133/10, 145/11, 157/12, 13, 181/14]], [1, [6, 25/3, 19/2, 51/5, 32/3, 11, 45/4, 103/9, 58/5, 129/11, 71/6, 155/13, 12]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 index12 reverse signed error with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute finite prefix point1 index12 function value with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute finite prefix point1 index12 signed correction with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute finite prefix point1 index12 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 index11 reverse signed error with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute finite prefix point1 index11 function value with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute finite prefix point1 index11 signed correction with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute finite prefix point1 index11 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 index10 reverse signed error with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute finite prefix point1 index10 function value with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute finite prefix point1 index10 signed correction with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute finite prefix point1 index10 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 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute finite prefix point1 index9 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 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute finite prefix point1 index8 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 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute finite prefix point1 index7 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 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute finite prefix point1 index6 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 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute finite prefix point1 index5 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 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute finite prefix point1 index4 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 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute finite prefix point1 index3 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 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute finite prefix point1 index2 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 point1 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute finite prefix point1 index1 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 point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute finite prefix point1 index0 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

(13, [[0, [37/2, 49/3, 61/4, 73/5, 85/6, 97/7, 109/8, 121/9, 133/10, 145/11, 157/12, 13, 181/14]], [1, [6, 25/3, 19/2, 51/5, 32/3, 11, 45/4, 103/9, 58/5, 129/11, 71/6, 155/13, 12]]], 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.