Question 17589: A Long Prefix Still Does Not Replace the All-n Tail

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0034-17589 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0089

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[13, 14], coefficients=[14, -15], 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=14, 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=[13, 14], coefficients=[14, -15], 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=14, 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

S001 · implies

Before

same COMPLETE finite domain Fin(2);original aligned limits=[13, 14];signed coefficients=[14, -15];positive shift=2

After

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)

Conditions: none

The complete common finite domain and all-n formula are declared premises,not extrapolated from samples.Function values and errors are not probabilities;they may be signed or exceed1.Do not clip or shift indices.

Step identifiers and dependencies

Rule: pointwise_explicit_common_domain_all_n_tail_not_probability · Uses: G001

S002 · calculation

Before

finite prefix point0 index0 denominator := sum(0, 2)

After

finite prefix point0 index0 denominator = 2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

finite prefix point0 index0 signed correction := div(14, 2)

After

finite prefix point0 index0 signed correction = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

finite prefix point0 index0 function value := sum(13, 7)

After

finite prefix point0 index0 function value = 20

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

finite prefix point0 index0 reverse signed error := sub(20, 13)

After

finite prefix point0 index0 reverse signed error = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

finite prefix point0 index0:original point=0;n=0;limit=13;signed coefficient=14

After

f_n(x)=20;f_n(x)-limit=7;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S005

S007 · calculation

Before

finite prefix point0 index1 denominator := sum(1, 2)

After

finite prefix point0 index1 denominator = 3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

finite prefix point0 index1 signed correction := div(14, 3)

After

finite prefix point0 index1 signed correction = 14/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

finite prefix point0 index1 function value := sum(13, 14/3)

After

finite prefix point0 index1 function value = 53/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

finite prefix point0 index1 reverse signed error := sub(53/3, 13)

After

finite prefix point0 index1 reverse signed error = 14/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

finite prefix point0 index1:original point=0;n=1;limit=13;signed coefficient=14

After

f_n(x)=53/3;f_n(x)-limit=14/3;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S010

S012 · calculation

Before

finite prefix point0 index2 denominator := sum(2, 2)

After

finite prefix point0 index2 denominator = 4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

finite prefix point0 index2 signed correction := div(14, 4)

After

finite prefix point0 index2 signed correction = 7/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

finite prefix point0 index2 function value := sum(13, 7/2)

After

finite prefix point0 index2 function value = 33/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

finite prefix point0 index2 reverse signed error := sub(33/2, 13)

After

finite prefix point0 index2 reverse signed error = 7/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

finite prefix point0 index2:original point=0;n=2;limit=13;signed coefficient=14

After

f_n(x)=33/2;f_n(x)-limit=7/2;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S015

S017 · calculation

Before

finite prefix point0 index3 denominator := sum(3, 2)

After

finite prefix point0 index3 denominator = 5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

finite prefix point0 index3 signed correction := div(14, 5)

After

finite prefix point0 index3 signed correction = 14/5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

finite prefix point0 index3 function value := sum(13, 14/5)

After

finite prefix point0 index3 function value = 79/5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

finite prefix point0 index3 reverse signed error := sub(79/5, 13)

After

finite prefix point0 index3 reverse signed error = 14/5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

finite prefix point0 index3:original point=0;n=3;limit=13;signed coefficient=14

After

f_n(x)=79/5;f_n(x)-limit=14/5;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S020

S022 · calculation

Before

finite prefix point0 index4 denominator := sum(4, 2)

After

finite prefix point0 index4 denominator = 6

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

finite prefix point0 index4 signed correction := div(14, 6)

After

finite prefix point0 index4 signed correction = 7/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

finite prefix point0 index4 function value := sum(13, 7/3)

After

finite prefix point0 index4 function value = 46/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

finite prefix point0 index4 reverse signed error := sub(46/3, 13)

After

finite prefix point0 index4 reverse signed error = 7/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

finite prefix point0 index4:original point=0;n=4;limit=13;signed coefficient=14

After

f_n(x)=46/3;f_n(x)-limit=7/3;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S025

S027 · calculation

Before

finite prefix point0 index5 denominator := sum(5, 2)

After

finite prefix point0 index5 denominator = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

finite prefix point0 index5 signed correction := div(14, 7)

After

finite prefix point0 index5 signed correction = 2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

finite prefix point0 index5 function value := sum(13, 2)

After

finite prefix point0 index5 function value = 15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

finite prefix point0 index5 reverse signed error := sub(15, 13)

After

finite prefix point0 index5 reverse signed error = 2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

finite prefix point0 index5:original point=0;n=5;limit=13;signed coefficient=14

After

f_n(x)=15;f_n(x)-limit=2;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S030

S032 · calculation

Before

finite prefix point0 index6 denominator := sum(6, 2)

After

finite prefix point0 index6 denominator = 8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

finite prefix point0 index6 signed correction := div(14, 8)

After

finite prefix point0 index6 signed correction = 7/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

finite prefix point0 index6 function value := sum(13, 7/4)

After

finite prefix point0 index6 function value = 59/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

finite prefix point0 index6 reverse signed error := sub(59/4, 13)

After

finite prefix point0 index6 reverse signed error = 7/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

finite prefix point0 index6:original point=0;n=6;limit=13;signed coefficient=14

After

f_n(x)=59/4;f_n(x)-limit=7/4;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S035

S037 · calculation

Before

finite prefix point0 index7 denominator := sum(7, 2)

After

finite prefix point0 index7 denominator = 9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

finite prefix point0 index7 signed correction := div(14, 9)

After

finite prefix point0 index7 signed correction = 14/9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

finite prefix point0 index7 function value := sum(13, 14/9)

After

finite prefix point0 index7 function value = 131/9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

finite prefix point0 index7 reverse signed error := sub(131/9, 13)

After

finite prefix point0 index7 reverse signed error = 14/9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · implies

Before

finite prefix point0 index7:original point=0;n=7;limit=13;signed coefficient=14

After

f_n(x)=131/9;f_n(x)-limit=14/9;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S040

S042 · calculation

Before

finite prefix point0 index8 denominator := sum(8, 2)

After

finite prefix point0 index8 denominator = 10

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

finite prefix point0 index8 signed correction := div(14, 10)

After

finite prefix point0 index8 signed correction = 7/5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

finite prefix point0 index8 function value := sum(13, 7/5)

After

finite prefix point0 index8 function value = 72/5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

finite prefix point0 index8 reverse signed error := sub(72/5, 13)

After

finite prefix point0 index8 reverse signed error = 7/5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

finite prefix point0 index8:original point=0;n=8;limit=13;signed coefficient=14

After

f_n(x)=72/5;f_n(x)-limit=7/5;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S045

S047 · calculation

Before

finite prefix point0 index9 denominator := sum(9, 2)

After

finite prefix point0 index9 denominator = 11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

finite prefix point0 index9 signed correction := div(14, 11)

After

finite prefix point0 index9 signed correction = 14/11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

finite prefix point0 index9 function value := sum(13, 14/11)

After

finite prefix point0 index9 function value = 157/11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

finite prefix point0 index9 reverse signed error := sub(157/11, 13)

After

finite prefix point0 index9 reverse signed error = 14/11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · implies

Before

finite prefix point0 index9:original point=0;n=9;limit=13;signed coefficient=14

After

f_n(x)=157/11;f_n(x)-limit=14/11;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S050

S052 · calculation

Before

finite prefix point0 index10 denominator := sum(10, 2)

After

finite prefix point0 index10 denominator = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

finite prefix point0 index10 signed correction := div(14, 12)

After

finite prefix point0 index10 signed correction = 7/6

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

finite prefix point0 index10 function value := sum(13, 7/6)

After

finite prefix point0 index10 function value = 85/6

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

finite prefix point0 index10 reverse signed error := sub(85/6, 13)

After

finite prefix point0 index10 reverse signed error = 7/6

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · implies

Before

finite prefix point0 index10:original point=0;n=10;limit=13;signed coefficient=14

After

f_n(x)=85/6;f_n(x)-limit=7/6;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S055

S057 · calculation

Before

finite prefix point0 index11 denominator := sum(11, 2)

After

finite prefix point0 index11 denominator = 13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

finite prefix point0 index11 signed correction := div(14, 13)

After

finite prefix point0 index11 signed correction = 14/13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

finite prefix point0 index11 function value := sum(13, 14/13)

After

finite prefix point0 index11 function value = 183/13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

finite prefix point0 index11 reverse signed error := sub(183/13, 13)

After

finite prefix point0 index11 reverse signed error = 14/13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · implies

Before

finite prefix point0 index11:original point=0;n=11;limit=13;signed coefficient=14

After

f_n(x)=183/13;f_n(x)-limit=14/13;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S060

S062 · calculation

Before

finite prefix point0 index12 denominator := sum(12, 2)

After

finite prefix point0 index12 denominator = 14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

finite prefix point0 index12 signed correction := div(14, 14)

After

finite prefix point0 index12 signed correction = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

finite prefix point0 index12 function value := sum(13, 1)

After

finite prefix point0 index12 function value = 14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

finite prefix point0 index12 reverse signed error := sub(14, 13)

After

finite prefix point0 index12 reverse signed error = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

finite prefix point0 index12:original point=0;n=12;limit=13;signed coefficient=14

After

f_n(x)=14;f_n(x)-limit=1;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S065

S067 · calculation

Before

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

After

finite prefix point0 index13 denominator = 15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

finite prefix point0 index13 signed correction := div(14, 15)

After

finite prefix point0 index13 signed correction = 14/15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

finite prefix point0 index13 function value := sum(13, 14/15)

After

finite prefix point0 index13 function value = 209/15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

finite prefix point0 index13 reverse signed error := sub(209/15, 13)

After

finite prefix point0 index13 reverse signed error = 14/15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · implies

Before

finite prefix point0 index13:original point=0;n=13;limit=13;signed coefficient=14

After

f_n(x)=209/15;f_n(x)-limit=14/15;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S070

S072 · calculation

Before

finite prefix point1 index0 denominator := sum(0, 2)

After

finite prefix point1 index0 denominator = 2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

finite prefix point1 index0 signed correction := div(-15, 2)

After

finite prefix point1 index0 signed correction = -15/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · calculation

Before

finite prefix point1 index0 function value := sum(14, -15/2)

After

finite prefix point1 index0 function value = 13/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · calculation

Before

finite prefix point1 index0 reverse signed error := sub(13/2, 14)

After

finite prefix point1 index0 reverse signed error = -15/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · implies

Before

finite prefix point1 index0:original point=1;n=0;limit=14;signed coefficient=-15

After

f_n(x)=13/2;f_n(x)-limit=-15/2;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S075

S077 · calculation

Before

finite prefix point1 index1 denominator := sum(1, 2)

After

finite prefix point1 index1 denominator = 3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · calculation

Before

finite prefix point1 index1 signed correction := div(-15, 3)

After

finite prefix point1 index1 signed correction = -5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · calculation

Before

finite prefix point1 index1 function value := sum(14, -5)

After

finite prefix point1 index1 function value = 9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · calculation

Before

finite prefix point1 index1 reverse signed error := sub(9, 14)

After

finite prefix point1 index1 reverse signed error = -5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · implies

Before

finite prefix point1 index1:original point=1;n=1;limit=14;signed coefficient=-15

After

f_n(x)=9;f_n(x)-limit=-5;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S080

S082 · calculation

Before

finite prefix point1 index2 denominator := sum(2, 2)

After

finite prefix point1 index2 denominator = 4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S081

S083 · calculation

Before

finite prefix point1 index2 signed correction := div(-15, 4)

After

finite prefix point1 index2 signed correction = -15/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · calculation

Before

finite prefix point1 index2 function value := sum(14, -15/4)

After

finite prefix point1 index2 function value = 41/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · calculation

Before

finite prefix point1 index2 reverse signed error := sub(41/4, 14)

After

finite prefix point1 index2 reverse signed error = -15/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S084

S086 · implies

Before

finite prefix point1 index2:original point=1;n=2;limit=14;signed coefficient=-15

After

f_n(x)=41/4;f_n(x)-limit=-15/4;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S085

S087 · calculation

Before

finite prefix point1 index3 denominator := sum(3, 2)

After

finite prefix point1 index3 denominator = 5

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S086

S088 · calculation

Before

finite prefix point1 index3 signed correction := div(-15, 5)

After

finite prefix point1 index3 signed correction = -3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S087

S089 · calculation

Before

finite prefix point1 index3 function value := sum(14, -3)

After

finite prefix point1 index3 function value = 11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · calculation

Before

finite prefix point1 index3 reverse signed error := sub(11, 14)

After

finite prefix point1 index3 reverse signed error = -3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S089

S091 · implies

Before

finite prefix point1 index3:original point=1;n=3;limit=14;signed coefficient=-15

After

f_n(x)=11;f_n(x)-limit=-3;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S090

S092 · calculation

Before

finite prefix point1 index4 denominator := sum(4, 2)

After

finite prefix point1 index4 denominator = 6

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S091

S093 · calculation

Before

finite prefix point1 index4 signed correction := div(-15, 6)

After

finite prefix point1 index4 signed correction = -5/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · calculation

Before

finite prefix point1 index4 function value := sum(14, -5/2)

After

finite prefix point1 index4 function value = 23/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S093

S095 · calculation

Before

finite prefix point1 index4 reverse signed error := sub(23/2, 14)

After

finite prefix point1 index4 reverse signed error = -5/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S094

S096 · implies

Before

finite prefix point1 index4:original point=1;n=4;limit=14;signed coefficient=-15

After

f_n(x)=23/2;f_n(x)-limit=-5/2;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S095

S097 · calculation

Before

finite prefix point1 index5 denominator := sum(5, 2)

After

finite prefix point1 index5 denominator = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · calculation

Before

finite prefix point1 index5 signed correction := div(-15, 7)

After

finite prefix point1 index5 signed correction = -15/7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S097

S099 · calculation

Before

finite prefix point1 index5 function value := sum(14, -15/7)

After

finite prefix point1 index5 function value = 83/7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · calculation

Before

finite prefix point1 index5 reverse signed error := sub(83/7, 14)

After

finite prefix point1 index5 reverse signed error = -15/7

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S099

S101 · implies

Before

finite prefix point1 index5:original point=1;n=5;limit=14;signed coefficient=-15

After

f_n(x)=83/7;f_n(x)-limit=-15/7;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S100

S102 · calculation

Before

finite prefix point1 index6 denominator := sum(6, 2)

After

finite prefix point1 index6 denominator = 8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S101

S103 · calculation

Before

finite prefix point1 index6 signed correction := div(-15, 8)

After

finite prefix point1 index6 signed correction = -15/8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S102

S104 · calculation

Before

finite prefix point1 index6 function value := sum(14, -15/8)

After

finite prefix point1 index6 function value = 97/8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S103

S105 · calculation

Before

finite prefix point1 index6 reverse signed error := sub(97/8, 14)

After

finite prefix point1 index6 reverse signed error = -15/8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · implies

Before

finite prefix point1 index6:original point=1;n=6;limit=14;signed coefficient=-15

After

f_n(x)=97/8;f_n(x)-limit=-15/8;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S105

S107 · calculation

Before

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

After

finite prefix point1 index7 denominator = 9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S106

S108 · calculation

Before

finite prefix point1 index7 signed correction := div(-15, 9)

After

finite prefix point1 index7 signed correction = -5/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S107

S109 · calculation

Before

finite prefix point1 index7 function value := sum(14, -5/3)

After

finite prefix point1 index7 function value = 37/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S108

S110 · calculation

Before

finite prefix point1 index7 reverse signed error := sub(37/3, 14)

After

finite prefix point1 index7 reverse signed error = -5/3

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S109

S111 · implies

Before

finite prefix point1 index7:original point=1;n=7;limit=14;signed coefficient=-15

After

f_n(x)=37/3;f_n(x)-limit=-5/3;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S110

S112 · calculation

Before

finite prefix point1 index8 denominator := sum(8, 2)

After

finite prefix point1 index8 denominator = 10

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S111

S113 · calculation

Before

finite prefix point1 index8 signed correction := div(-15, 10)

After

finite prefix point1 index8 signed correction = -3/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S112

S114 · calculation

Before

finite prefix point1 index8 function value := sum(14, -3/2)

After

finite prefix point1 index8 function value = 25/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S113

S115 · calculation

Before

finite prefix point1 index8 reverse signed error := sub(25/2, 14)

After

finite prefix point1 index8 reverse signed error = -3/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S114

S116 · implies

Before

finite prefix point1 index8:original point=1;n=8;limit=14;signed coefficient=-15

After

f_n(x)=25/2;f_n(x)-limit=-3/2;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S115

S117 · calculation

Before

finite prefix point1 index9 denominator := sum(9, 2)

After

finite prefix point1 index9 denominator = 11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S116

S118 · calculation

Before

finite prefix point1 index9 signed correction := div(-15, 11)

After

finite prefix point1 index9 signed correction = -15/11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S117

S119 · calculation

Before

finite prefix point1 index9 function value := sum(14, -15/11)

After

finite prefix point1 index9 function value = 139/11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S118

S120 · calculation

Before

finite prefix point1 index9 reverse signed error := sub(139/11, 14)

After

finite prefix point1 index9 reverse signed error = -15/11

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S119

S121 · implies

Before

finite prefix point1 index9:original point=1;n=9;limit=14;signed coefficient=-15

After

f_n(x)=139/11;f_n(x)-limit=-15/11;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S120

S122 · calculation

Before

finite prefix point1 index10 denominator := sum(10, 2)

After

finite prefix point1 index10 denominator = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S121

S123 · calculation

Before

finite prefix point1 index10 signed correction := div(-15, 12)

After

finite prefix point1 index10 signed correction = -5/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S122

S124 · calculation

Before

finite prefix point1 index10 function value := sum(14, -5/4)

After

finite prefix point1 index10 function value = 51/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S123

S125 · calculation

Before

finite prefix point1 index10 reverse signed error := sub(51/4, 14)

After

finite prefix point1 index10 reverse signed error = -5/4

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S124

S126 · implies

Before

finite prefix point1 index10:original point=1;n=10;limit=14;signed coefficient=-15

After

f_n(x)=51/4;f_n(x)-limit=-5/4;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S125

S127 · calculation

Before

finite prefix point1 index11 denominator := sum(11, 2)

After

finite prefix point1 index11 denominator = 13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S126

S128 · calculation

Before

finite prefix point1 index11 signed correction := div(-15, 13)

After

finite prefix point1 index11 signed correction = -15/13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S127

S129 · calculation

Before

finite prefix point1 index11 function value := sum(14, -15/13)

After

finite prefix point1 index11 function value = 167/13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S128

S130 · calculation

Before

finite prefix point1 index11 reverse signed error := sub(167/13, 14)

After

finite prefix point1 index11 reverse signed error = -15/13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S129

S131 · implies

Before

finite prefix point1 index11:original point=1;n=11;limit=14;signed coefficient=-15

After

f_n(x)=167/13;f_n(x)-limit=-15/13;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S130

S132 · calculation

Before

finite prefix point1 index12 denominator := sum(12, 2)

After

finite prefix point1 index12 denominator = 14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S131

S133 · calculation

Before

finite prefix point1 index12 signed correction := div(-15, 14)

After

finite prefix point1 index12 signed correction = -15/14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S132

S134 · calculation

Before

finite prefix point1 index12 function value := sum(14, -15/14)

After

finite prefix point1 index12 function value = 181/14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S133

S135 · calculation

Before

finite prefix point1 index12 reverse signed error := sub(181/14, 14)

After

finite prefix point1 index12 reverse signed error = -15/14

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S134

S136 · implies

Before

finite prefix point1 index12:original point=1;n=12;limit=14;signed coefficient=-15

After

f_n(x)=181/14;f_n(x)-limit=-15/14;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S135

S137 · calculation

Before

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

After

finite prefix point1 index13 denominator = 15

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S136

S138 · calculation

Before

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

After

finite prefix point1 index13 signed correction = -1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S137

S139 · calculation

Before

finite prefix point1 index13 function value := sum(14, -1)

After

finite prefix point1 index13 function value = 13

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S138

S140 · calculation

Before

finite prefix point1 index13 reverse signed error := sub(13, 14)

After

finite prefix point1 index13 reverse signed error = -1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S139

S141 · implies

Before

finite prefix point1 index13:original point=1;n=13;limit=14;signed coefficient=-15

After

f_n(x)=13;f_n(x)-limit=-1;this error field remains SIGNED

Conditions: none

Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value.

Step identifiers and dependencies

Rule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S140

S142 · implies

Before

finite observed length=14;complete domain-order rows=[[0, [20, 53/3, 33/2, 79/5, 46/3, 15, 59/4, 131/9, 72/5, 157/11, 85/6, 183/13, 14, 209/15]], [1, [13/2, 9, 41/4, 11, 23/2, 83/7, 97/8, 37/3, 25/2, 139/11, 51/4, 167/13, 181/14, 13]]]

After

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Conditions: none

This No limits inference from the finite prefix alone,not the separate declared all-n formula.A pointwise witness may depend on x and epsilon and must cover every later n.

Step identifiers and dependencies

Rule: pointwise_finite_prefix_no_not_declared_model_failure · Uses: G001, S141

Reverse checks

S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 index13 reverse signed error with exact rational arithmetic, not rounded decimals.
S139 · premise_and_result_audit — Recompute finite prefix point1 index13 function value with exact rational arithmetic, not rounded decimals.
S138 · premise_and_result_audit — Recompute finite prefix point1 index13 signed correction with exact rational arithmetic, not rounded decimals.
S137 · premise_and_result_audit — Recompute finite prefix point1 index13 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 index12 reverse signed error with exact rational arithmetic, not rounded decimals.
S134 · premise_and_result_audit — Recompute finite prefix point1 index12 function value with exact rational arithmetic, not rounded decimals.
S133 · premise_and_result_audit — Recompute finite prefix point1 index12 signed correction with exact rational arithmetic, not rounded decimals.
S132 · premise_and_result_audit — Recompute finite prefix point1 index12 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 index11 reverse signed error with exact rational arithmetic, not rounded decimals.
S129 · premise_and_result_audit — Recompute finite prefix point1 index11 function value with exact rational arithmetic, not rounded decimals.
S128 · premise_and_result_audit — Recompute finite prefix point1 index11 signed correction with exact rational arithmetic, not rounded decimals.
S127 · premise_and_result_audit — Recompute finite prefix point1 index11 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 index10 reverse signed error with exact rational arithmetic, not rounded decimals.
S124 · premise_and_result_audit — Recompute finite prefix point1 index10 function value with exact rational arithmetic, not rounded decimals.
S123 · premise_and_result_audit — Recompute finite prefix point1 index10 signed correction with exact rational arithmetic, not rounded decimals.
S122 · premise_and_result_audit — Recompute finite prefix point1 index10 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 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
S119 · premise_and_result_audit — Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals.
S118 · premise_and_result_audit — Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals.
S117 · premise_and_result_audit — Recompute finite prefix point1 index9 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 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
S114 · premise_and_result_audit — Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals.
S113 · premise_and_result_audit — Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals.
S112 · premise_and_result_audit — Recompute finite prefix point1 index8 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 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
S109 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals.
S108 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals.
S107 · premise_and_result_audit — Recompute finite prefix point1 index7 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 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
S104 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
S103 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
S102 · premise_and_result_audit — Recompute finite prefix point1 index6 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 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
S099 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
S098 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
S097 · premise_and_result_audit — Recompute finite prefix point1 index5 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 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
S094 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
S093 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
S092 · premise_and_result_audit — Recompute finite prefix point1 index4 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 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
S089 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
S088 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
S087 · premise_and_result_audit — Recompute finite prefix point1 index3 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 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
S084 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
S083 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
S082 · premise_and_result_audit — Recompute finite prefix point1 index2 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 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
S079 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
S078 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
S077 · premise_and_result_audit — Recompute finite prefix point1 index1 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 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
S074 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
S073 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
S072 · premise_and_result_audit — Recompute finite prefix point1 index0 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

(14, [[0, [20, 53/3, 33/2, 79/5, 46/3, 15, 59/4, 131/9, 72/5, 157/11, 85/6, 183/13, 14, 209/15]], [1, [13/2, 9, 41/4, 11, 23/2, 83/7, 97/8, 37/3, 25/2, 139/11, 51/4, 167/13, 181/14, 13]]], 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.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 9600a1dcd7ff5670c678062bd900d510d6c37b93621e3f09c55690b8ae466b94
candidate record SHA-256: 74e2f090c92ddbf00cf60f91e1bf1d49889b026c1de491da30eefac3e0c41dd6
semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336
question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON