# Question 3871: Exact Moment Integral Minus a Finite Left Sum [Return to catalogue page 20](https://ocme.evemisslab.com/en/questions/page/20/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r06-definite-integral-foundations/#q-20261007-advance-20261009-r06-definite-integral-0071) 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. R0006-03871 · q-20261007-advance-20261009-r06-definite-integral-0071 ## Original question and givens On all R, f(x)=(21)x+(8). On a=-2≤b=1, use N=21 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers. - G001 — On all R, f(x)=(21)x+(8). On a=-2≤b=1, use N=21 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers. ## 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-definite-integral](https://ocme.evemisslab.com/en/mko/mko-definite-integral/) ## Step-by-step derivation S001 · impliesBefore explicit real polynomial moment or constant integrand model After oriented I(a,b)=H(b)−H(a);do not sort endpoints Conditions: none Real polynomials are continuous and integrable;use termwise primitive endpoint differences for exact oriented integrals,not absolute area or a finite sum. Step identifiers and dependenciesRule: polynomial_moment_scope · Uses: G001 S002 · calculationBefore interval width := sub(1, -2) After interval width = 3 Conditions: none Compute the nonnegative interval width,allowing zero width. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore partition step := div(3, 21) After partition step = 1/7 Conditions: none Divide by positive integer N to get the exact step. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore left sample0 offset := mul(0, 1/7) After left sample0 offset = 0 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore left sample0 x := sum(-2, 0) After left sample0 x = -2 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore left sample0 mx := mul(21, -2) After left sample0 mx = -42 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore left sample0 value := sum(-42, 8) After left sample0 value = -34 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore left sample0 rectangle := mul(-34, 1/7) After left sample0 rectangle = -34/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore [-2, -34, -34/7] After finite left-sum term index=0 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S008 S010 · calculationBefore left sample1 offset := mul(1, 1/7) After left sample1 offset = 1/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore left sample1 x := sum(-2, 1/7) After left sample1 x = -13/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore left sample1 mx := mul(21, -13/7) After left sample1 mx = -39 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore left sample1 value := sum(-39, 8) After left sample1 value = -31 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore left sample1 rectangle := mul(-31, 1/7) After left sample1 rectangle = -31/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore [-13/7, -31, -31/7] After finite left-sum term index=1 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S014 S016 · calculationBefore left sample2 offset := mul(2, 1/7) After left sample2 offset = 2/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore left sample2 x := sum(-2, 2/7) After left sample2 x = -12/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore left sample2 mx := mul(21, -12/7) After left sample2 mx = -36 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore left sample2 value := sum(-36, 8) After left sample2 value = -28 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore left sample2 rectangle := mul(-28, 1/7) After left sample2 rectangle = -4 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore [-12/7, -28, -4] After finite left-sum term index=2 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S020 S022 · calculationBefore left sample3 offset := mul(3, 1/7) After left sample3 offset = 3/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore left sample3 x := sum(-2, 3/7) After left sample3 x = -11/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore left sample3 mx := mul(21, -11/7) After left sample3 mx = -33 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore left sample3 value := sum(-33, 8) After left sample3 value = -25 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore left sample3 rectangle := mul(-25, 1/7) After left sample3 rectangle = -25/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore [-11/7, -25, -25/7] After finite left-sum term index=3 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S026 S028 · calculationBefore left sample4 offset := mul(4, 1/7) After left sample4 offset = 4/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore left sample4 x := sum(-2, 4/7) After left sample4 x = -10/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore left sample4 mx := mul(21, -10/7) After left sample4 mx = -30 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore left sample4 value := sum(-30, 8) After left sample4 value = -22 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore left sample4 rectangle := mul(-22, 1/7) After left sample4 rectangle = -22/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore [-10/7, -22, -22/7] After finite left-sum term index=4 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S032 S034 · calculationBefore left sample5 offset := mul(5, 1/7) After left sample5 offset = 5/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore left sample5 x := sum(-2, 5/7) After left sample5 x = -9/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore left sample5 mx := mul(21, -9/7) After left sample5 mx = -27 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore left sample5 value := sum(-27, 8) After left sample5 value = -19 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore left sample5 rectangle := mul(-19, 1/7) After left sample5 rectangle = -19/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore [-9/7, -19, -19/7] After finite left-sum term index=5 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S038 S040 · calculationBefore left sample6 offset := mul(6, 1/7) After left sample6 offset = 6/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore left sample6 x := sum(-2, 6/7) After left sample6 x = -8/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore left sample6 mx := mul(21, -8/7) After left sample6 mx = -24 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore left sample6 value := sum(-24, 8) After left sample6 value = -16 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore left sample6 rectangle := mul(-16, 1/7) After left sample6 rectangle = -16/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · impliesBefore [-8/7, -16, -16/7] After finite left-sum term index=6 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S044 S046 · calculationBefore left sample7 offset := mul(7, 1/7) After left sample7 offset = 1 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore left sample7 x := sum(-2, 1) After left sample7 x = -1 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore left sample7 mx := mul(21, -1) After left sample7 mx = -21 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore left sample7 value := sum(-21, 8) After left sample7 value = -13 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore left sample7 rectangle := mul(-13, 1/7) After left sample7 rectangle = -13/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · impliesBefore [-1, -13, -13/7] After finite left-sum term index=7 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S050 S052 · calculationBefore left sample8 offset := mul(8, 1/7) After left sample8 offset = 8/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore left sample8 x := sum(-2, 8/7) After left sample8 x = -6/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore left sample8 mx := mul(21, -6/7) After left sample8 mx = -18 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore left sample8 value := sum(-18, 8) After left sample8 value = -10 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore left sample8 rectangle := mul(-10, 1/7) After left sample8 rectangle = -10/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · impliesBefore [-6/7, -10, -10/7] After finite left-sum term index=8 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S056 S058 · calculationBefore left sample9 offset := mul(9, 1/7) After left sample9 offset = 9/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore left sample9 x := sum(-2, 9/7) After left sample9 x = -5/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore left sample9 mx := mul(21, -5/7) After left sample9 mx = -15 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore left sample9 value := sum(-15, 8) After left sample9 value = -7 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore left sample9 rectangle := mul(-7, 1/7) After left sample9 rectangle = -1 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · impliesBefore [-5/7, -7, -1] After finite left-sum term index=9 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S062 S064 · calculationBefore left sample10 offset := mul(10, 1/7) After left sample10 offset = 10/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore left sample10 x := sum(-2, 10/7) After left sample10 x = -4/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore left sample10 mx := mul(21, -4/7) After left sample10 mx = -12 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore left sample10 value := sum(-12, 8) After left sample10 value = -4 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore left sample10 rectangle := mul(-4, 1/7) After left sample10 rectangle = -4/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · impliesBefore [-4/7, -4, -4/7] After finite left-sum term index=10 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S068 S070 · calculationBefore left sample11 offset := mul(11, 1/7) After left sample11 offset = 11/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore left sample11 x := sum(-2, 11/7) After left sample11 x = -3/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore left sample11 mx := mul(21, -3/7) After left sample11 mx = -9 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore left sample11 value := sum(-9, 8) After left sample11 value = -1 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore left sample11 rectangle := mul(-1, 1/7) After left sample11 rectangle = -1/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · impliesBefore [-3/7, -1, -1/7] After finite left-sum term index=11 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S074 S076 · calculationBefore left sample12 offset := mul(12, 1/7) After left sample12 offset = 12/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore left sample12 x := sum(-2, 12/7) After left sample12 x = -2/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore left sample12 mx := mul(21, -2/7) After left sample12 mx = -6 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · calculationBefore left sample12 value := sum(-6, 8) After left sample12 value = 2 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · calculationBefore left sample12 rectangle := mul(2, 1/7) After left sample12 rectangle = 2/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · impliesBefore [-2/7, 2, 2/7] After finite left-sum term index=12 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S080 S082 · calculationBefore left sample13 offset := mul(13, 1/7) After left sample13 offset = 13/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore left sample13 x := sum(-2, 13/7) After left sample13 x = -1/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore left sample13 mx := mul(21, -1/7) After left sample13 mx = -3 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore left sample13 value := sum(-3, 8) After left sample13 value = 5 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · calculationBefore left sample13 rectangle := mul(5, 1/7) After left sample13 rectangle = 5/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085 S087 · impliesBefore [-1/7, 5, 5/7] After finite left-sum term index=13 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S086 S088 · calculationBefore left sample14 offset := mul(14, 1/7) After left sample14 offset = 2 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · calculationBefore left sample14 x := sum(-2, 2) After left sample14 x = 0 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · calculationBefore left sample14 mx := mul(21, 0) After left sample14 mx = 0 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089 S091 · calculationBefore left sample14 value := sum(0, 8) After left sample14 value = 8 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S090 S092 · calculationBefore left sample14 rectangle := mul(8, 1/7) After left sample14 rectangle = 8/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · impliesBefore [0, 8, 8/7] After finite left-sum term index=14 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S092 S094 · calculationBefore left sample15 offset := mul(15, 1/7) After left sample15 offset = 15/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S093 S095 · calculationBefore left sample15 x := sum(-2, 15/7) After left sample15 x = 1/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S094 S096 · calculationBefore left sample15 mx := mul(21, 1/7) After left sample15 mx = 3 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S095 S097 · calculationBefore left sample15 value := sum(3, 8) After left sample15 value = 11 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · calculationBefore left sample15 rectangle := mul(11, 1/7) After left sample15 rectangle = 11/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097 S099 · impliesBefore [1/7, 11, 11/7] After finite left-sum term index=15 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S098 S100 · calculationBefore left sample16 offset := mul(16, 1/7) After left sample16 offset = 16/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099 S101 · calculationBefore left sample16 x := sum(-2, 16/7) After left sample16 x = 2/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S100 S102 · calculationBefore left sample16 mx := mul(21, 2/7) After left sample16 mx = 6 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · calculationBefore left sample16 value := sum(6, 8) After left sample16 value = 14 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · calculationBefore left sample16 rectangle := mul(14, 1/7) After left sample16 rectangle = 2 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · impliesBefore [2/7, 14, 2] After finite left-sum term index=16 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S104 S106 · calculationBefore left sample17 offset := mul(17, 1/7) After left sample17 offset = 17/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105 S107 · calculationBefore left sample17 x := sum(-2, 17/7) After left sample17 x = 3/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106 S108 · calculationBefore left sample17 mx := mul(21, 3/7) After left sample17 mx = 9 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107 S109 · calculationBefore left sample17 value := sum(9, 8) After left sample17 value = 17 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S108 S110 · calculationBefore left sample17 rectangle := mul(17, 1/7) After left sample17 rectangle = 17/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S109 S111 · impliesBefore [3/7, 17, 17/7] After finite left-sum term index=17 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S110 S112 · calculationBefore left sample18 offset := mul(18, 1/7) After left sample18 offset = 18/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111 S113 · calculationBefore left sample18 x := sum(-2, 18/7) After left sample18 x = 4/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S112 S114 · calculationBefore left sample18 mx := mul(21, 4/7) After left sample18 mx = 12 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S113 S115 · calculationBefore left sample18 value := sum(12, 8) After left sample18 value = 20 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S114 S116 · calculationBefore left sample18 rectangle := mul(20, 1/7) After left sample18 rectangle = 20/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S115 S117 · impliesBefore [4/7, 20, 20/7] After finite left-sum term index=18 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S116 S118 · calculationBefore left sample19 offset := mul(19, 1/7) After left sample19 offset = 19/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117 S119 · calculationBefore left sample19 x := sum(-2, 19/7) After left sample19 x = 5/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S118 S120 · calculationBefore left sample19 mx := mul(21, 5/7) After left sample19 mx = 15 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119 S121 · calculationBefore left sample19 value := sum(15, 8) After left sample19 value = 23 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S120 S122 · calculationBefore left sample19 rectangle := mul(23, 1/7) After left sample19 rectangle = 23/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S121 S123 · impliesBefore [5/7, 23, 23/7] After finite left-sum term index=19 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S122 S124 · calculationBefore left sample20 offset := mul(20, 1/7) After left sample20 offset = 20/7 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123 S125 · calculationBefore left sample20 x := sum(-2, 20/7) After left sample20 x = 6/7 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124 S126 · calculationBefore left sample20 mx := mul(21, 6/7) After left sample20 mx = 18 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125 S127 · calculationBefore left sample20 value := sum(18, 8) After left sample20 value = 26 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S126 S128 · calculationBefore left sample20 rectangle := mul(26, 1/7) After left sample20 rectangle = 26/7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127 S129 · impliesBefore [6/7, 26, 26/7] After finite left-sum term index=20 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S128 S130 · calculationBefore finite left sum := sum(-34/7, -31/7, -4, -25/7, -22/7, -19/7, -16/7, -13/7, -10/7, -1, -4/7, -1/7, 2/7, 5/7, 8/7, 11/7, 2, 17/7, 20/7, 23/7, 26/7) After finite left sum = -12 Conditions: none Add all N contributions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S129 S131 · calculationBefore affine integral primitive coefficient1 := div(8, 1) After affine integral primitive coefficient1 = 8 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S130 S132 · calculationBefore affine integral primitive coefficient2 := div(21, 2) After affine integral primitive coefficient2 = 21/2 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S131 S133 · impliesBefore [0, 8, 21/2] After H′=the supplied polynomial;integrate from -2 to 1 Conditions: none Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference. Step identifiers and dependenciesRule: polynomial_primitive_identity · Uses: G001, S132 S134 · impliesBefore affine integral H(b) ascending coefficients=[0, 8, 21/2] After evaluate at x=1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S133 S135 · calculationBefore affine integral H(b) Horner multiply before degree1 := mul(21/2, 1) After affine integral H(b) Horner multiply before degree1 = 21/2 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S134 S136 · calculationBefore affine integral H(b) Horner add degree1 := sum(21/2, 8) After affine integral H(b) Horner add degree1 = 37/2 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S135 S137 · calculationBefore affine integral H(b) Horner multiply before degree0 := mul(37/2, 1) After affine integral H(b) Horner multiply before degree0 = 37/2 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S136 S138 · calculationBefore affine integral H(b) Horner add degree0 := sum(37/2, 0) After affine integral H(b) Horner add degree0 = 37/2 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S137 S139 · impliesBefore affine integral H(a) ascending coefficients=[0, 8, 21/2] After evaluate at x=-2 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S138 S140 · calculationBefore affine integral H(a) Horner multiply before degree1 := mul(21/2, -2) After affine integral H(a) Horner multiply before degree1 = -21 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S139 S141 · calculationBefore affine integral H(a) Horner add degree1 := sum(-21, 8) After affine integral H(a) Horner add degree1 = -13 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S140 S142 · calculationBefore affine integral H(a) Horner multiply before degree0 := mul(-13, -2) After affine integral H(a) Horner multiply before degree0 = 26 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S141 S143 · calculationBefore affine integral H(a) Horner add degree0 := sum(26, 0) After affine integral H(a) Horner add degree0 = 26 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S142 S144 · calculationBefore affine integral oriented moment := sub(37/2, 26) After affine integral oriented moment = -15/2 Conditions: none Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S143 S145 · calculationBefore integral minus left sum := sub(-15/2, -12) After integral minus left sum = 9/2 Conditions: none The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S144 S146 · impliesBefore I=-15/2;S_N=-12;N=21 After signed finite-sum difference=9/2 Conditions: none This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions. Step identifiers and dependenciesRule: finite_sum_vs_integral · Uses: G001, S145 ## Reverse checks - 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 integral minus left sum with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S142 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S140 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S138 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S136 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S135 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S132 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals. - S131 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals. - S130 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S128 · premise_and_result_audit — Recompute left sample20 rectangle with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Recompute left sample20 value with exact rational arithmetic, not rounded decimals. - S126 · premise_and_result_audit — Recompute left sample20 mx with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Recompute left sample20 x with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute left sample20 offset with exact rational arithmetic, not rounded decimals. - S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S122 · premise_and_result_audit — Recompute left sample19 rectangle with exact rational arithmetic, not rounded decimals. - S121 · premise_and_result_audit — Recompute left sample19 value with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute left sample19 mx with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute left sample19 x with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute left sample19 offset with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S116 · premise_and_result_audit — Recompute left sample18 rectangle with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Recompute left sample18 value with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute left sample18 mx with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute left sample18 x with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute left sample18 offset 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 left sample17 rectangle with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute left sample17 value with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Recompute left sample17 mx with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Recompute left sample17 x with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Recompute left sample17 offset with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S104 · premise_and_result_audit — Recompute left sample16 rectangle with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute left sample16 value with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Recompute left sample16 mx with exact rational arithmetic, not rounded decimals. - S101 · premise_and_result_audit — Recompute left sample16 x with exact rational arithmetic, not rounded decimals. - S100 · premise_and_result_audit — Recompute left sample16 offset with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S098 · premise_and_result_audit — Recompute left sample15 rectangle with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute left sample15 value with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute left sample15 mx with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Recompute left sample15 x with exact rational arithmetic, not rounded decimals. - S094 · premise_and_result_audit — Recompute left sample15 offset with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S092 · premise_and_result_audit — Recompute left sample14 rectangle with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute left sample14 value with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute left sample14 mx with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute left sample14 x with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute left sample14 offset with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S086 · premise_and_result_audit — Recompute left sample13 rectangle with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute left sample13 value with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute left sample13 mx with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute left sample13 x with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute left sample13 offset 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 left sample12 rectangle with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute left sample12 value with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute left sample12 mx with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute left sample12 x with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute left sample12 offset with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S074 · premise_and_result_audit — Recompute left sample11 rectangle with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute left sample11 value with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute left sample11 mx with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute left sample11 x with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute left sample11 offset with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S068 · premise_and_result_audit — Recompute left sample10 rectangle with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute left sample10 value with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute left sample10 mx with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute left sample10 x with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute left sample10 offset with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Recompute left sample9 rectangle with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute left sample9 value with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute left sample9 mx with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute left sample9 x with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute left sample9 offset with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Recompute left sample8 rectangle with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute left sample8 value with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute left sample8 mx with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute left sample8 x with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute left sample8 offset 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 left sample7 rectangle with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute left sample7 value with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute left sample7 mx with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute left sample7 x with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute left sample7 offset with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Recompute left sample6 rectangle with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute left sample6 value with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute left sample6 mx with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute left sample6 x with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute left sample6 offset with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Recompute left sample5 rectangle with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute left sample5 value with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute left sample5 mx with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute left sample5 x with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute left sample5 offset with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute left sample4 rectangle with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute left sample4 value with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute left sample4 mx with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute left sample4 x with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute left sample4 offset with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute left sample3 rectangle with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute left sample3 value with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute left sample3 mx with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute left sample3 x with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute left sample3 offset 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 left sample2 rectangle with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute left sample2 value with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute left sample2 mx with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute left sample2 x with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute left sample2 offset with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute left sample1 rectangle with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute left sample1 value with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute left sample1 mx with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute left sample1 x with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute left sample1 offset with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute left sample0 rectangle with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute left sample0 value with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute left sample0 mx with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute left sample0 x with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute left sample0 offset with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute partition step with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute interval width 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 9/2 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: f78b389290b4c722060e58f98a03b77a0e5049e17726453210931554cf4c29d6 - candidate record SHA-256: b2f6d0685ee603d93658ae5d151a5f6966a914228f8059f763f03a607df8b89a - semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d - question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r06-definite-integral-0071.json)