# Question 17598: A Long Prefix Still Does Not Replace the All-n Tail [Return to catalogue page 88](https://ocme.evemisslab.com/en/questions/page/88/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations/#q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0098) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0034-17598 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0098 ## Original question and givens Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[22, 23], coefficients=[23, -24], 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=23, 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=[22, 23], coefficients=[23, -24], 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=23, 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](https://ocme.evemisslab.com/en/mko/mko-pointwise-convergence-of-functions/) ## Step-by-step derivation S001 · impliesBefore same COMPLETE finite domain Fin(2);original aligned limits=[22, 23];signed coefficients=[23, -24];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 dependenciesRule: pointwise_explicit_common_domain_all_n_tail_not_probability · Uses: G001 S002 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore finite prefix point0 index0 signed correction := div(23, 2) After finite prefix point0 index0 signed correction = 23/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore finite prefix point0 index0 function value := sum(22, 23/2) After finite prefix point0 index0 function value = 67/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore finite prefix point0 index0 reverse signed error := sub(67/2, 22) After finite prefix point0 index0 reverse signed error = 23/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore finite prefix point0 index0:original point=0;n=0;limit=22;signed coefficient=23 After f_n(x)=67/2;f_n(x)-limit=23/2;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S005 S007 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore finite prefix point0 index1 signed correction := div(23, 3) After finite prefix point0 index1 signed correction = 23/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore finite prefix point0 index1 function value := sum(22, 23/3) After finite prefix point0 index1 function value = 89/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore finite prefix point0 index1 reverse signed error := sub(89/3, 22) After finite prefix point0 index1 reverse signed error = 23/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore finite prefix point0 index1:original point=0;n=1;limit=22;signed coefficient=23 After f_n(x)=89/3;f_n(x)-limit=23/3;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S010 S012 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore finite prefix point0 index2 signed correction := div(23, 4) After finite prefix point0 index2 signed correction = 23/4 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore finite prefix point0 index2 function value := sum(22, 23/4) After finite prefix point0 index2 function value = 111/4 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore finite prefix point0 index2 reverse signed error := sub(111/4, 22) After finite prefix point0 index2 reverse signed error = 23/4 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore finite prefix point0 index2:original point=0;n=2;limit=22;signed coefficient=23 After f_n(x)=111/4;f_n(x)-limit=23/4;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S015 S017 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore finite prefix point0 index3 signed correction := div(23, 5) After finite prefix point0 index3 signed correction = 23/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore finite prefix point0 index3 function value := sum(22, 23/5) After finite prefix point0 index3 function value = 133/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore finite prefix point0 index3 reverse signed error := sub(133/5, 22) After finite prefix point0 index3 reverse signed error = 23/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore finite prefix point0 index3:original point=0;n=3;limit=22;signed coefficient=23 After f_n(x)=133/5;f_n(x)-limit=23/5;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S020 S022 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore finite prefix point0 index4 signed correction := div(23, 6) After finite prefix point0 index4 signed correction = 23/6 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore finite prefix point0 index4 function value := sum(22, 23/6) After finite prefix point0 index4 function value = 155/6 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore finite prefix point0 index4 reverse signed error := sub(155/6, 22) After finite prefix point0 index4 reverse signed error = 23/6 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore finite prefix point0 index4:original point=0;n=4;limit=22;signed coefficient=23 After f_n(x)=155/6;f_n(x)-limit=23/6;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S025 S027 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore finite prefix point0 index5 signed correction := div(23, 7) After finite prefix point0 index5 signed correction = 23/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore finite prefix point0 index5 function value := sum(22, 23/7) After finite prefix point0 index5 function value = 177/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore finite prefix point0 index5 reverse signed error := sub(177/7, 22) After finite prefix point0 index5 reverse signed error = 23/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore finite prefix point0 index5:original point=0;n=5;limit=22;signed coefficient=23 After f_n(x)=177/7;f_n(x)-limit=23/7;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S030 S032 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore finite prefix point0 index6 signed correction := div(23, 8) After finite prefix point0 index6 signed correction = 23/8 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore finite prefix point0 index6 function value := sum(22, 23/8) After finite prefix point0 index6 function value = 199/8 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore finite prefix point0 index6 reverse signed error := sub(199/8, 22) After finite prefix point0 index6 reverse signed error = 23/8 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · impliesBefore finite prefix point0 index6:original point=0;n=6;limit=22;signed coefficient=23 After f_n(x)=199/8;f_n(x)-limit=23/8;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S035 S037 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore finite prefix point0 index7 signed correction := div(23, 9) After finite prefix point0 index7 signed correction = 23/9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore finite prefix point0 index7 function value := sum(22, 23/9) After finite prefix point0 index7 function value = 221/9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore finite prefix point0 index7 reverse signed error := sub(221/9, 22) After finite prefix point0 index7 reverse signed error = 23/9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · impliesBefore finite prefix point0 index7:original point=0;n=7;limit=22;signed coefficient=23 After f_n(x)=221/9;f_n(x)-limit=23/9;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S040 S042 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore finite prefix point0 index8 signed correction := div(23, 10) After finite prefix point0 index8 signed correction = 23/10 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore finite prefix point0 index8 function value := sum(22, 23/10) After finite prefix point0 index8 function value = 243/10 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore finite prefix point0 index8 reverse signed error := sub(243/10, 22) After finite prefix point0 index8 reverse signed error = 23/10 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · impliesBefore finite prefix point0 index8:original point=0;n=8;limit=22;signed coefficient=23 After f_n(x)=243/10;f_n(x)-limit=23/10;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S045 S047 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore finite prefix point0 index9 signed correction := div(23, 11) After finite prefix point0 index9 signed correction = 23/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore finite prefix point0 index9 function value := sum(22, 23/11) After finite prefix point0 index9 function value = 265/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore finite prefix point0 index9 reverse signed error := sub(265/11, 22) After finite prefix point0 index9 reverse signed error = 23/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · impliesBefore finite prefix point0 index9:original point=0;n=9;limit=22;signed coefficient=23 After f_n(x)=265/11;f_n(x)-limit=23/11;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S050 S052 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore finite prefix point0 index10 signed correction := div(23, 12) After finite prefix point0 index10 signed correction = 23/12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore finite prefix point0 index10 function value := sum(22, 23/12) After finite prefix point0 index10 function value = 287/12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore finite prefix point0 index10 reverse signed error := sub(287/12, 22) After finite prefix point0 index10 reverse signed error = 23/12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · impliesBefore finite prefix point0 index10:original point=0;n=10;limit=22;signed coefficient=23 After f_n(x)=287/12;f_n(x)-limit=23/12;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S055 S057 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore finite prefix point0 index11 signed correction := div(23, 13) After finite prefix point0 index11 signed correction = 23/13 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore finite prefix point0 index11 function value := sum(22, 23/13) After finite prefix point0 index11 function value = 309/13 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore finite prefix point0 index11 reverse signed error := sub(309/13, 22) After finite prefix point0 index11 reverse signed error = 23/13 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · impliesBefore finite prefix point0 index11:original point=0;n=11;limit=22;signed coefficient=23 After f_n(x)=309/13;f_n(x)-limit=23/13;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S060 S062 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore finite prefix point0 index12 signed correction := div(23, 14) After finite prefix point0 index12 signed correction = 23/14 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · calculationBefore finite prefix point0 index12 function value := sum(22, 23/14) After finite prefix point0 index12 function value = 331/14 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore finite prefix point0 index12 reverse signed error := sub(331/14, 22) After finite prefix point0 index12 reverse signed error = 23/14 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · impliesBefore finite prefix point0 index12:original point=0;n=12;limit=22;signed coefficient=23 After f_n(x)=331/14;f_n(x)-limit=23/14;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S065 S067 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore finite prefix point0 index13 signed correction := div(23, 15) After finite prefix point0 index13 signed correction = 23/15 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore finite prefix point0 index13 function value := sum(22, 23/15) After finite prefix point0 index13 function value = 353/15 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore finite prefix point0 index13 reverse signed error := sub(353/15, 22) After finite prefix point0 index13 reverse signed error = 23/15 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · impliesBefore finite prefix point0 index13:original point=0;n=13;limit=22;signed coefficient=23 After f_n(x)=353/15;f_n(x)-limit=23/15;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S070 S072 · calculationBefore finite prefix point0 index14 denominator := sum(14, 2) After finite prefix point0 index14 denominator = 16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore finite prefix point0 index14 signed correction := div(23, 16) After finite prefix point0 index14 signed correction = 23/16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore finite prefix point0 index14 function value := sum(22, 23/16) After finite prefix point0 index14 function value = 375/16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · calculationBefore finite prefix point0 index14 reverse signed error := sub(375/16, 22) After finite prefix point0 index14 reverse signed error = 23/16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · impliesBefore finite prefix point0 index14:original point=0;n=14;limit=22;signed coefficient=23 After f_n(x)=375/16;f_n(x)-limit=23/16;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S075 S077 · calculationBefore finite prefix point0 index15 denominator := sum(15, 2) After finite prefix point0 index15 denominator = 17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore finite prefix point0 index15 signed correction := div(23, 17) After finite prefix point0 index15 signed correction = 23/17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · calculationBefore finite prefix point0 index15 function value := sum(22, 23/17) After finite prefix point0 index15 function value = 397/17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · calculationBefore finite prefix point0 index15 reverse signed error := sub(397/17, 22) After finite prefix point0 index15 reverse signed error = 23/17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · impliesBefore finite prefix point0 index15:original point=0;n=15;limit=22;signed coefficient=23 After f_n(x)=397/17;f_n(x)-limit=23/17;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S080 S082 · calculationBefore finite prefix point0 index16 denominator := sum(16, 2) After finite prefix point0 index16 denominator = 18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore finite prefix point0 index16 signed correction := div(23, 18) After finite prefix point0 index16 signed correction = 23/18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore finite prefix point0 index16 function value := sum(22, 23/18) After finite prefix point0 index16 function value = 419/18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore finite prefix point0 index16 reverse signed error := sub(419/18, 22) After finite prefix point0 index16 reverse signed error = 23/18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · impliesBefore finite prefix point0 index16:original point=0;n=16;limit=22;signed coefficient=23 After f_n(x)=419/18;f_n(x)-limit=23/18;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S085 S087 · calculationBefore finite prefix point0 index17 denominator := sum(17, 2) After finite prefix point0 index17 denominator = 19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · calculationBefore finite prefix point0 index17 signed correction := div(23, 19) After finite prefix point0 index17 signed correction = 23/19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · calculationBefore finite prefix point0 index17 function value := sum(22, 23/19) After finite prefix point0 index17 function value = 441/19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · calculationBefore finite prefix point0 index17 reverse signed error := sub(441/19, 22) After finite prefix point0 index17 reverse signed error = 23/19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089 S091 · impliesBefore finite prefix point0 index17:original point=0;n=17;limit=22;signed coefficient=23 After f_n(x)=441/19;f_n(x)-limit=23/19;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S090 S092 · calculationBefore finite prefix point0 index18 denominator := sum(18, 2) After finite prefix point0 index18 denominator = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · calculationBefore finite prefix point0 index18 signed correction := div(23, 20) After finite prefix point0 index18 signed correction = 23/20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S092 S094 · calculationBefore finite prefix point0 index18 function value := sum(22, 23/20) After finite prefix point0 index18 function value = 463/20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S093 S095 · calculationBefore finite prefix point0 index18 reverse signed error := sub(463/20, 22) After finite prefix point0 index18 reverse signed error = 23/20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S094 S096 · impliesBefore finite prefix point0 index18:original point=0;n=18;limit=22;signed coefficient=23 After f_n(x)=463/20;f_n(x)-limit=23/20;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S095 S097 · calculationBefore finite prefix point0 index19 denominator := sum(19, 2) After finite prefix point0 index19 denominator = 21 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · calculationBefore finite prefix point0 index19 signed correction := div(23, 21) After finite prefix point0 index19 signed correction = 23/21 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097 S099 · calculationBefore finite prefix point0 index19 function value := sum(22, 23/21) After finite prefix point0 index19 function value = 485/21 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S098 S100 · calculationBefore finite prefix point0 index19 reverse signed error := sub(485/21, 22) After finite prefix point0 index19 reverse signed error = 23/21 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099 S101 · impliesBefore finite prefix point0 index19:original point=0;n=19;limit=22;signed coefficient=23 After f_n(x)=485/21;f_n(x)-limit=23/21;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S100 S102 · calculationBefore finite prefix point0 index20 denominator := sum(20, 2) After finite prefix point0 index20 denominator = 22 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · calculationBefore finite prefix point0 index20 signed correction := div(23, 22) After finite prefix point0 index20 signed correction = 23/22 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · calculationBefore finite prefix point0 index20 function value := sum(22, 23/22) After finite prefix point0 index20 function value = 507/22 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · calculationBefore finite prefix point0 index20 reverse signed error := sub(507/22, 22) After finite prefix point0 index20 reverse signed error = 23/22 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S104 S106 · impliesBefore finite prefix point0 index20:original point=0;n=20;limit=22;signed coefficient=23 After f_n(x)=507/22;f_n(x)-limit=23/22;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S105 S107 · calculationBefore finite prefix point0 index21 denominator := sum(21, 2) After finite prefix point0 index21 denominator = 23 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106 S108 · calculationBefore finite prefix point0 index21 signed correction := div(23, 23) After finite prefix point0 index21 signed correction = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107 S109 · calculationBefore finite prefix point0 index21 function value := sum(22, 1) After finite prefix point0 index21 function value = 23 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S108 S110 · calculationBefore finite prefix point0 index21 reverse signed error := sub(23, 22) After finite prefix point0 index21 reverse signed error = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S109 S111 · impliesBefore finite prefix point0 index21:original point=0;n=21;limit=22;signed coefficient=23 After f_n(x)=23;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S110 S112 · calculationBefore finite prefix point0 index22 denominator := sum(22, 2) After finite prefix point0 index22 denominator = 24 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111 S113 · calculationBefore finite prefix point0 index22 signed correction := div(23, 24) After finite prefix point0 index22 signed correction = 23/24 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S112 S114 · calculationBefore finite prefix point0 index22 function value := sum(22, 23/24) After finite prefix point0 index22 function value = 551/24 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S113 S115 · calculationBefore finite prefix point0 index22 reverse signed error := sub(551/24, 22) After finite prefix point0 index22 reverse signed error = 23/24 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S114 S116 · impliesBefore finite prefix point0 index22:original point=0;n=22;limit=22;signed coefficient=23 After f_n(x)=551/24;f_n(x)-limit=23/24;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S115 S117 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S116 S118 · calculationBefore finite prefix point1 index0 signed correction := div(-24, 2) After finite prefix point1 index0 signed correction = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117 S119 · calculationBefore finite prefix point1 index0 function value := sum(23, -12) After finite prefix point1 index0 function value = 11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S118 S120 · calculationBefore finite prefix point1 index0 reverse signed error := sub(11, 23) After finite prefix point1 index0 reverse signed error = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119 S121 · impliesBefore finite prefix point1 index0:original point=1;n=0;limit=23;signed coefficient=-24 After f_n(x)=11;f_n(x)-limit=-12;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S120 S122 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S121 S123 · calculationBefore finite prefix point1 index1 signed correction := div(-24, 3) After finite prefix point1 index1 signed correction = -8 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S122 S124 · calculationBefore finite prefix point1 index1 function value := sum(23, -8) After finite prefix point1 index1 function value = 15 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123 S125 · calculationBefore finite prefix point1 index1 reverse signed error := sub(15, 23) After finite prefix point1 index1 reverse signed error = -8 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124 S126 · impliesBefore finite prefix point1 index1:original point=1;n=1;limit=23;signed coefficient=-24 After f_n(x)=15;f_n(x)-limit=-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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S125 S127 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S126 S128 · calculationBefore finite prefix point1 index2 signed correction := div(-24, 4) After finite prefix point1 index2 signed correction = -6 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127 S129 · calculationBefore finite prefix point1 index2 function value := sum(23, -6) After finite prefix point1 index2 function value = 17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S128 S130 · calculationBefore finite prefix point1 index2 reverse signed error := sub(17, 23) After finite prefix point1 index2 reverse signed error = -6 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S129 S131 · impliesBefore finite prefix point1 index2:original point=1;n=2;limit=23;signed coefficient=-24 After f_n(x)=17;f_n(x)-limit=-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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S130 S132 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S131 S133 · calculationBefore finite prefix point1 index3 signed correction := div(-24, 5) After finite prefix point1 index3 signed correction = -24/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S132 S134 · calculationBefore finite prefix point1 index3 function value := sum(23, -24/5) After finite prefix point1 index3 function value = 91/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S133 S135 · calculationBefore finite prefix point1 index3 reverse signed error := sub(91/5, 23) After finite prefix point1 index3 reverse signed error = -24/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S134 S136 · impliesBefore finite prefix point1 index3:original point=1;n=3;limit=23;signed coefficient=-24 After f_n(x)=91/5;f_n(x)-limit=-24/5;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S135 S137 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S136 S138 · calculationBefore finite prefix point1 index4 signed correction := div(-24, 6) After finite prefix point1 index4 signed correction = -4 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S137 S139 · calculationBefore finite prefix point1 index4 function value := sum(23, -4) After finite prefix point1 index4 function value = 19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S138 S140 · calculationBefore finite prefix point1 index4 reverse signed error := sub(19, 23) After finite prefix point1 index4 reverse signed error = -4 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S139 S141 · impliesBefore finite prefix point1 index4:original point=1;n=4;limit=23;signed coefficient=-24 After f_n(x)=19;f_n(x)-limit=-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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S140 S142 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S141 S143 · calculationBefore finite prefix point1 index5 signed correction := div(-24, 7) After finite prefix point1 index5 signed correction = -24/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S142 S144 · calculationBefore finite prefix point1 index5 function value := sum(23, -24/7) After finite prefix point1 index5 function value = 137/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S143 S145 · calculationBefore finite prefix point1 index5 reverse signed error := sub(137/7, 23) After finite prefix point1 index5 reverse signed error = -24/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S144 S146 · impliesBefore finite prefix point1 index5:original point=1;n=5;limit=23;signed coefficient=-24 After f_n(x)=137/7;f_n(x)-limit=-24/7;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S145 S147 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S146 S148 · calculationBefore finite prefix point1 index6 signed correction := div(-24, 8) After finite prefix point1 index6 signed correction = -3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S147 S149 · calculationBefore finite prefix point1 index6 function value := sum(23, -3) After finite prefix point1 index6 function value = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S148 S150 · calculationBefore finite prefix point1 index6 reverse signed error := sub(20, 23) After finite prefix point1 index6 reverse signed error = -3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S149 S151 · impliesBefore finite prefix point1 index6:original point=1;n=6;limit=23;signed coefficient=-24 After f_n(x)=20;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S150 S152 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S151 S153 · calculationBefore finite prefix point1 index7 signed correction := div(-24, 9) After finite prefix point1 index7 signed correction = -8/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S152 S154 · calculationBefore finite prefix point1 index7 function value := sum(23, -8/3) After finite prefix point1 index7 function value = 61/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S153 S155 · calculationBefore finite prefix point1 index7 reverse signed error := sub(61/3, 23) After finite prefix point1 index7 reverse signed error = -8/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S154 S156 · impliesBefore finite prefix point1 index7:original point=1;n=7;limit=23;signed coefficient=-24 After f_n(x)=61/3;f_n(x)-limit=-8/3;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S155 S157 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S156 S158 · calculationBefore finite prefix point1 index8 signed correction := div(-24, 10) After finite prefix point1 index8 signed correction = -12/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S157 S159 · calculationBefore finite prefix point1 index8 function value := sum(23, -12/5) After finite prefix point1 index8 function value = 103/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S158 S160 · calculationBefore finite prefix point1 index8 reverse signed error := sub(103/5, 23) After finite prefix point1 index8 reverse signed error = -12/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S159 S161 · impliesBefore finite prefix point1 index8:original point=1;n=8;limit=23;signed coefficient=-24 After f_n(x)=103/5;f_n(x)-limit=-12/5;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S160 S162 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S161 S163 · calculationBefore finite prefix point1 index9 signed correction := div(-24, 11) After finite prefix point1 index9 signed correction = -24/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S162 S164 · calculationBefore finite prefix point1 index9 function value := sum(23, -24/11) After finite prefix point1 index9 function value = 229/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S163 S165 · calculationBefore finite prefix point1 index9 reverse signed error := sub(229/11, 23) After finite prefix point1 index9 reverse signed error = -24/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S164 S166 · impliesBefore finite prefix point1 index9:original point=1;n=9;limit=23;signed coefficient=-24 After f_n(x)=229/11;f_n(x)-limit=-24/11;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S165 S167 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S166 S168 · calculationBefore finite prefix point1 index10 signed correction := div(-24, 12) After finite prefix point1 index10 signed correction = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S167 S169 · calculationBefore finite prefix point1 index10 function value := sum(23, -2) After finite prefix point1 index10 function value = 21 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S168 S170 · calculationBefore finite prefix point1 index10 reverse signed error := sub(21, 23) After finite prefix point1 index10 reverse signed error = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S169 S171 · impliesBefore finite prefix point1 index10:original point=1;n=10;limit=23;signed coefficient=-24 After f_n(x)=21;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S170 S172 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S171 S173 · calculationBefore finite prefix point1 index11 signed correction := div(-24, 13) After finite prefix point1 index11 signed correction = -24/13 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S172 S174 · calculationBefore finite prefix point1 index11 function value := sum(23, -24/13) After finite prefix point1 index11 function value = 275/13 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S173 S175 · calculationBefore finite prefix point1 index11 reverse signed error := sub(275/13, 23) After finite prefix point1 index11 reverse signed error = -24/13 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S174 S176 · impliesBefore finite prefix point1 index11:original point=1;n=11;limit=23;signed coefficient=-24 After f_n(x)=275/13;f_n(x)-limit=-24/13;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S175 S177 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S176 S178 · calculationBefore finite prefix point1 index12 signed correction := div(-24, 14) After finite prefix point1 index12 signed correction = -12/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S177 S179 · calculationBefore finite prefix point1 index12 function value := sum(23, -12/7) After finite prefix point1 index12 function value = 149/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S178 S180 · calculationBefore finite prefix point1 index12 reverse signed error := sub(149/7, 23) After finite prefix point1 index12 reverse signed error = -12/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S179 S181 · impliesBefore finite prefix point1 index12:original point=1;n=12;limit=23;signed coefficient=-24 After f_n(x)=149/7;f_n(x)-limit=-12/7;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S180 S182 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S181 S183 · calculationBefore finite prefix point1 index13 signed correction := div(-24, 15) After finite prefix point1 index13 signed correction = -8/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S182 S184 · calculationBefore finite prefix point1 index13 function value := sum(23, -8/5) After finite prefix point1 index13 function value = 107/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S183 S185 · calculationBefore finite prefix point1 index13 reverse signed error := sub(107/5, 23) After finite prefix point1 index13 reverse signed error = -8/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S184 S186 · impliesBefore finite prefix point1 index13:original point=1;n=13;limit=23;signed coefficient=-24 After f_n(x)=107/5;f_n(x)-limit=-8/5;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S185 S187 · calculationBefore finite prefix point1 index14 denominator := sum(14, 2) After finite prefix point1 index14 denominator = 16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S186 S188 · calculationBefore finite prefix point1 index14 signed correction := div(-24, 16) After finite prefix point1 index14 signed correction = -3/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S187 S189 · calculationBefore finite prefix point1 index14 function value := sum(23, -3/2) After finite prefix point1 index14 function value = 43/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S188 S190 · calculationBefore finite prefix point1 index14 reverse signed error := sub(43/2, 23) After finite prefix point1 index14 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S189 S191 · impliesBefore finite prefix point1 index14:original point=1;n=14;limit=23;signed coefficient=-24 After f_n(x)=43/2;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S190 S192 · calculationBefore finite prefix point1 index15 denominator := sum(15, 2) After finite prefix point1 index15 denominator = 17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S191 S193 · calculationBefore finite prefix point1 index15 signed correction := div(-24, 17) After finite prefix point1 index15 signed correction = -24/17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S192 S194 · calculationBefore finite prefix point1 index15 function value := sum(23, -24/17) After finite prefix point1 index15 function value = 367/17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S193 S195 · calculationBefore finite prefix point1 index15 reverse signed error := sub(367/17, 23) After finite prefix point1 index15 reverse signed error = -24/17 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S194 S196 · impliesBefore finite prefix point1 index15:original point=1;n=15;limit=23;signed coefficient=-24 After f_n(x)=367/17;f_n(x)-limit=-24/17;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S195 S197 · calculationBefore finite prefix point1 index16 denominator := sum(16, 2) After finite prefix point1 index16 denominator = 18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S196 S198 · calculationBefore finite prefix point1 index16 signed correction := div(-24, 18) After finite prefix point1 index16 signed correction = -4/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S197 S199 · calculationBefore finite prefix point1 index16 function value := sum(23, -4/3) After finite prefix point1 index16 function value = 65/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S198 S200 · calculationBefore finite prefix point1 index16 reverse signed error := sub(65/3, 23) After finite prefix point1 index16 reverse signed error = -4/3 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S199 S201 · impliesBefore finite prefix point1 index16:original point=1;n=16;limit=23;signed coefficient=-24 After f_n(x)=65/3;f_n(x)-limit=-4/3;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S200 S202 · calculationBefore finite prefix point1 index17 denominator := sum(17, 2) After finite prefix point1 index17 denominator = 19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S201 S203 · calculationBefore finite prefix point1 index17 signed correction := div(-24, 19) After finite prefix point1 index17 signed correction = -24/19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S202 S204 · calculationBefore finite prefix point1 index17 function value := sum(23, -24/19) After finite prefix point1 index17 function value = 413/19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S203 S205 · calculationBefore finite prefix point1 index17 reverse signed error := sub(413/19, 23) After finite prefix point1 index17 reverse signed error = -24/19 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S204 S206 · impliesBefore finite prefix point1 index17:original point=1;n=17;limit=23;signed coefficient=-24 After f_n(x)=413/19;f_n(x)-limit=-24/19;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S205 S207 · calculationBefore finite prefix point1 index18 denominator := sum(18, 2) After finite prefix point1 index18 denominator = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S206 S208 · calculationBefore finite prefix point1 index18 signed correction := div(-24, 20) After finite prefix point1 index18 signed correction = -6/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S207 S209 · calculationBefore finite prefix point1 index18 function value := sum(23, -6/5) After finite prefix point1 index18 function value = 109/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S208 S210 · calculationBefore finite prefix point1 index18 reverse signed error := sub(109/5, 23) After finite prefix point1 index18 reverse signed error = -6/5 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S209 S211 · impliesBefore finite prefix point1 index18:original point=1;n=18;limit=23;signed coefficient=-24 After f_n(x)=109/5;f_n(x)-limit=-6/5;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S210 S212 · calculationBefore finite prefix point1 index19 denominator := sum(19, 2) After finite prefix point1 index19 denominator = 21 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S211 S213 · calculationBefore finite prefix point1 index19 signed correction := div(-24, 21) After finite prefix point1 index19 signed correction = -8/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S212 S214 · calculationBefore finite prefix point1 index19 function value := sum(23, -8/7) After finite prefix point1 index19 function value = 153/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S213 S215 · calculationBefore finite prefix point1 index19 reverse signed error := sub(153/7, 23) After finite prefix point1 index19 reverse signed error = -8/7 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S214 S216 · impliesBefore finite prefix point1 index19:original point=1;n=19;limit=23;signed coefficient=-24 After f_n(x)=153/7;f_n(x)-limit=-8/7;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S215 S217 · calculationBefore finite prefix point1 index20 denominator := sum(20, 2) After finite prefix point1 index20 denominator = 22 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S216 S218 · calculationBefore finite prefix point1 index20 signed correction := div(-24, 22) After finite prefix point1 index20 signed correction = -12/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S217 S219 · calculationBefore finite prefix point1 index20 function value := sum(23, -12/11) After finite prefix point1 index20 function value = 241/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S218 S220 · calculationBefore finite prefix point1 index20 reverse signed error := sub(241/11, 23) After finite prefix point1 index20 reverse signed error = -12/11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S219 S221 · impliesBefore finite prefix point1 index20:original point=1;n=20;limit=23;signed coefficient=-24 After f_n(x)=241/11;f_n(x)-limit=-12/11;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S220 S222 · calculationBefore finite prefix point1 index21 denominator := sum(21, 2) After finite prefix point1 index21 denominator = 23 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S221 S223 · calculationBefore finite prefix point1 index21 signed correction := div(-24, 23) After finite prefix point1 index21 signed correction = -24/23 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S222 S224 · calculationBefore finite prefix point1 index21 function value := sum(23, -24/23) After finite prefix point1 index21 function value = 505/23 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S223 S225 · calculationBefore finite prefix point1 index21 reverse signed error := sub(505/23, 23) After finite prefix point1 index21 reverse signed error = -24/23 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S224 S226 · impliesBefore finite prefix point1 index21:original point=1;n=21;limit=23;signed coefficient=-24 After f_n(x)=505/23;f_n(x)-limit=-24/23;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S225 S227 · calculationBefore finite prefix point1 index22 denominator := sum(22, 2) After finite prefix point1 index22 denominator = 24 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S226 S228 · calculationBefore finite prefix point1 index22 signed correction := div(-24, 24) After finite prefix point1 index22 signed correction = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S227 S229 · calculationBefore finite prefix point1 index22 function value := sum(23, -1) After finite prefix point1 index22 function value = 22 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S228 S230 · calculationBefore finite prefix point1 index22 reverse signed error := sub(22, 23) After finite prefix point1 index22 reverse signed error = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S229 S231 · impliesBefore finite prefix point1 index22:original point=1;n=22;limit=23;signed coefficient=-24 After f_n(x)=22;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 dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S230 S232 · impliesBefore finite observed length=23;complete domain-order rows=[[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]] 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 dependenciesRule: pointwise_finite_prefix_no_not_declared_model_failure · Uses: G001, S231 ## Reverse checks - S232 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S231 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S230 · premise_and_result_audit — Recompute finite prefix point1 index22 reverse signed error with exact rational arithmetic, not rounded decimals. - S229 · premise_and_result_audit — Recompute finite prefix point1 index22 function value with exact rational arithmetic, not rounded decimals. - S228 · premise_and_result_audit — Recompute finite prefix point1 index22 signed correction with exact rational arithmetic, not rounded decimals. - S227 · premise_and_result_audit — Recompute finite prefix point1 index22 denominator with exact rational arithmetic, not rounded decimals. - S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S225 · premise_and_result_audit — Recompute finite prefix point1 index21 reverse signed error with exact rational arithmetic, not rounded decimals. - S224 · premise_and_result_audit — Recompute finite prefix point1 index21 function value with exact rational arithmetic, not rounded decimals. - S223 · premise_and_result_audit — Recompute finite prefix point1 index21 signed correction with exact rational arithmetic, not rounded decimals. - S222 · premise_and_result_audit — Recompute finite prefix point1 index21 denominator with exact rational arithmetic, not rounded decimals. - S221 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S220 · premise_and_result_audit — Recompute finite prefix point1 index20 reverse signed error with exact rational arithmetic, not rounded decimals. - S219 · premise_and_result_audit — Recompute finite prefix point1 index20 function value with exact rational arithmetic, not rounded decimals. - S218 · premise_and_result_audit — Recompute finite prefix point1 index20 signed correction with exact rational arithmetic, not rounded decimals. - S217 · premise_and_result_audit — Recompute finite prefix point1 index20 denominator with exact rational arithmetic, not rounded decimals. - S216 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S215 · premise_and_result_audit — Recompute finite prefix point1 index19 reverse signed error with exact rational arithmetic, not rounded decimals. - S214 · premise_and_result_audit — Recompute finite prefix point1 index19 function value with exact rational arithmetic, not rounded decimals. - S213 · premise_and_result_audit — Recompute finite prefix point1 index19 signed correction with exact rational arithmetic, not rounded decimals. - S212 · premise_and_result_audit — Recompute finite prefix point1 index19 denominator with exact rational arithmetic, not rounded decimals. - S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S210 · premise_and_result_audit — Recompute finite prefix point1 index18 reverse signed error with exact rational arithmetic, not rounded decimals. - S209 · premise_and_result_audit — Recompute finite prefix point1 index18 function value with exact rational arithmetic, not rounded decimals. - S208 · premise_and_result_audit — Recompute finite prefix point1 index18 signed correction with exact rational arithmetic, not rounded decimals. - S207 · premise_and_result_audit — Recompute finite prefix point1 index18 denominator with exact rational arithmetic, not rounded decimals. - S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S205 · premise_and_result_audit — Recompute finite prefix point1 index17 reverse signed error with exact rational arithmetic, not rounded decimals. - S204 · premise_and_result_audit — Recompute finite prefix point1 index17 function value with exact rational arithmetic, not rounded decimals. - S203 · premise_and_result_audit — Recompute finite prefix point1 index17 signed correction with exact rational arithmetic, not rounded decimals. - S202 · premise_and_result_audit — Recompute finite prefix point1 index17 denominator with exact rational arithmetic, not rounded decimals. - S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S200 · premise_and_result_audit — Recompute finite prefix point1 index16 reverse signed error with exact rational arithmetic, not rounded decimals. - S199 · premise_and_result_audit — Recompute finite prefix point1 index16 function value with exact rational arithmetic, not rounded decimals. - S198 · premise_and_result_audit — Recompute finite prefix point1 index16 signed correction with exact rational arithmetic, not rounded decimals. - S197 · premise_and_result_audit — Recompute finite prefix point1 index16 denominator with exact rational arithmetic, not rounded decimals. - S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S195 · premise_and_result_audit — Recompute finite prefix point1 index15 reverse signed error with exact rational arithmetic, not rounded decimals. - S194 · premise_and_result_audit — Recompute finite prefix point1 index15 function value with exact rational arithmetic, not rounded decimals. - S193 · premise_and_result_audit — Recompute finite prefix point1 index15 signed correction with exact rational arithmetic, not rounded decimals. - S192 · premise_and_result_audit — Recompute finite prefix point1 index15 denominator with exact rational arithmetic, not rounded decimals. - S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S190 · premise_and_result_audit — Recompute finite prefix point1 index14 reverse signed error with exact rational arithmetic, not rounded decimals. - S189 · premise_and_result_audit — Recompute finite prefix point1 index14 function value with exact rational arithmetic, not rounded decimals. - S188 · premise_and_result_audit — Recompute finite prefix point1 index14 signed correction with exact rational arithmetic, not rounded decimals. - S187 · premise_and_result_audit — Recompute finite prefix point1 index14 denominator with exact rational arithmetic, not rounded decimals. - S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S185 · premise_and_result_audit — Recompute finite prefix point1 index13 reverse signed error with exact rational arithmetic, not rounded decimals. - S184 · premise_and_result_audit — Recompute finite prefix point1 index13 function value with exact rational arithmetic, not rounded decimals. - S183 · premise_and_result_audit — Recompute finite prefix point1 index13 signed correction with exact rational arithmetic, not rounded decimals. - S182 · premise_and_result_audit — Recompute finite prefix point1 index13 denominator with exact rational arithmetic, not rounded decimals. - S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S180 · premise_and_result_audit — Recompute finite prefix point1 index12 reverse signed error with exact rational arithmetic, not rounded decimals. - S179 · premise_and_result_audit — Recompute finite prefix point1 index12 function value with exact rational arithmetic, not rounded decimals. - S178 · premise_and_result_audit — Recompute finite prefix point1 index12 signed correction with exact rational arithmetic, not rounded decimals. - S177 · premise_and_result_audit — Recompute finite prefix point1 index12 denominator with exact rational arithmetic, not rounded decimals. - S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S175 · premise_and_result_audit — Recompute finite prefix point1 index11 reverse signed error with exact rational arithmetic, not rounded decimals. - S174 · premise_and_result_audit — Recompute finite prefix point1 index11 function value with exact rational arithmetic, not rounded decimals. - S173 · premise_and_result_audit — Recompute finite prefix point1 index11 signed correction with exact rational arithmetic, not rounded decimals. - S172 · premise_and_result_audit — Recompute finite prefix point1 index11 denominator with exact rational arithmetic, not rounded decimals. - S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S170 · premise_and_result_audit — Recompute finite prefix point1 index10 reverse signed error with exact rational arithmetic, not rounded decimals. - S169 · premise_and_result_audit — Recompute finite prefix point1 index10 function value with exact rational arithmetic, not rounded decimals. - S168 · premise_and_result_audit — Recompute finite prefix point1 index10 signed correction with exact rational arithmetic, not rounded decimals. - S167 · premise_and_result_audit — Recompute finite prefix point1 index10 denominator with exact rational arithmetic, not rounded decimals. - S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S165 · premise_and_result_audit — Recompute finite prefix point1 index9 reverse signed error with exact rational arithmetic, not rounded decimals. - S164 · premise_and_result_audit — Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals. - S163 · premise_and_result_audit — Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals. - S162 · premise_and_result_audit — Recompute finite prefix point1 index9 denominator with exact rational arithmetic, not rounded decimals. - S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S160 · premise_and_result_audit — Recompute finite prefix point1 index8 reverse signed error with exact rational arithmetic, not rounded decimals. - S159 · premise_and_result_audit — Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals. - S158 · premise_and_result_audit — Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals. - S157 · premise_and_result_audit — Recompute finite prefix point1 index8 denominator with exact rational arithmetic, not rounded decimals. - S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S155 · premise_and_result_audit — Recompute finite prefix point1 index7 reverse signed error with exact rational arithmetic, not rounded decimals. - S154 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals. - S153 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals. - S152 · premise_and_result_audit — Recompute finite prefix point1 index7 denominator with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S150 · premise_and_result_audit — Recompute finite prefix point1 index6 reverse signed error with exact rational arithmetic, not rounded decimals. - S149 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals. - S148 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Recompute finite prefix point1 index6 denominator with exact rational arithmetic, not rounded decimals. - S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S145 · premise_and_result_audit — Recompute finite prefix point1 index5 reverse signed error with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals. - S142 · premise_and_result_audit — Recompute finite prefix point1 index5 denominator with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S140 · premise_and_result_audit — Recompute finite prefix point1 index4 reverse signed error with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals. - S138 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Recompute finite prefix point1 index4 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 index3 reverse signed error with exact rational arithmetic, not rounded decimals. - S134 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals. - S133 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals. - S132 · premise_and_result_audit — Recompute finite prefix point1 index3 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 index2 reverse signed error with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals. - S128 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Recompute finite prefix point1 index2 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 index1 reverse signed error with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals. - S123 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals. - S122 · premise_and_result_audit — Recompute finite prefix point1 index1 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 index0 reverse signed error with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute finite prefix point1 index0 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 point0 index22 reverse signed error with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute finite prefix point0 index22 function value with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute finite prefix point0 index22 signed correction with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute finite prefix point0 index22 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 point0 index21 reverse signed error with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute finite prefix point0 index21 function value with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Recompute finite prefix point0 index21 signed correction with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Recompute finite prefix point0 index21 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 point0 index20 reverse signed error with exact rational arithmetic, not rounded decimals. - S104 · premise_and_result_audit — Recompute finite prefix point0 index20 function value with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute finite prefix point0 index20 signed correction with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Recompute finite prefix point0 index20 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 point0 index19 reverse signed error with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Recompute finite prefix point0 index19 function value with exact rational arithmetic, not rounded decimals. - S098 · premise_and_result_audit — Recompute finite prefix point0 index19 signed correction with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute finite prefix point0 index19 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 point0 index18 reverse signed error with exact rational arithmetic, not rounded decimals. - S094 · premise_and_result_audit — Recompute finite prefix point0 index18 function value with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Recompute finite prefix point0 index18 signed correction with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute finite prefix point0 index18 denominator with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S090 · premise_and_result_audit — Recompute finite prefix point0 index17 reverse signed error with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute finite prefix point0 index17 function value with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute finite prefix point0 index17 signed correction with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Recompute finite prefix point0 index17 denominator with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S085 · premise_and_result_audit — Recompute finite prefix point0 index16 reverse signed error with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute finite prefix point0 index16 function value with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute finite prefix point0 index16 signed correction with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute finite prefix point0 index16 denominator with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S080 · premise_and_result_audit — Recompute finite prefix point0 index15 reverse signed error with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute finite prefix point0 index15 function value with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute finite prefix point0 index15 signed correction with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute finite prefix point0 index15 denominator with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S075 · premise_and_result_audit — Recompute finite prefix point0 index14 reverse signed error with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute finite prefix point0 index14 function value with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute finite prefix point0 index14 signed correction with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute finite prefix point0 index14 denominator with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S070 · premise_and_result_audit — Recompute finite prefix point0 index13 reverse signed error with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute finite prefix point0 index13 function value with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute finite prefix point0 index13 signed correction with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute finite prefix point0 index13 denominator with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Recompute finite prefix point0 index12 reverse signed error with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute finite prefix point0 index12 function value with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute finite prefix point0 index12 signed correction with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute finite prefix point0 index12 denominator with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Recompute finite prefix point0 index11 reverse signed error with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute finite prefix point0 index11 function value with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute finite prefix point0 index11 signed correction with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute finite prefix point0 index11 denominator with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Recompute finite prefix point0 index10 reverse signed error with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute finite prefix point0 index10 function value with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute finite prefix point0 index10 signed correction with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute finite prefix point0 index10 denominator with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S050 · premise_and_result_audit — Recompute finite prefix point0 index9 reverse signed error with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute finite prefix point0 index9 function value with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute finite prefix point0 index9 signed correction with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute finite prefix point0 index9 denominator with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Recompute finite prefix point0 index8 reverse signed error with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute finite prefix point0 index8 function value with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute finite prefix point0 index8 signed correction with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute finite prefix point0 index8 denominator with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Recompute finite prefix point0 index7 reverse signed error with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute finite prefix point0 index7 function value with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute finite prefix point0 index7 signed correction with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute finite prefix point0 index7 denominator with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Recompute finite prefix point0 index6 reverse signed error with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute finite prefix point0 index6 function value with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute finite prefix point0 index6 signed correction with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute finite prefix point0 index6 denominator with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Recompute finite prefix point0 index5 reverse signed error with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute finite prefix point0 index5 function value with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute finite prefix point0 index5 signed correction with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute finite prefix point0 index5 denominator with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute finite prefix point0 index4 reverse signed error with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute finite prefix point0 index4 function value with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute finite prefix point0 index4 signed correction with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute finite prefix point0 index4 denominator with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute finite prefix point0 index3 reverse signed error with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute finite prefix point0 index3 function value with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute finite prefix point0 index3 signed correction with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute finite prefix point0 index3 denominator with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute finite prefix point0 index2 reverse signed error with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute finite prefix point0 index2 function value with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute finite prefix point0 index2 signed correction with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute finite prefix point0 index2 denominator with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute finite prefix point0 index1 reverse signed error with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute finite prefix point0 index1 function value with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute finite prefix point0 index1 signed correction with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute finite prefix point0 index1 denominator with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute finite prefix point0 index0 reverse signed error with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute finite prefix point0 index0 function value with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute finite prefix point0 index0 signed correction with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute finite prefix point0 index0 denominator with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], 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: 86e48b0fe10fcc38349cfe16999186513f3bf328010666780100c5f19e443601 - candidate record SHA-256: d0a98936528e2333507955a0301791a018e30a90324d9309cf4fc73f611027ab - semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336 - question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0098.json)