Question 17586: 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=[10, 11], coefficients=[11, -12], 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=11, 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=[10, 11], coefficients=[11, -12], 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=11, 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=[10, 11];signed coefficients=[11, -12];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(11, 2)

finite prefix point0 index0 signed correction = 11/2

finite prefix point0 index0 function value := sum(10, 11/2)

finite prefix point0 index0 function value = 31/2

finite prefix point0 index0 reverse signed error := sub(31/2, 10)

finite prefix point0 index0 reverse signed error = 11/2

finite prefix point0 index0:original point=0;n=0;limit=10;signed coefficient=11

f_n(x)=31/2;f_n(x)-limit=11/2;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(11, 3)

finite prefix point0 index1 signed correction = 11/3

finite prefix point0 index1 function value := sum(10, 11/3)

finite prefix point0 index1 function value = 41/3

finite prefix point0 index1 reverse signed error := sub(41/3, 10)

finite prefix point0 index1 reverse signed error = 11/3

finite prefix point0 index1:original point=0;n=1;limit=10;signed coefficient=11

f_n(x)=41/3;f_n(x)-limit=11/3;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(11, 4)

finite prefix point0 index2 signed correction = 11/4

finite prefix point0 index2 function value := sum(10, 11/4)

finite prefix point0 index2 function value = 51/4

finite prefix point0 index2 reverse signed error := sub(51/4, 10)

finite prefix point0 index2 reverse signed error = 11/4

finite prefix point0 index2:original point=0;n=2;limit=10;signed coefficient=11

f_n(x)=51/4;f_n(x)-limit=11/4;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(11, 5)

finite prefix point0 index3 signed correction = 11/5

finite prefix point0 index3 function value := sum(10, 11/5)

finite prefix point0 index3 function value = 61/5

finite prefix point0 index3 reverse signed error := sub(61/5, 10)

finite prefix point0 index3 reverse signed error = 11/5

finite prefix point0 index3:original point=0;n=3;limit=10;signed coefficient=11

f_n(x)=61/5;f_n(x)-limit=11/5;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(11, 6)

finite prefix point0 index4 signed correction = 11/6

finite prefix point0 index4 function value := sum(10, 11/6)

finite prefix point0 index4 function value = 71/6

finite prefix point0 index4 reverse signed error := sub(71/6, 10)

finite prefix point0 index4 reverse signed error = 11/6

finite prefix point0 index4:original point=0;n=4;limit=10;signed coefficient=11

f_n(x)=71/6;f_n(x)-limit=11/6;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(11, 7)

finite prefix point0 index5 signed correction = 11/7

finite prefix point0 index5 function value := sum(10, 11/7)

finite prefix point0 index5 function value = 81/7

finite prefix point0 index5 reverse signed error := sub(81/7, 10)

finite prefix point0 index5 reverse signed error = 11/7

finite prefix point0 index5:original point=0;n=5;limit=10;signed coefficient=11

f_n(x)=81/7;f_n(x)-limit=11/7;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(11, 8)

finite prefix point0 index6 signed correction = 11/8

finite prefix point0 index6 function value := sum(10, 11/8)

finite prefix point0 index6 function value = 91/8

finite prefix point0 index6 reverse signed error := sub(91/8, 10)

finite prefix point0 index6 reverse signed error = 11/8

finite prefix point0 index6:original point=0;n=6;limit=10;signed coefficient=11

f_n(x)=91/8;f_n(x)-limit=11/8;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(11, 9)

finite prefix point0 index7 signed correction = 11/9

finite prefix point0 index7 function value := sum(10, 11/9)

finite prefix point0 index7 function value = 101/9

finite prefix point0 index7 reverse signed error := sub(101/9, 10)

finite prefix point0 index7 reverse signed error = 11/9

finite prefix point0 index7:original point=0;n=7;limit=10;signed coefficient=11

f_n(x)=101/9;f_n(x)-limit=11/9;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(11, 10)

finite prefix point0 index8 signed correction = 11/10

finite prefix point0 index8 function value := sum(10, 11/10)

finite prefix point0 index8 function value = 111/10

finite prefix point0 index8 reverse signed error := sub(111/10, 10)

finite prefix point0 index8 reverse signed error = 11/10

finite prefix point0 index8:original point=0;n=8;limit=10;signed coefficient=11

f_n(x)=111/10;f_n(x)-limit=11/10;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(11, 11)

finite prefix point0 index9 signed correction = 1

finite prefix point0 index9 function value := sum(10, 1)

finite prefix point0 index9 function value = 11

finite prefix point0 index9 reverse signed error := sub(11, 10)

finite prefix point0 index9 reverse signed error = 1

finite prefix point0 index9:original point=0;n=9;limit=10;signed coefficient=11

f_n(x)=11;f_n(x)-limit=1;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(11, 12)

finite prefix point0 index10 signed correction = 11/12

finite prefix point0 index10 function value := sum(10, 11/12)

finite prefix point0 index10 function value = 131/12

finite prefix point0 index10 reverse signed error := sub(131/12, 10)

finite prefix point0 index10 reverse signed error = 11/12

finite prefix point0 index10:original point=0;n=10;limit=10;signed coefficient=11

f_n(x)=131/12;f_n(x)-limit=11/12;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(-12, 2)

finite prefix point1 index0 signed correction = -6

finite prefix point1 index0 function value := sum(11, -6)

finite prefix point1 index0 function value = 5

finite prefix point1 index0 reverse signed error := sub(5, 11)

finite prefix point1 index0 reverse signed error = -6

finite prefix point1 index0:original point=1;n=0;limit=11;signed coefficient=-12

f_n(x)=5;f_n(x)-limit=-6;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(-12, 3)

finite prefix point1 index1 signed correction = -4

finite prefix point1 index1 function value := sum(11, -4)

finite prefix point1 index1 function value = 7

finite prefix point1 index1 reverse signed error := sub(7, 11)

finite prefix point1 index1 reverse signed error = -4

finite prefix point1 index1:original point=1;n=1;limit=11;signed coefficient=-12

f_n(x)=7;f_n(x)-limit=-4;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(-12, 4)

finite prefix point1 index2 signed correction = -3

finite prefix point1 index2 function value := sum(11, -3)

finite prefix point1 index2 function value = 8

finite prefix point1 index2 reverse signed error := sub(8, 11)

finite prefix point1 index2 reverse signed error = -3

finite prefix point1 index2:original point=1;n=2;limit=11;signed coefficient=-12

f_n(x)=8;f_n(x)-limit=-3;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(-12, 5)

finite prefix point1 index3 signed correction = -12/5

finite prefix point1 index3 function value := sum(11, -12/5)

finite prefix point1 index3 function value = 43/5

finite prefix point1 index3 reverse signed error := sub(43/5, 11)

finite prefix point1 index3 reverse signed error = -12/5

finite prefix point1 index3:original point=1;n=3;limit=11;signed coefficient=-12

f_n(x)=43/5;f_n(x)-limit=-12/5;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(-12, 6)

finite prefix point1 index4 signed correction = -2

finite prefix point1 index4 function value := sum(11, -2)

finite prefix point1 index4 function value = 9

finite prefix point1 index4 reverse signed error := sub(9, 11)

finite prefix point1 index4 reverse signed error = -2

finite prefix point1 index4:original point=1;n=4;limit=11;signed coefficient=-12

f_n(x)=9;f_n(x)-limit=-2;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(-12, 7)

finite prefix point1 index5 signed correction = -12/7

finite prefix point1 index5 function value := sum(11, -12/7)

finite prefix point1 index5 function value = 65/7

finite prefix point1 index5 reverse signed error := sub(65/7, 11)

finite prefix point1 index5 reverse signed error = -12/7

finite prefix point1 index5:original point=1;n=5;limit=11;signed coefficient=-12

f_n(x)=65/7;f_n(x)-limit=-12/7;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(-12, 8)

finite prefix point1 index6 signed correction = -3/2

finite prefix point1 index6 function value := sum(11, -3/2)

finite prefix point1 index6 function value = 19/2

finite prefix point1 index6 reverse signed error := sub(19/2, 11)

finite prefix point1 index6 reverse signed error = -3/2

finite prefix point1 index6:original point=1;n=6;limit=11;signed coefficient=-12

f_n(x)=19/2;f_n(x)-limit=-3/2;this error field remains SIGNED

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

finite prefix point1 index7 denominator = 9

finite prefix point1 index7 signed correction := div(-12, 9)

finite prefix point1 index7 signed correction = -4/3

finite prefix point1 index7 function value := sum(11, -4/3)

finite prefix point1 index7 function value = 29/3

finite prefix point1 index7 reverse signed error := sub(29/3, 11)

finite prefix point1 index7 reverse signed error = -4/3

finite prefix point1 index7:original point=1;n=7;limit=11;signed coefficient=-12

f_n(x)=29/3;f_n(x)-limit=-4/3;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(-12, 10)

finite prefix point1 index8 signed correction = -6/5

finite prefix point1 index8 function value := sum(11, -6/5)

finite prefix point1 index8 function value = 49/5

finite prefix point1 index8 reverse signed error := sub(49/5, 11)

finite prefix point1 index8 reverse signed error = -6/5

finite prefix point1 index8:original point=1;n=8;limit=11;signed coefficient=-12

f_n(x)=49/5;f_n(x)-limit=-6/5;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(-12, 11)

finite prefix point1 index9 signed correction = -12/11

finite prefix point1 index9 function value := sum(11, -12/11)

finite prefix point1 index9 function value = 109/11

finite prefix point1 index9 reverse signed error := sub(109/11, 11)

finite prefix point1 index9 reverse signed error = -12/11

finite prefix point1 index9:original point=1;n=9;limit=11;signed coefficient=-12

f_n(x)=109/11;f_n(x)-limit=-12/11;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(-12, 12)

finite prefix point1 index10 signed correction = -1

finite prefix point1 index10 function value := sum(11, -1)

finite prefix point1 index10 function value = 10

finite prefix point1 index10 reverse signed error := sub(10, 11)

finite prefix point1 index10 reverse signed error = -1

finite prefix point1 index10:original point=1;n=10;limit=11;signed coefficient=-12

f_n(x)=10;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=11;complete domain-order rows=[[0, [31/2, 41/3, 51/4, 61/5, 71/6, 81/7, 91/8, 101/9, 111/10, 11, 131/12]], [1, [5, 7, 8, 43/5, 9, 65/7, 19/2, 29/3, 49/5, 109/11, 10]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 index10 reverse signed error with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute finite prefix point1 index10 function value with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute finite prefix point1 index10 signed correction with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute finite prefix point1 index10 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 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute finite prefix point1 index9 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 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute finite prefix point1 index8 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 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute finite prefix point1 index7 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 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute finite prefix point1 index6 denominator with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Recompute finite prefix point1 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute finite prefix point1 index5 denominator with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Recompute finite prefix point1 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute finite prefix point1 index4 denominator with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Recompute finite prefix point1 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute finite prefix point1 index3 denominator with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Recompute finite prefix point1 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute finite prefix point1 index2 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 point1 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute finite prefix point1 index1 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 point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute finite prefix point1 index0 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

(11, [[0, [31/2, 41/3, 51/4, 61/5, 71/6, 81/7, 91/8, 101/9, 111/10, 11, 131/12]], [1, [5, 7, 8, 43/5, 9, 65/7, 19/2, 29/3, 49/5, 109/11, 10]]], 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.