Question 17593: 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=[17, 18], coefficients=[18, -19], 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=18, 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=[17, 18], coefficients=[18, -19], 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=18, 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=[17, 18];signed coefficients=[18, -19];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(18, 2)

finite prefix point0 index0 signed correction = 9

finite prefix point0 index0 function value := sum(17, 9)

finite prefix point0 index0 function value = 26

finite prefix point0 index0 reverse signed error := sub(26, 17)

finite prefix point0 index0 reverse signed error = 9

finite prefix point0 index0:original point=0;n=0;limit=17;signed coefficient=18

f_n(x)=26;f_n(x)-limit=9;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(18, 3)

finite prefix point0 index1 signed correction = 6

finite prefix point0 index1 function value := sum(17, 6)

finite prefix point0 index1 function value = 23

finite prefix point0 index1 reverse signed error := sub(23, 17)

finite prefix point0 index1 reverse signed error = 6

finite prefix point0 index1:original point=0;n=1;limit=17;signed coefficient=18

f_n(x)=23;f_n(x)-limit=6;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(18, 4)

finite prefix point0 index2 signed correction = 9/2

finite prefix point0 index2 function value := sum(17, 9/2)

finite prefix point0 index2 function value = 43/2

finite prefix point0 index2 reverse signed error := sub(43/2, 17)

finite prefix point0 index2 reverse signed error = 9/2

finite prefix point0 index2:original point=0;n=2;limit=17;signed coefficient=18

f_n(x)=43/2;f_n(x)-limit=9/2;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(18, 5)

finite prefix point0 index3 signed correction = 18/5

finite prefix point0 index3 function value := sum(17, 18/5)

finite prefix point0 index3 function value = 103/5

finite prefix point0 index3 reverse signed error := sub(103/5, 17)

finite prefix point0 index3 reverse signed error = 18/5

finite prefix point0 index3:original point=0;n=3;limit=17;signed coefficient=18

f_n(x)=103/5;f_n(x)-limit=18/5;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(18, 6)

finite prefix point0 index4 signed correction = 3

finite prefix point0 index4 function value := sum(17, 3)

finite prefix point0 index4 function value = 20

finite prefix point0 index4 reverse signed error := sub(20, 17)

finite prefix point0 index4 reverse signed error = 3

finite prefix point0 index4:original point=0;n=4;limit=17;signed coefficient=18

f_n(x)=20;f_n(x)-limit=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(18, 7)

finite prefix point0 index5 signed correction = 18/7

finite prefix point0 index5 function value := sum(17, 18/7)

finite prefix point0 index5 function value = 137/7

finite prefix point0 index5 reverse signed error := sub(137/7, 17)

finite prefix point0 index5 reverse signed error = 18/7

finite prefix point0 index5:original point=0;n=5;limit=17;signed coefficient=18

f_n(x)=137/7;f_n(x)-limit=18/7;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(18, 8)

finite prefix point0 index6 signed correction = 9/4

finite prefix point0 index6 function value := sum(17, 9/4)

finite prefix point0 index6 function value = 77/4

finite prefix point0 index6 reverse signed error := sub(77/4, 17)

finite prefix point0 index6 reverse signed error = 9/4

finite prefix point0 index6:original point=0;n=6;limit=17;signed coefficient=18

f_n(x)=77/4;f_n(x)-limit=9/4;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(18, 9)

finite prefix point0 index7 signed correction = 2

finite prefix point0 index7 function value := sum(17, 2)

finite prefix point0 index7 function value = 19

finite prefix point0 index7 reverse signed error := sub(19, 17)

finite prefix point0 index7 reverse signed error = 2

finite prefix point0 index7:original point=0;n=7;limit=17;signed coefficient=18

f_n(x)=19;f_n(x)-limit=2;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(18, 10)

finite prefix point0 index8 signed correction = 9/5

finite prefix point0 index8 function value := sum(17, 9/5)

finite prefix point0 index8 function value = 94/5

finite prefix point0 index8 reverse signed error := sub(94/5, 17)

finite prefix point0 index8 reverse signed error = 9/5

finite prefix point0 index8:original point=0;n=8;limit=17;signed coefficient=18

f_n(x)=94/5;f_n(x)-limit=9/5;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(18, 11)

finite prefix point0 index9 signed correction = 18/11

finite prefix point0 index9 function value := sum(17, 18/11)

finite prefix point0 index9 function value = 205/11

finite prefix point0 index9 reverse signed error := sub(205/11, 17)

finite prefix point0 index9 reverse signed error = 18/11

finite prefix point0 index9:original point=0;n=9;limit=17;signed coefficient=18

f_n(x)=205/11;f_n(x)-limit=18/11;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(18, 12)

finite prefix point0 index10 signed correction = 3/2

finite prefix point0 index10 function value := sum(17, 3/2)

finite prefix point0 index10 function value = 37/2

finite prefix point0 index10 reverse signed error := sub(37/2, 17)

finite prefix point0 index10 reverse signed error = 3/2

finite prefix point0 index10:original point=0;n=10;limit=17;signed coefficient=18

f_n(x)=37/2;f_n(x)-limit=3/2;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(18, 13)

finite prefix point0 index11 signed correction = 18/13

finite prefix point0 index11 function value := sum(17, 18/13)

finite prefix point0 index11 function value = 239/13

finite prefix point0 index11 reverse signed error := sub(239/13, 17)

finite prefix point0 index11 reverse signed error = 18/13

finite prefix point0 index11:original point=0;n=11;limit=17;signed coefficient=18

f_n(x)=239/13;f_n(x)-limit=18/13;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(18, 14)

finite prefix point0 index12 signed correction = 9/7

finite prefix point0 index12 function value := sum(17, 9/7)

finite prefix point0 index12 function value = 128/7

finite prefix point0 index12 reverse signed error := sub(128/7, 17)

finite prefix point0 index12 reverse signed error = 9/7

finite prefix point0 index12:original point=0;n=12;limit=17;signed coefficient=18

f_n(x)=128/7;f_n(x)-limit=9/7;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(18, 15)

finite prefix point0 index13 signed correction = 6/5

finite prefix point0 index13 function value := sum(17, 6/5)

finite prefix point0 index13 function value = 91/5

finite prefix point0 index13 reverse signed error := sub(91/5, 17)

finite prefix point0 index13 reverse signed error = 6/5

finite prefix point0 index13:original point=0;n=13;limit=17;signed coefficient=18

f_n(x)=91/5;f_n(x)-limit=6/5;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(18, 16)

finite prefix point0 index14 signed correction = 9/8

finite prefix point0 index14 function value := sum(17, 9/8)

finite prefix point0 index14 function value = 145/8

finite prefix point0 index14 reverse signed error := sub(145/8, 17)

finite prefix point0 index14 reverse signed error = 9/8

finite prefix point0 index14:original point=0;n=14;limit=17;signed coefficient=18

f_n(x)=145/8;f_n(x)-limit=9/8;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(18, 17)

finite prefix point0 index15 signed correction = 18/17

finite prefix point0 index15 function value := sum(17, 18/17)

finite prefix point0 index15 function value = 307/17

finite prefix point0 index15 reverse signed error := sub(307/17, 17)

finite prefix point0 index15 reverse signed error = 18/17

finite prefix point0 index15:original point=0;n=15;limit=17;signed coefficient=18

f_n(x)=307/17;f_n(x)-limit=18/17;this error field remains SIGNED

finite prefix point0 index16 denominator := sum(16, 2)

finite prefix point0 index16 denominator = 18

finite prefix point0 index16 signed correction := div(18, 18)

finite prefix point0 index16 signed correction = 1

finite prefix point0 index16 function value := sum(17, 1)

finite prefix point0 index16 function value = 18

finite prefix point0 index16 reverse signed error := sub(18, 17)

finite prefix point0 index16 reverse signed error = 1

finite prefix point0 index16:original point=0;n=16;limit=17;signed coefficient=18

f_n(x)=18;f_n(x)-limit=1;this error field remains SIGNED

finite prefix point0 index17 denominator := sum(17, 2)

finite prefix point0 index17 denominator = 19

finite prefix point0 index17 signed correction := div(18, 19)

finite prefix point0 index17 signed correction = 18/19

finite prefix point0 index17 function value := sum(17, 18/19)

finite prefix point0 index17 function value = 341/19

finite prefix point0 index17 reverse signed error := sub(341/19, 17)

finite prefix point0 index17 reverse signed error = 18/19

finite prefix point0 index17:original point=0;n=17;limit=17;signed coefficient=18

f_n(x)=341/19;f_n(x)-limit=18/19;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(-19, 2)

finite prefix point1 index0 signed correction = -19/2

finite prefix point1 index0 function value := sum(18, -19/2)

finite prefix point1 index0 function value = 17/2

finite prefix point1 index0 reverse signed error := sub(17/2, 18)

finite prefix point1 index0 reverse signed error = -19/2

finite prefix point1 index0:original point=1;n=0;limit=18;signed coefficient=-19

f_n(x)=17/2;f_n(x)-limit=-19/2;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(-19, 3)

finite prefix point1 index1 signed correction = -19/3

finite prefix point1 index1 function value := sum(18, -19/3)

finite prefix point1 index1 function value = 35/3

finite prefix point1 index1 reverse signed error := sub(35/3, 18)

finite prefix point1 index1 reverse signed error = -19/3

finite prefix point1 index1:original point=1;n=1;limit=18;signed coefficient=-19

f_n(x)=35/3;f_n(x)-limit=-19/3;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(-19, 4)

finite prefix point1 index2 signed correction = -19/4

finite prefix point1 index2 function value := sum(18, -19/4)

finite prefix point1 index2 function value = 53/4

finite prefix point1 index2 reverse signed error := sub(53/4, 18)

finite prefix point1 index2 reverse signed error = -19/4

finite prefix point1 index2:original point=1;n=2;limit=18;signed coefficient=-19

f_n(x)=53/4;f_n(x)-limit=-19/4;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(-19, 5)

finite prefix point1 index3 signed correction = -19/5

finite prefix point1 index3 function value := sum(18, -19/5)

finite prefix point1 index3 function value = 71/5

finite prefix point1 index3 reverse signed error := sub(71/5, 18)

finite prefix point1 index3 reverse signed error = -19/5

finite prefix point1 index3:original point=1;n=3;limit=18;signed coefficient=-19

f_n(x)=71/5;f_n(x)-limit=-19/5;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(-19, 6)

finite prefix point1 index4 signed correction = -19/6

finite prefix point1 index4 function value := sum(18, -19/6)

finite prefix point1 index4 function value = 89/6

finite prefix point1 index4 reverse signed error := sub(89/6, 18)

finite prefix point1 index4 reverse signed error = -19/6

finite prefix point1 index4:original point=1;n=4;limit=18;signed coefficient=-19

f_n(x)=89/6;f_n(x)-limit=-19/6;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(-19, 7)

finite prefix point1 index5 signed correction = -19/7

finite prefix point1 index5 function value := sum(18, -19/7)

finite prefix point1 index5 function value = 107/7

finite prefix point1 index5 reverse signed error := sub(107/7, 18)

finite prefix point1 index5 reverse signed error = -19/7

finite prefix point1 index5:original point=1;n=5;limit=18;signed coefficient=-19

f_n(x)=107/7;f_n(x)-limit=-19/7;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(-19, 8)

finite prefix point1 index6 signed correction = -19/8

finite prefix point1 index6 function value := sum(18, -19/8)

finite prefix point1 index6 function value = 125/8

finite prefix point1 index6 reverse signed error := sub(125/8, 18)

finite prefix point1 index6 reverse signed error = -19/8

finite prefix point1 index6:original point=1;n=6;limit=18;signed coefficient=-19

f_n(x)=125/8;f_n(x)-limit=-19/8;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(-19, 9)

finite prefix point1 index7 signed correction = -19/9

finite prefix point1 index7 function value := sum(18, -19/9)

finite prefix point1 index7 function value = 143/9

finite prefix point1 index7 reverse signed error := sub(143/9, 18)

finite prefix point1 index7 reverse signed error = -19/9

finite prefix point1 index7:original point=1;n=7;limit=18;signed coefficient=-19

f_n(x)=143/9;f_n(x)-limit=-19/9;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(-19, 10)

finite prefix point1 index8 signed correction = -19/10

finite prefix point1 index8 function value := sum(18, -19/10)

finite prefix point1 index8 function value = 161/10

finite prefix point1 index8 reverse signed error := sub(161/10, 18)

finite prefix point1 index8 reverse signed error = -19/10

finite prefix point1 index8:original point=1;n=8;limit=18;signed coefficient=-19

f_n(x)=161/10;f_n(x)-limit=-19/10;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(-19, 11)

finite prefix point1 index9 signed correction = -19/11

finite prefix point1 index9 function value := sum(18, -19/11)

finite prefix point1 index9 function value = 179/11

finite prefix point1 index9 reverse signed error := sub(179/11, 18)

finite prefix point1 index9 reverse signed error = -19/11

finite prefix point1 index9:original point=1;n=9;limit=18;signed coefficient=-19

f_n(x)=179/11;f_n(x)-limit=-19/11;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(-19, 12)

finite prefix point1 index10 signed correction = -19/12

finite prefix point1 index10 function value := sum(18, -19/12)

finite prefix point1 index10 function value = 197/12

finite prefix point1 index10 reverse signed error := sub(197/12, 18)

finite prefix point1 index10 reverse signed error = -19/12

finite prefix point1 index10:original point=1;n=10;limit=18;signed coefficient=-19

f_n(x)=197/12;f_n(x)-limit=-19/12;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(-19, 13)

finite prefix point1 index11 signed correction = -19/13

finite prefix point1 index11 function value := sum(18, -19/13)

finite prefix point1 index11 function value = 215/13

finite prefix point1 index11 reverse signed error := sub(215/13, 18)

finite prefix point1 index11 reverse signed error = -19/13

finite prefix point1 index11:original point=1;n=11;limit=18;signed coefficient=-19

f_n(x)=215/13;f_n(x)-limit=-19/13;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(-19, 14)

finite prefix point1 index12 signed correction = -19/14

finite prefix point1 index12 function value := sum(18, -19/14)

finite prefix point1 index12 function value = 233/14

finite prefix point1 index12 reverse signed error := sub(233/14, 18)

finite prefix point1 index12 reverse signed error = -19/14

finite prefix point1 index12:original point=1;n=12;limit=18;signed coefficient=-19

f_n(x)=233/14;f_n(x)-limit=-19/14;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(-19, 15)

finite prefix point1 index13 signed correction = -19/15

finite prefix point1 index13 function value := sum(18, -19/15)

finite prefix point1 index13 function value = 251/15

finite prefix point1 index13 reverse signed error := sub(251/15, 18)

finite prefix point1 index13 reverse signed error = -19/15

finite prefix point1 index13:original point=1;n=13;limit=18;signed coefficient=-19

f_n(x)=251/15;f_n(x)-limit=-19/15;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(-19, 16)

finite prefix point1 index14 signed correction = -19/16

finite prefix point1 index14 function value := sum(18, -19/16)

finite prefix point1 index14 function value = 269/16

finite prefix point1 index14 reverse signed error := sub(269/16, 18)

finite prefix point1 index14 reverse signed error = -19/16

finite prefix point1 index14:original point=1;n=14;limit=18;signed coefficient=-19

f_n(x)=269/16;f_n(x)-limit=-19/16;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(-19, 17)

finite prefix point1 index15 signed correction = -19/17

finite prefix point1 index15 function value := sum(18, -19/17)

finite prefix point1 index15 function value = 287/17

finite prefix point1 index15 reverse signed error := sub(287/17, 18)

finite prefix point1 index15 reverse signed error = -19/17

finite prefix point1 index15:original point=1;n=15;limit=18;signed coefficient=-19

f_n(x)=287/17;f_n(x)-limit=-19/17;this error field remains SIGNED

finite prefix point1 index16 denominator := sum(16, 2)

finite prefix point1 index16 denominator = 18

finite prefix point1 index16 signed correction := div(-19, 18)

finite prefix point1 index16 signed correction = -19/18

finite prefix point1 index16 function value := sum(18, -19/18)

finite prefix point1 index16 function value = 305/18

finite prefix point1 index16 reverse signed error := sub(305/18, 18)

finite prefix point1 index16 reverse signed error = -19/18

finite prefix point1 index16:original point=1;n=16;limit=18;signed coefficient=-19

f_n(x)=305/18;f_n(x)-limit=-19/18;this error field remains SIGNED

finite prefix point1 index17 denominator := sum(17, 2)

finite prefix point1 index17 denominator = 19

finite prefix point1 index17 signed correction := div(-19, 19)

finite prefix point1 index17 signed correction = -1

finite prefix point1 index17 function value := sum(18, -1)

finite prefix point1 index17 function value = 17

finite prefix point1 index17 reverse signed error := sub(17, 18)

finite prefix point1 index17 reverse signed error = -1

finite prefix point1 index17:original point=1;n=17;limit=18;signed coefficient=-19

f_n(x)=17;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=18;complete domain-order rows=[[0, [26, 23, 43/2, 103/5, 20, 137/7, 77/4, 19, 94/5, 205/11, 37/2, 239/13, 128/7, 91/5, 145/8, 307/17, 18, 341/19]], [1, [17/2, 35/3, 53/4, 71/5, 89/6, 107/7, 125/8, 143/9, 161/10, 179/11, 197/12, 215/13, 233/14, 251/15, 269/16, 287/17, 305/18, 17]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S180 · premise_and_result_audit — Recompute finite prefix point1 index17 reverse signed error with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute finite prefix point1 index17 function value with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute finite prefix point1 index17 signed correction with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute finite prefix point1 index17 denominator with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S175 · premise_and_result_audit — Recompute finite prefix point1 index16 reverse signed error with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Recompute finite prefix point1 index16 function value with exact rational arithmetic, not rounded decimals.
• S173 · premise_and_result_audit — Recompute finite prefix point1 index16 signed correction with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Recompute finite prefix point1 index16 denominator with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S170 · premise_and_result_audit — Recompute finite prefix point1 index15 reverse signed error with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Recompute finite prefix point1 index15 function value with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Recompute finite prefix point1 index15 signed correction with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Recompute finite prefix point1 index15 denominator with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S165 · premise_and_result_audit — Recompute finite prefix point1 index14 reverse signed error with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Recompute finite prefix point1 index14 function value with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Recompute finite prefix point1 index14 signed correction with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute finite prefix point1 index14 denominator with exact rational arithmetic, not rounded decimals.
• 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 index13 reverse signed error with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute finite prefix point1 index13 function value with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute finite prefix point1 index13 signed correction with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute finite prefix point1 index13 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 index12 reverse signed error with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute finite prefix point1 index12 function value with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute finite prefix point1 index12 signed correction with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute finite prefix point1 index12 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 index11 reverse signed error with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute finite prefix point1 index11 function value with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute finite prefix point1 index11 signed correction with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute finite prefix point1 index11 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 index10 reverse signed error with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute finite prefix point1 index10 function value with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute finite prefix point1 index10 signed correction with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute finite prefix point1 index10 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 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute finite prefix point1 index9 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 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute finite prefix point1 index8 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 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute finite prefix point1 index7 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 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute finite prefix point1 index6 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 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute finite prefix point1 index5 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 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute finite prefix point1 index4 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 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute finite prefix point1 index3 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 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute finite prefix point1 index2 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 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute finite prefix point1 index1 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 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute finite prefix point1 index0 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 point0 index17 reverse signed error with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute finite prefix point0 index17 function value with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute finite prefix point0 index17 signed correction with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute finite prefix point0 index17 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 point0 index16 reverse signed error with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute finite prefix point0 index16 function value with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute finite prefix point0 index16 signed correction with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute finite prefix point0 index16 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

(18, [[0, [26, 23, 43/2, 103/5, 20, 137/7, 77/4, 19, 94/5, 205/11, 37/2, 239/13, 128/7, 91/5, 145/8, 307/17, 18, 341/19]], [1, [17/2, 35/3, 53/4, 71/5, 89/6, 107/7, 125/8, 143/9, 161/10, 179/11, 197/12, 215/13, 233/14, 251/15, 269/16, 287/17, 305/18, 17]]], 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.