Question 17592: 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=[16, 17], coefficients=[17, -18], 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=17, 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=[16, 17], coefficients=[17, -18], 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=17, 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=[16, 17];signed coefficients=[17, -18];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(17, 2)

finite prefix point0 index0 signed correction = 17/2

finite prefix point0 index0 function value := sum(16, 17/2)

finite prefix point0 index0 function value = 49/2

finite prefix point0 index0 reverse signed error := sub(49/2, 16)

finite prefix point0 index0 reverse signed error = 17/2

finite prefix point0 index0:original point=0;n=0;limit=16;signed coefficient=17

f_n(x)=49/2;f_n(x)-limit=17/2;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(17, 3)

finite prefix point0 index1 signed correction = 17/3

finite prefix point0 index1 function value := sum(16, 17/3)

finite prefix point0 index1 function value = 65/3

finite prefix point0 index1 reverse signed error := sub(65/3, 16)

finite prefix point0 index1 reverse signed error = 17/3

finite prefix point0 index1:original point=0;n=1;limit=16;signed coefficient=17

f_n(x)=65/3;f_n(x)-limit=17/3;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(17, 4)

finite prefix point0 index2 signed correction = 17/4

finite prefix point0 index2 function value := sum(16, 17/4)

finite prefix point0 index2 function value = 81/4

finite prefix point0 index2 reverse signed error := sub(81/4, 16)

finite prefix point0 index2 reverse signed error = 17/4

finite prefix point0 index2:original point=0;n=2;limit=16;signed coefficient=17

f_n(x)=81/4;f_n(x)-limit=17/4;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(17, 5)

finite prefix point0 index3 signed correction = 17/5

finite prefix point0 index3 function value := sum(16, 17/5)

finite prefix point0 index3 function value = 97/5

finite prefix point0 index3 reverse signed error := sub(97/5, 16)

finite prefix point0 index3 reverse signed error = 17/5

finite prefix point0 index3:original point=0;n=3;limit=16;signed coefficient=17

f_n(x)=97/5;f_n(x)-limit=17/5;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(17, 6)

finite prefix point0 index4 signed correction = 17/6

finite prefix point0 index4 function value := sum(16, 17/6)

finite prefix point0 index4 function value = 113/6

finite prefix point0 index4 reverse signed error := sub(113/6, 16)

finite prefix point0 index4 reverse signed error = 17/6

finite prefix point0 index4:original point=0;n=4;limit=16;signed coefficient=17

f_n(x)=113/6;f_n(x)-limit=17/6;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(17, 7)

finite prefix point0 index5 signed correction = 17/7

finite prefix point0 index5 function value := sum(16, 17/7)

finite prefix point0 index5 function value = 129/7

finite prefix point0 index5 reverse signed error := sub(129/7, 16)

finite prefix point0 index5 reverse signed error = 17/7

finite prefix point0 index5:original point=0;n=5;limit=16;signed coefficient=17

f_n(x)=129/7;f_n(x)-limit=17/7;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(17, 8)

finite prefix point0 index6 signed correction = 17/8

finite prefix point0 index6 function value := sum(16, 17/8)

finite prefix point0 index6 function value = 145/8

finite prefix point0 index6 reverse signed error := sub(145/8, 16)

finite prefix point0 index6 reverse signed error = 17/8

finite prefix point0 index6:original point=0;n=6;limit=16;signed coefficient=17

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

finite prefix point0 index7 signed correction = 17/9

finite prefix point0 index7 function value := sum(16, 17/9)

finite prefix point0 index7 function value = 161/9

finite prefix point0 index7 reverse signed error := sub(161/9, 16)

finite prefix point0 index7 reverse signed error = 17/9

finite prefix point0 index7:original point=0;n=7;limit=16;signed coefficient=17

f_n(x)=161/9;f_n(x)-limit=17/9;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(17, 10)

finite prefix point0 index8 signed correction = 17/10

finite prefix point0 index8 function value := sum(16, 17/10)

finite prefix point0 index8 function value = 177/10

finite prefix point0 index8 reverse signed error := sub(177/10, 16)

finite prefix point0 index8 reverse signed error = 17/10

finite prefix point0 index8:original point=0;n=8;limit=16;signed coefficient=17

f_n(x)=177/10;f_n(x)-limit=17/10;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(17, 11)

finite prefix point0 index9 signed correction = 17/11

finite prefix point0 index9 function value := sum(16, 17/11)

finite prefix point0 index9 function value = 193/11

finite prefix point0 index9 reverse signed error := sub(193/11, 16)

finite prefix point0 index9 reverse signed error = 17/11

finite prefix point0 index9:original point=0;n=9;limit=16;signed coefficient=17

f_n(x)=193/11;f_n(x)-limit=17/11;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(17, 12)

finite prefix point0 index10 signed correction = 17/12

finite prefix point0 index10 function value := sum(16, 17/12)

finite prefix point0 index10 function value = 209/12

finite prefix point0 index10 reverse signed error := sub(209/12, 16)

finite prefix point0 index10 reverse signed error = 17/12

finite prefix point0 index10:original point=0;n=10;limit=16;signed coefficient=17

f_n(x)=209/12;f_n(x)-limit=17/12;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(17, 13)

finite prefix point0 index11 signed correction = 17/13

finite prefix point0 index11 function value := sum(16, 17/13)

finite prefix point0 index11 function value = 225/13

finite prefix point0 index11 reverse signed error := sub(225/13, 16)

finite prefix point0 index11 reverse signed error = 17/13

finite prefix point0 index11:original point=0;n=11;limit=16;signed coefficient=17

f_n(x)=225/13;f_n(x)-limit=17/13;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(17, 14)

finite prefix point0 index12 signed correction = 17/14

finite prefix point0 index12 function value := sum(16, 17/14)

finite prefix point0 index12 function value = 241/14

finite prefix point0 index12 reverse signed error := sub(241/14, 16)

finite prefix point0 index12 reverse signed error = 17/14

finite prefix point0 index12:original point=0;n=12;limit=16;signed coefficient=17

f_n(x)=241/14;f_n(x)-limit=17/14;this error field remains SIGNED

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

finite prefix point0 index13 denominator = 15

finite prefix point0 index13 signed correction := div(17, 15)

finite prefix point0 index13 signed correction = 17/15

finite prefix point0 index13 function value := sum(16, 17/15)

finite prefix point0 index13 function value = 257/15

finite prefix point0 index13 reverse signed error := sub(257/15, 16)

finite prefix point0 index13 reverse signed error = 17/15

finite prefix point0 index13:original point=0;n=13;limit=16;signed coefficient=17

f_n(x)=257/15;f_n(x)-limit=17/15;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(17, 16)

finite prefix point0 index14 signed correction = 17/16

finite prefix point0 index14 function value := sum(16, 17/16)

finite prefix point0 index14 function value = 273/16

finite prefix point0 index14 reverse signed error := sub(273/16, 16)

finite prefix point0 index14 reverse signed error = 17/16

finite prefix point0 index14:original point=0;n=14;limit=16;signed coefficient=17

f_n(x)=273/16;f_n(x)-limit=17/16;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(17, 17)

finite prefix point0 index15 signed correction = 1

finite prefix point0 index15 function value := sum(16, 1)

finite prefix point0 index15 function value = 17

finite prefix point0 index15 reverse signed error := sub(17, 16)

finite prefix point0 index15 reverse signed error = 1

finite prefix point0 index15:original point=0;n=15;limit=16;signed coefficient=17

f_n(x)=17;f_n(x)-limit=1;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(17, 18)

finite prefix point0 index16 signed correction = 17/18

finite prefix point0 index16 function value := sum(16, 17/18)

finite prefix point0 index16 function value = 305/18

finite prefix point0 index16 reverse signed error := sub(305/18, 16)

finite prefix point0 index16 reverse signed error = 17/18

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

f_n(x)=305/18;f_n(x)-limit=17/18;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(-18, 2)

finite prefix point1 index0 signed correction = -9

finite prefix point1 index0 function value := sum(17, -9)

finite prefix point1 index0 function value = 8

finite prefix point1 index0 reverse signed error := sub(8, 17)

finite prefix point1 index0 reverse signed error = -9

finite prefix point1 index0:original point=1;n=0;limit=17;signed coefficient=-18

f_n(x)=8;f_n(x)-limit=-9;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(-18, 3)

finite prefix point1 index1 signed correction = -6

finite prefix point1 index1 function value := sum(17, -6)

finite prefix point1 index1 function value = 11

finite prefix point1 index1 reverse signed error := sub(11, 17)

finite prefix point1 index1 reverse signed error = -6

finite prefix point1 index1:original point=1;n=1;limit=17;signed coefficient=-18

f_n(x)=11;f_n(x)-limit=-6;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(-18, 4)

finite prefix point1 index2 signed correction = -9/2

finite prefix point1 index2 function value := sum(17, -9/2)

finite prefix point1 index2 function value = 25/2

finite prefix point1 index2 reverse signed error := sub(25/2, 17)

finite prefix point1 index2 reverse signed error = -9/2

finite prefix point1 index2:original point=1;n=2;limit=17;signed coefficient=-18

f_n(x)=25/2;f_n(x)-limit=-9/2;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(-18, 5)

finite prefix point1 index3 signed correction = -18/5

finite prefix point1 index3 function value := sum(17, -18/5)

finite prefix point1 index3 function value = 67/5

finite prefix point1 index3 reverse signed error := sub(67/5, 17)

finite prefix point1 index3 reverse signed error = -18/5

finite prefix point1 index3:original point=1;n=3;limit=17;signed coefficient=-18

f_n(x)=67/5;f_n(x)-limit=-18/5;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(-18, 6)

finite prefix point1 index4 signed correction = -3

finite prefix point1 index4 function value := sum(17, -3)

finite prefix point1 index4 function value = 14

finite prefix point1 index4 reverse signed error := sub(14, 17)

finite prefix point1 index4 reverse signed error = -3

finite prefix point1 index4:original point=1;n=4;limit=17;signed coefficient=-18

f_n(x)=14;f_n(x)-limit=-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(-18, 7)

finite prefix point1 index5 signed correction = -18/7

finite prefix point1 index5 function value := sum(17, -18/7)

finite prefix point1 index5 function value = 101/7

finite prefix point1 index5 reverse signed error := sub(101/7, 17)

finite prefix point1 index5 reverse signed error = -18/7

finite prefix point1 index5:original point=1;n=5;limit=17;signed coefficient=-18

f_n(x)=101/7;f_n(x)-limit=-18/7;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(-18, 8)

finite prefix point1 index6 signed correction = -9/4

finite prefix point1 index6 function value := sum(17, -9/4)

finite prefix point1 index6 function value = 59/4

finite prefix point1 index6 reverse signed error := sub(59/4, 17)

finite prefix point1 index6 reverse signed error = -9/4

finite prefix point1 index6:original point=1;n=6;limit=17;signed coefficient=-18

f_n(x)=59/4;f_n(x)-limit=-9/4;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(-18, 9)

finite prefix point1 index7 signed correction = -2

finite prefix point1 index7 function value := sum(17, -2)

finite prefix point1 index7 function value = 15

finite prefix point1 index7 reverse signed error := sub(15, 17)

finite prefix point1 index7 reverse signed error = -2

finite prefix point1 index7:original point=1;n=7;limit=17;signed coefficient=-18

f_n(x)=15;f_n(x)-limit=-2;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(-18, 10)

finite prefix point1 index8 signed correction = -9/5

finite prefix point1 index8 function value := sum(17, -9/5)

finite prefix point1 index8 function value = 76/5

finite prefix point1 index8 reverse signed error := sub(76/5, 17)

finite prefix point1 index8 reverse signed error = -9/5

finite prefix point1 index8:original point=1;n=8;limit=17;signed coefficient=-18

f_n(x)=76/5;f_n(x)-limit=-9/5;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(-18, 11)

finite prefix point1 index9 signed correction = -18/11

finite prefix point1 index9 function value := sum(17, -18/11)

finite prefix point1 index9 function value = 169/11

finite prefix point1 index9 reverse signed error := sub(169/11, 17)

finite prefix point1 index9 reverse signed error = -18/11

finite prefix point1 index9:original point=1;n=9;limit=17;signed coefficient=-18

f_n(x)=169/11;f_n(x)-limit=-18/11;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(-18, 12)

finite prefix point1 index10 signed correction = -3/2

finite prefix point1 index10 function value := sum(17, -3/2)

finite prefix point1 index10 function value = 31/2

finite prefix point1 index10 reverse signed error := sub(31/2, 17)

finite prefix point1 index10 reverse signed error = -3/2

finite prefix point1 index10:original point=1;n=10;limit=17;signed coefficient=-18

f_n(x)=31/2;f_n(x)-limit=-3/2;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(-18, 13)

finite prefix point1 index11 signed correction = -18/13

finite prefix point1 index11 function value := sum(17, -18/13)

finite prefix point1 index11 function value = 203/13

finite prefix point1 index11 reverse signed error := sub(203/13, 17)

finite prefix point1 index11 reverse signed error = -18/13

finite prefix point1 index11:original point=1;n=11;limit=17;signed coefficient=-18

f_n(x)=203/13;f_n(x)-limit=-18/13;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(-18, 14)

finite prefix point1 index12 signed correction = -9/7

finite prefix point1 index12 function value := sum(17, -9/7)

finite prefix point1 index12 function value = 110/7

finite prefix point1 index12 reverse signed error := sub(110/7, 17)

finite prefix point1 index12 reverse signed error = -9/7

finite prefix point1 index12:original point=1;n=12;limit=17;signed coefficient=-18

f_n(x)=110/7;f_n(x)-limit=-9/7;this error field remains SIGNED

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

finite prefix point1 index13 denominator = 15

finite prefix point1 index13 signed correction := div(-18, 15)

finite prefix point1 index13 signed correction = -6/5

finite prefix point1 index13 function value := sum(17, -6/5)

finite prefix point1 index13 function value = 79/5

finite prefix point1 index13 reverse signed error := sub(79/5, 17)

finite prefix point1 index13 reverse signed error = -6/5

finite prefix point1 index13:original point=1;n=13;limit=17;signed coefficient=-18

f_n(x)=79/5;f_n(x)-limit=-6/5;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(-18, 16)

finite prefix point1 index14 signed correction = -9/8

finite prefix point1 index14 function value := sum(17, -9/8)

finite prefix point1 index14 function value = 127/8

finite prefix point1 index14 reverse signed error := sub(127/8, 17)

finite prefix point1 index14 reverse signed error = -9/8

finite prefix point1 index14:original point=1;n=14;limit=17;signed coefficient=-18

f_n(x)=127/8;f_n(x)-limit=-9/8;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(-18, 17)

finite prefix point1 index15 signed correction = -18/17

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

finite prefix point1 index15 function value = 271/17

finite prefix point1 index15 reverse signed error := sub(271/17, 17)

finite prefix point1 index15 reverse signed error = -18/17

finite prefix point1 index15:original point=1;n=15;limit=17;signed coefficient=-18

f_n(x)=271/17;f_n(x)-limit=-18/17;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(-18, 18)

finite prefix point1 index16 signed correction = -1

finite prefix point1 index16 function value := sum(17, -1)

finite prefix point1 index16 function value = 16

finite prefix point1 index16 reverse signed error := sub(16, 17)

finite prefix point1 index16 reverse signed error = -1

finite prefix point1 index16:original point=1;n=16;limit=17;signed coefficient=-18

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

finite observed length=17;complete domain-order rows=[[0, [49/2, 65/3, 81/4, 97/5, 113/6, 129/7, 145/8, 161/9, 177/10, 193/11, 209/12, 225/13, 241/14, 257/15, 273/16, 17, 305/18]], [1, [8, 11, 25/2, 67/5, 14, 101/7, 59/4, 15, 76/5, 169/11, 31/2, 203/13, 110/7, 79/5, 127/8, 271/17, 16]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

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

(17, [[0, [49/2, 65/3, 81/4, 97/5, 113/6, 129/7, 145/8, 161/9, 177/10, 193/11, 209/12, 225/13, 241/14, 257/15, 273/16, 17, 305/18]], [1, [8, 11, 25/2, 67/5, 14, 101/7, 59/4, 15, 76/5, 169/11, 31/2, 203/13, 110/7, 79/5, 127/8, 271/17, 16]]], 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.