Question 3869: Exact Moment Integral Minus a Finite Left Sum

Original question and givens

On all R, f(x)=(19)x+(6). On a=0≤b=1, use N=19 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)=(19)x+(6). On a=0≤b=1, use N=19 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

Step-by-step derivation

explicit real polynomial moment or constant integrand model

oriented I(a,b)=H(b)−H(a);do not sort endpoints

interval width := sub(1, 0)

interval width = 1

partition step := div(1, 19)

partition step = 1/19

left sample0 offset := mul(0, 1/19)

left sample0 offset = 0

left sample0 x := sum(0, 0)

left sample0 x = 0

left sample0 mx := mul(19, 0)

left sample0 mx = 0

left sample0 value := sum(0, 6)

left sample0 value = 6

left sample0 rectangle := mul(6, 1/19)

left sample0 rectangle = 6/19

[0, 6, 6/19]

finite left-sum term index=0

left sample1 offset := mul(1, 1/19)

left sample1 offset = 1/19

left sample1 x := sum(0, 1/19)

left sample1 x = 1/19

left sample1 mx := mul(19, 1/19)

left sample1 mx = 1

left sample1 value := sum(1, 6)

left sample1 value = 7

left sample1 rectangle := mul(7, 1/19)

left sample1 rectangle = 7/19

[1/19, 7, 7/19]

finite left-sum term index=1

left sample2 offset := mul(2, 1/19)

left sample2 offset = 2/19

left sample2 x := sum(0, 2/19)

left sample2 x = 2/19

left sample2 mx := mul(19, 2/19)

left sample2 mx = 2

left sample2 value := sum(2, 6)

left sample2 value = 8

left sample2 rectangle := mul(8, 1/19)

left sample2 rectangle = 8/19

[2/19, 8, 8/19]

finite left-sum term index=2

left sample3 offset := mul(3, 1/19)

left sample3 offset = 3/19

left sample3 x := sum(0, 3/19)

left sample3 x = 3/19

left sample3 mx := mul(19, 3/19)

left sample3 mx = 3

left sample3 value := sum(3, 6)

left sample3 value = 9

left sample3 rectangle := mul(9, 1/19)

left sample3 rectangle = 9/19

[3/19, 9, 9/19]

finite left-sum term index=3

left sample4 offset := mul(4, 1/19)

left sample4 offset = 4/19

left sample4 x := sum(0, 4/19)

left sample4 x = 4/19

left sample4 mx := mul(19, 4/19)

left sample4 mx = 4

left sample4 value := sum(4, 6)

left sample4 value = 10

left sample4 rectangle := mul(10, 1/19)

left sample4 rectangle = 10/19

[4/19, 10, 10/19]

finite left-sum term index=4

left sample5 offset := mul(5, 1/19)

left sample5 offset = 5/19

left sample5 x := sum(0, 5/19)

left sample5 x = 5/19

left sample5 mx := mul(19, 5/19)

left sample5 mx = 5

left sample5 value := sum(5, 6)

left sample5 value = 11

left sample5 rectangle := mul(11, 1/19)

left sample5 rectangle = 11/19

[5/19, 11, 11/19]

finite left-sum term index=5

left sample6 offset := mul(6, 1/19)

left sample6 offset = 6/19

left sample6 x := sum(0, 6/19)

left sample6 x = 6/19

left sample6 mx := mul(19, 6/19)

left sample6 mx = 6

left sample6 value := sum(6, 6)

left sample6 value = 12

left sample6 rectangle := mul(12, 1/19)

left sample6 rectangle = 12/19

[6/19, 12, 12/19]

finite left-sum term index=6

left sample7 offset := mul(7, 1/19)

left sample7 offset = 7/19

left sample7 x := sum(0, 7/19)

left sample7 x = 7/19

left sample7 mx := mul(19, 7/19)

left sample7 mx = 7

left sample7 value := sum(7, 6)

left sample7 value = 13

left sample7 rectangle := mul(13, 1/19)

left sample7 rectangle = 13/19

[7/19, 13, 13/19]

finite left-sum term index=7

left sample8 offset := mul(8, 1/19)

left sample8 offset = 8/19

left sample8 x := sum(0, 8/19)

left sample8 x = 8/19

left sample8 mx := mul(19, 8/19)

left sample8 mx = 8

left sample8 value := sum(8, 6)

left sample8 value = 14

left sample8 rectangle := mul(14, 1/19)

left sample8 rectangle = 14/19

[8/19, 14, 14/19]

finite left-sum term index=8

left sample9 offset := mul(9, 1/19)

left sample9 offset = 9/19

left sample9 x := sum(0, 9/19)

left sample9 x = 9/19

left sample9 mx := mul(19, 9/19)

left sample9 mx = 9

left sample9 value := sum(9, 6)

left sample9 value = 15

left sample9 rectangle := mul(15, 1/19)

left sample9 rectangle = 15/19

[9/19, 15, 15/19]

finite left-sum term index=9

left sample10 offset := mul(10, 1/19)

left sample10 offset = 10/19

left sample10 x := sum(0, 10/19)

left sample10 x = 10/19

left sample10 mx := mul(19, 10/19)

left sample10 mx = 10

left sample10 value := sum(10, 6)

left sample10 value = 16

left sample10 rectangle := mul(16, 1/19)

left sample10 rectangle = 16/19

[10/19, 16, 16/19]

finite left-sum term index=10

left sample11 offset := mul(11, 1/19)

left sample11 offset = 11/19

left sample11 x := sum(0, 11/19)

left sample11 x = 11/19

left sample11 mx := mul(19, 11/19)

left sample11 mx = 11

left sample11 value := sum(11, 6)

left sample11 value = 17

left sample11 rectangle := mul(17, 1/19)

left sample11 rectangle = 17/19

[11/19, 17, 17/19]

finite left-sum term index=11

left sample12 offset := mul(12, 1/19)

left sample12 offset = 12/19

left sample12 x := sum(0, 12/19)

left sample12 x = 12/19

left sample12 mx := mul(19, 12/19)

left sample12 mx = 12

left sample12 value := sum(12, 6)

left sample12 value = 18

left sample12 rectangle := mul(18, 1/19)

left sample12 rectangle = 18/19

[12/19, 18, 18/19]

finite left-sum term index=12

left sample13 offset := mul(13, 1/19)

left sample13 offset = 13/19

left sample13 x := sum(0, 13/19)

left sample13 x = 13/19

left sample13 mx := mul(19, 13/19)

left sample13 mx = 13

left sample13 value := sum(13, 6)

left sample13 value = 19

left sample13 rectangle := mul(19, 1/19)

left sample13 rectangle = 1

[13/19, 19, 1]

finite left-sum term index=13

left sample14 offset := mul(14, 1/19)

left sample14 offset = 14/19

left sample14 x := sum(0, 14/19)

left sample14 x = 14/19

left sample14 mx := mul(19, 14/19)

left sample14 mx = 14

left sample14 value := sum(14, 6)

left sample14 value = 20

left sample14 rectangle := mul(20, 1/19)

left sample14 rectangle = 20/19

[14/19, 20, 20/19]

finite left-sum term index=14

left sample15 offset := mul(15, 1/19)

left sample15 offset = 15/19

left sample15 x := sum(0, 15/19)

left sample15 x = 15/19

left sample15 mx := mul(19, 15/19)

left sample15 mx = 15

left sample15 value := sum(15, 6)

left sample15 value = 21

left sample15 rectangle := mul(21, 1/19)

left sample15 rectangle = 21/19

[15/19, 21, 21/19]

finite left-sum term index=15

left sample16 offset := mul(16, 1/19)

left sample16 offset = 16/19

left sample16 x := sum(0, 16/19)

left sample16 x = 16/19

left sample16 mx := mul(19, 16/19)

left sample16 mx = 16

left sample16 value := sum(16, 6)

left sample16 value = 22

left sample16 rectangle := mul(22, 1/19)

left sample16 rectangle = 22/19

[16/19, 22, 22/19]

finite left-sum term index=16

left sample17 offset := mul(17, 1/19)

left sample17 offset = 17/19

left sample17 x := sum(0, 17/19)

left sample17 x = 17/19

left sample17 mx := mul(19, 17/19)

left sample17 mx = 17

left sample17 value := sum(17, 6)

left sample17 value = 23

left sample17 rectangle := mul(23, 1/19)

left sample17 rectangle = 23/19

[17/19, 23, 23/19]

finite left-sum term index=17

left sample18 offset := mul(18, 1/19)

left sample18 offset = 18/19

left sample18 x := sum(0, 18/19)

left sample18 x = 18/19

left sample18 mx := mul(19, 18/19)

left sample18 mx = 18

left sample18 value := sum(18, 6)

left sample18 value = 24

left sample18 rectangle := mul(24, 1/19)

left sample18 rectangle = 24/19

[18/19, 24, 24/19]

finite left-sum term index=18

finite left sum := sum(6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19, 1, 20/19, 21/19, 22/19, 23/19, 24/19)

finite left sum = 15

affine integral primitive coefficient1 := div(6, 1)

affine integral primitive coefficient1 = 6

affine integral primitive coefficient2 := div(19, 2)

affine integral primitive coefficient2 = 19/2

[0, 6, 19/2]

H′=the supplied polynomial;integrate from 0 to 1

affine integral H(b) ascending coefficients=[0, 6, 19/2]

evaluate at x=1

affine integral H(b) Horner multiply before degree1 := mul(19/2, 1)

affine integral H(b) Horner multiply before degree1 = 19/2

affine integral H(b) Horner add degree1 := sum(19/2, 6)

affine integral H(b) Horner add degree1 = 31/2

affine integral H(b) Horner multiply before degree0 := mul(31/2, 1)

affine integral H(b) Horner multiply before degree0 = 31/2

affine integral H(b) Horner add degree0 := sum(31/2, 0)

affine integral H(b) Horner add degree0 = 31/2

affine integral H(a) ascending coefficients=[0, 6, 19/2]

evaluate at x=0

affine integral H(a) Horner multiply before degree1 := mul(19/2, 0)

affine integral H(a) Horner multiply before degree1 = 0

affine integral H(a) Horner add degree1 := sum(0, 6)

affine integral H(a) Horner add degree1 = 6

affine integral H(a) Horner multiply before degree0 := mul(6, 0)

affine integral H(a) Horner multiply before degree0 = 0

affine integral H(a) Horner add degree0 := sum(0, 0)

affine integral H(a) Horner add degree0 = 0

affine integral oriented moment := sub(31/2, 0)

affine integral oriented moment = 31/2

integral minus left sum := sub(31/2, 15)

integral minus left sum = 1/2

I=31/2;S_N=15;N=19

signed finite-sum difference=1/2

Reverse checks

• 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 — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute finite left sum 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

1/2

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.