Question 32621: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(21)x+(8). For a=-2≤b=-1 and N=21 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f(x)=(21)x+(8). For a=-2≤b=-1 and N=21 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, 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

source roles = [qf-integral-affine-left-sum, affine_left_sum, reals, []]

derived = [[finite_left_sum_fraction_RETAIN1]]

Left sum ORIGINAL upper minus lower endpoint := sub(-1, -2)

Left sum ORIGINAL upper minus lower endpoint = 1

Left sum ORIGINAL width over positive cell count := div(1, 21)

Left sum ORIGINAL width over positive cell count = 1/21

Left sum ORIGINAL sample0 index times width := mul(0, 1/21)

Left sum ORIGINAL sample0 index times width = 0

Left sum ORIGINAL sample0 lower plus offset := sum(-2, 0)

Left sum ORIGINAL sample0 lower plus offset = -2

Left sum ORIGINAL sample0 slope times x := mul(21, -2)

Left sum ORIGINAL sample0 slope times x = -42

Left sum ORIGINAL sample0 affine value plus c := sum(-42, 8)

Left sum ORIGINAL sample0 affine value plus c = -34

Left sum ORIGINAL sample0 value times cell width := mul(-34, 1/21)

Left sum ORIGINAL sample0 value times cell width = -34/21

Left sum ORIGINAL accumulated term0 := sum(0, -34/21)

Left sum ORIGINAL accumulated term0 = -34/21

source roles = [0, 21, 21, 8, -2, -1, 1/21]

derived = [-2/1, -34/1, -34/21, -34/21]

Left sum ORIGINAL sample1 index times width := mul(1, 1/21)

Left sum ORIGINAL sample1 index times width = 1/21

Left sum ORIGINAL sample1 lower plus offset := sum(-2, 1/21)

Left sum ORIGINAL sample1 lower plus offset = -41/21

Left sum ORIGINAL sample1 slope times x := mul(21, -41/21)

Left sum ORIGINAL sample1 slope times x = -41

Left sum ORIGINAL sample1 affine value plus c := sum(-41, 8)

Left sum ORIGINAL sample1 affine value plus c = -33

Left sum ORIGINAL sample1 value times cell width := mul(-33, 1/21)

Left sum ORIGINAL sample1 value times cell width = -11/7

Left sum ORIGINAL accumulated term1 := sum(-34/21, -11/7)

Left sum ORIGINAL accumulated term1 = -67/21

source roles = [1, 21, 21, 8, -2, -1, 1/21]

derived = [-41/21, -33/1, -11/7, -67/21]

Left sum ORIGINAL sample2 index times width := mul(2, 1/21)

Left sum ORIGINAL sample2 index times width = 2/21

Left sum ORIGINAL sample2 lower plus offset := sum(-2, 2/21)

Left sum ORIGINAL sample2 lower plus offset = -40/21

Left sum ORIGINAL sample2 slope times x := mul(21, -40/21)

Left sum ORIGINAL sample2 slope times x = -40

Left sum ORIGINAL sample2 affine value plus c := sum(-40, 8)

Left sum ORIGINAL sample2 affine value plus c = -32

Left sum ORIGINAL sample2 value times cell width := mul(-32, 1/21)

Left sum ORIGINAL sample2 value times cell width = -32/21

Left sum ORIGINAL accumulated term2 := sum(-67/21, -32/21)

Left sum ORIGINAL accumulated term2 = -33/7

source roles = [2, 21, 21, 8, -2, -1, 1/21]

derived = [-40/21, -32/1, -32/21, -33/7]

Left sum ORIGINAL sample3 index times width := mul(3, 1/21)

Left sum ORIGINAL sample3 index times width = 1/7

Left sum ORIGINAL sample3 lower plus offset := sum(-2, 1/7)

Left sum ORIGINAL sample3 lower plus offset = -13/7

Left sum ORIGINAL sample3 slope times x := mul(21, -13/7)

Left sum ORIGINAL sample3 slope times x = -39

Left sum ORIGINAL sample3 affine value plus c := sum(-39, 8)

Left sum ORIGINAL sample3 affine value plus c = -31

Left sum ORIGINAL sample3 value times cell width := mul(-31, 1/21)

Left sum ORIGINAL sample3 value times cell width = -31/21

Left sum ORIGINAL accumulated term3 := sum(-33/7, -31/21)

Left sum ORIGINAL accumulated term3 = -130/21

source roles = [3, 21, 21, 8, -2, -1, 1/21]

derived = [-13/7, -31/1, -31/21, -130/21]

Left sum ORIGINAL sample4 index times width := mul(4, 1/21)

Left sum ORIGINAL sample4 index times width = 4/21

Left sum ORIGINAL sample4 lower plus offset := sum(-2, 4/21)

Left sum ORIGINAL sample4 lower plus offset = -38/21

Left sum ORIGINAL sample4 slope times x := mul(21, -38/21)

Left sum ORIGINAL sample4 slope times x = -38

Left sum ORIGINAL sample4 affine value plus c := sum(-38, 8)

Left sum ORIGINAL sample4 affine value plus c = -30

Left sum ORIGINAL sample4 value times cell width := mul(-30, 1/21)

Left sum ORIGINAL sample4 value times cell width = -10/7

Left sum ORIGINAL accumulated term4 := sum(-130/21, -10/7)

Left sum ORIGINAL accumulated term4 = -160/21

source roles = [4, 21, 21, 8, -2, -1, 1/21]

derived = [-38/21, -30/1, -10/7, -160/21]

Left sum ORIGINAL sample5 index times width := mul(5, 1/21)

Left sum ORIGINAL sample5 index times width = 5/21

Left sum ORIGINAL sample5 lower plus offset := sum(-2, 5/21)

Left sum ORIGINAL sample5 lower plus offset = -37/21

Left sum ORIGINAL sample5 slope times x := mul(21, -37/21)

Left sum ORIGINAL sample5 slope times x = -37

Left sum ORIGINAL sample5 affine value plus c := sum(-37, 8)

Left sum ORIGINAL sample5 affine value plus c = -29

Left sum ORIGINAL sample5 value times cell width := mul(-29, 1/21)

Left sum ORIGINAL sample5 value times cell width = -29/21

Left sum ORIGINAL accumulated term5 := sum(-160/21, -29/21)

Left sum ORIGINAL accumulated term5 = -9

source roles = [5, 21, 21, 8, -2, -1, 1/21]

derived = [-37/21, -29/1, -29/21, -9/1]

Left sum ORIGINAL sample6 index times width := mul(6, 1/21)

Left sum ORIGINAL sample6 index times width = 2/7

Left sum ORIGINAL sample6 lower plus offset := sum(-2, 2/7)

Left sum ORIGINAL sample6 lower plus offset = -12/7

Left sum ORIGINAL sample6 slope times x := mul(21, -12/7)

Left sum ORIGINAL sample6 slope times x = -36

Left sum ORIGINAL sample6 affine value plus c := sum(-36, 8)

Left sum ORIGINAL sample6 affine value plus c = -28

Left sum ORIGINAL sample6 value times cell width := mul(-28, 1/21)

Left sum ORIGINAL sample6 value times cell width = -4/3

Left sum ORIGINAL accumulated term6 := sum(-9, -4/3)

Left sum ORIGINAL accumulated term6 = -31/3

source roles = [6, 21, 21, 8, -2, -1, 1/21]

derived = [-12/7, -28/1, -4/3, -31/3]

Left sum ORIGINAL sample7 index times width := mul(7, 1/21)

Left sum ORIGINAL sample7 index times width = 1/3

Left sum ORIGINAL sample7 lower plus offset := sum(-2, 1/3)

Left sum ORIGINAL sample7 lower plus offset = -5/3

Left sum ORIGINAL sample7 slope times x := mul(21, -5/3)

Left sum ORIGINAL sample7 slope times x = -35

Left sum ORIGINAL sample7 affine value plus c := sum(-35, 8)

Left sum ORIGINAL sample7 affine value plus c = -27

Left sum ORIGINAL sample7 value times cell width := mul(-27, 1/21)

Left sum ORIGINAL sample7 value times cell width = -9/7

Left sum ORIGINAL accumulated term7 := sum(-31/3, -9/7)

Left sum ORIGINAL accumulated term7 = -244/21

source roles = [7, 21, 21, 8, -2, -1, 1/21]

derived = [-5/3, -27/1, -9/7, -244/21]

Left sum ORIGINAL sample8 index times width := mul(8, 1/21)

Left sum ORIGINAL sample8 index times width = 8/21

Left sum ORIGINAL sample8 lower plus offset := sum(-2, 8/21)

Left sum ORIGINAL sample8 lower plus offset = -34/21

Left sum ORIGINAL sample8 slope times x := mul(21, -34/21)

Left sum ORIGINAL sample8 slope times x = -34

Left sum ORIGINAL sample8 affine value plus c := sum(-34, 8)

Left sum ORIGINAL sample8 affine value plus c = -26

Left sum ORIGINAL sample8 value times cell width := mul(-26, 1/21)

Left sum ORIGINAL sample8 value times cell width = -26/21

Left sum ORIGINAL accumulated term8 := sum(-244/21, -26/21)

Left sum ORIGINAL accumulated term8 = -90/7

source roles = [8, 21, 21, 8, -2, -1, 1/21]

derived = [-34/21, -26/1, -26/21, -90/7]

Left sum ORIGINAL sample9 index times width := mul(9, 1/21)

Left sum ORIGINAL sample9 index times width = 3/7

Left sum ORIGINAL sample9 lower plus offset := sum(-2, 3/7)

Left sum ORIGINAL sample9 lower plus offset = -11/7

Left sum ORIGINAL sample9 slope times x := mul(21, -11/7)

Left sum ORIGINAL sample9 slope times x = -33

Left sum ORIGINAL sample9 affine value plus c := sum(-33, 8)

Left sum ORIGINAL sample9 affine value plus c = -25

Left sum ORIGINAL sample9 value times cell width := mul(-25, 1/21)

Left sum ORIGINAL sample9 value times cell width = -25/21

Left sum ORIGINAL accumulated term9 := sum(-90/7, -25/21)

Left sum ORIGINAL accumulated term9 = -295/21

source roles = [9, 21, 21, 8, -2, -1, 1/21]

derived = [-11/7, -25/1, -25/21, -295/21]

Left sum ORIGINAL sample10 index times width := mul(10, 1/21)

Left sum ORIGINAL sample10 index times width = 10/21

Left sum ORIGINAL sample10 lower plus offset := sum(-2, 10/21)

Left sum ORIGINAL sample10 lower plus offset = -32/21

Left sum ORIGINAL sample10 slope times x := mul(21, -32/21)

Left sum ORIGINAL sample10 slope times x = -32

Left sum ORIGINAL sample10 affine value plus c := sum(-32, 8)

Left sum ORIGINAL sample10 affine value plus c = -24

Left sum ORIGINAL sample10 value times cell width := mul(-24, 1/21)

Left sum ORIGINAL sample10 value times cell width = -8/7

Left sum ORIGINAL accumulated term10 := sum(-295/21, -8/7)

Left sum ORIGINAL accumulated term10 = -319/21

source roles = [10, 21, 21, 8, -2, -1, 1/21]

derived = [-32/21, -24/1, -8/7, -319/21]

Left sum ORIGINAL sample11 index times width := mul(11, 1/21)

Left sum ORIGINAL sample11 index times width = 11/21

Left sum ORIGINAL sample11 lower plus offset := sum(-2, 11/21)

Left sum ORIGINAL sample11 lower plus offset = -31/21

Left sum ORIGINAL sample11 slope times x := mul(21, -31/21)

Left sum ORIGINAL sample11 slope times x = -31

Left sum ORIGINAL sample11 affine value plus c := sum(-31, 8)

Left sum ORIGINAL sample11 affine value plus c = -23

Left sum ORIGINAL sample11 value times cell width := mul(-23, 1/21)

Left sum ORIGINAL sample11 value times cell width = -23/21

Left sum ORIGINAL accumulated term11 := sum(-319/21, -23/21)

Left sum ORIGINAL accumulated term11 = -114/7

source roles = [11, 21, 21, 8, -2, -1, 1/21]

derived = [-31/21, -23/1, -23/21, -114/7]

Left sum ORIGINAL sample12 index times width := mul(12, 1/21)

Left sum ORIGINAL sample12 index times width = 4/7

Left sum ORIGINAL sample12 lower plus offset := sum(-2, 4/7)

Left sum ORIGINAL sample12 lower plus offset = -10/7

Left sum ORIGINAL sample12 slope times x := mul(21, -10/7)

Left sum ORIGINAL sample12 slope times x = -30

Left sum ORIGINAL sample12 affine value plus c := sum(-30, 8)

Left sum ORIGINAL sample12 affine value plus c = -22

Left sum ORIGINAL sample12 value times cell width := mul(-22, 1/21)

Left sum ORIGINAL sample12 value times cell width = -22/21

Left sum ORIGINAL accumulated term12 := sum(-114/7, -22/21)

Left sum ORIGINAL accumulated term12 = -52/3

source roles = [12, 21, 21, 8, -2, -1, 1/21]

derived = [-10/7, -22/1, -22/21, -52/3]

Left sum ORIGINAL sample13 index times width := mul(13, 1/21)

Left sum ORIGINAL sample13 index times width = 13/21

Left sum ORIGINAL sample13 lower plus offset := sum(-2, 13/21)

Left sum ORIGINAL sample13 lower plus offset = -29/21

Left sum ORIGINAL sample13 slope times x := mul(21, -29/21)

Left sum ORIGINAL sample13 slope times x = -29

Left sum ORIGINAL sample13 affine value plus c := sum(-29, 8)

Left sum ORIGINAL sample13 affine value plus c = -21

Left sum ORIGINAL sample13 value times cell width := mul(-21, 1/21)

Left sum ORIGINAL sample13 value times cell width = -1

Left sum ORIGINAL accumulated term13 := sum(-52/3, -1)

Left sum ORIGINAL accumulated term13 = -55/3

source roles = [13, 21, 21, 8, -2, -1, 1/21]

derived = [-29/21, -21/1, -1/1, -55/3]

Left sum ORIGINAL sample14 index times width := mul(14, 1/21)

Left sum ORIGINAL sample14 index times width = 2/3

Left sum ORIGINAL sample14 lower plus offset := sum(-2, 2/3)

Left sum ORIGINAL sample14 lower plus offset = -4/3

Left sum ORIGINAL sample14 slope times x := mul(21, -4/3)

Left sum ORIGINAL sample14 slope times x = -28

Left sum ORIGINAL sample14 affine value plus c := sum(-28, 8)

Left sum ORIGINAL sample14 affine value plus c = -20

Left sum ORIGINAL sample14 value times cell width := mul(-20, 1/21)

Left sum ORIGINAL sample14 value times cell width = -20/21

Left sum ORIGINAL accumulated term14 := sum(-55/3, -20/21)

Left sum ORIGINAL accumulated term14 = -135/7

source roles = [14, 21, 21, 8, -2, -1, 1/21]

derived = [-4/3, -20/1, -20/21, -135/7]

Left sum ORIGINAL sample15 index times width := mul(15, 1/21)

Left sum ORIGINAL sample15 index times width = 5/7

Left sum ORIGINAL sample15 lower plus offset := sum(-2, 5/7)

Left sum ORIGINAL sample15 lower plus offset = -9/7

Left sum ORIGINAL sample15 slope times x := mul(21, -9/7)

Left sum ORIGINAL sample15 slope times x = -27

Left sum ORIGINAL sample15 affine value plus c := sum(-27, 8)

Left sum ORIGINAL sample15 affine value plus c = -19

Left sum ORIGINAL sample15 value times cell width := mul(-19, 1/21)

Left sum ORIGINAL sample15 value times cell width = -19/21

Left sum ORIGINAL accumulated term15 := sum(-135/7, -19/21)

Left sum ORIGINAL accumulated term15 = -424/21

source roles = [15, 21, 21, 8, -2, -1, 1/21]

derived = [-9/7, -19/1, -19/21, -424/21]

Left sum ORIGINAL sample16 index times width := mul(16, 1/21)

Left sum ORIGINAL sample16 index times width = 16/21

Left sum ORIGINAL sample16 lower plus offset := sum(-2, 16/21)

Left sum ORIGINAL sample16 lower plus offset = -26/21

Left sum ORIGINAL sample16 slope times x := mul(21, -26/21)

Left sum ORIGINAL sample16 slope times x = -26

Left sum ORIGINAL sample16 affine value plus c := sum(-26, 8)

Left sum ORIGINAL sample16 affine value plus c = -18

Left sum ORIGINAL sample16 value times cell width := mul(-18, 1/21)

Left sum ORIGINAL sample16 value times cell width = -6/7

Left sum ORIGINAL accumulated term16 := sum(-424/21, -6/7)

Left sum ORIGINAL accumulated term16 = -442/21

source roles = [16, 21, 21, 8, -2, -1, 1/21]

derived = [-26/21, -18/1, -6/7, -442/21]

Left sum ORIGINAL sample17 index times width := mul(17, 1/21)

Left sum ORIGINAL sample17 index times width = 17/21

Left sum ORIGINAL sample17 lower plus offset := sum(-2, 17/21)

Left sum ORIGINAL sample17 lower plus offset = -25/21

Left sum ORIGINAL sample17 slope times x := mul(21, -25/21)

Left sum ORIGINAL sample17 slope times x = -25

Left sum ORIGINAL sample17 affine value plus c := sum(-25, 8)

Left sum ORIGINAL sample17 affine value plus c = -17

Left sum ORIGINAL sample17 value times cell width := mul(-17, 1/21)

Left sum ORIGINAL sample17 value times cell width = -17/21

Left sum ORIGINAL accumulated term17 := sum(-442/21, -17/21)

Left sum ORIGINAL accumulated term17 = -153/7

source roles = [17, 21, 21, 8, -2, -1, 1/21]

derived = [-25/21, -17/1, -17/21, -153/7]

Left sum ORIGINAL sample18 index times width := mul(18, 1/21)

Left sum ORIGINAL sample18 index times width = 6/7

Left sum ORIGINAL sample18 lower plus offset := sum(-2, 6/7)

Left sum ORIGINAL sample18 lower plus offset = -8/7

Left sum ORIGINAL sample18 slope times x := mul(21, -8/7)

Left sum ORIGINAL sample18 slope times x = -24

Left sum ORIGINAL sample18 affine value plus c := sum(-24, 8)

Left sum ORIGINAL sample18 affine value plus c = -16

Left sum ORIGINAL sample18 value times cell width := mul(-16, 1/21)

Left sum ORIGINAL sample18 value times cell width = -16/21

Left sum ORIGINAL accumulated term18 := sum(-153/7, -16/21)

Left sum ORIGINAL accumulated term18 = -475/21

source roles = [18, 21, 21, 8, -2, -1, 1/21]

derived = [-8/7, -16/1, -16/21, -475/21]

Left sum ORIGINAL sample19 index times width := mul(19, 1/21)

Left sum ORIGINAL sample19 index times width = 19/21

Left sum ORIGINAL sample19 lower plus offset := sum(-2, 19/21)

Left sum ORIGINAL sample19 lower plus offset = -23/21

Left sum ORIGINAL sample19 slope times x := mul(21, -23/21)

Left sum ORIGINAL sample19 slope times x = -23

Left sum ORIGINAL sample19 affine value plus c := sum(-23, 8)

Left sum ORIGINAL sample19 affine value plus c = -15

Left sum ORIGINAL sample19 value times cell width := mul(-15, 1/21)

Left sum ORIGINAL sample19 value times cell width = -5/7

Left sum ORIGINAL accumulated term19 := sum(-475/21, -5/7)

Left sum ORIGINAL accumulated term19 = -70/3

source roles = [19, 21, 21, 8, -2, -1, 1/21]

derived = [-23/21, -15/1, -5/7, -70/3]

Left sum ORIGINAL sample20 index times width := mul(20, 1/21)

Left sum ORIGINAL sample20 index times width = 20/21

Left sum ORIGINAL sample20 lower plus offset := sum(-2, 20/21)

Left sum ORIGINAL sample20 lower plus offset = -22/21

Left sum ORIGINAL sample20 slope times x := mul(21, -22/21)

Left sum ORIGINAL sample20 slope times x = -22

Left sum ORIGINAL sample20 affine value plus c := sum(-22, 8)

Left sum ORIGINAL sample20 affine value plus c = -14

Left sum ORIGINAL sample20 value times cell width := mul(-14, 1/21)

Left sum ORIGINAL sample20 value times cell width = -2/3

Left sum ORIGINAL accumulated term20 := sum(-70/3, -2/3)

Left sum ORIGINAL accumulated term20 = -24

source roles = [20, 21, 21, 8, -2, -1, 1/21]

derived = [-22/21, -14/1, -2/3, -24/1]

source roles = [21, 21, 8, -2, -1]

derived = [1/21, -24/1]

Reverse checks

• S151 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S150 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S149 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term20 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 value times cell width with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 affine value plus c with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 slope times x with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 lower plus offset with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 index times width with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S142 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term19 with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 value times cell width with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 affine value plus c with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 slope times x with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 lower plus offset with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 index times width with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S135 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term18 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 value times cell width with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 affine value plus c with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 slope times x with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 lower plus offset with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 index times width with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S128 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term17 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 value times cell width with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 affine value plus c with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 slope times x with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 lower plus offset with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 index times width with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S121 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term16 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 value times cell width with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 affine value plus c with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 slope times x with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 lower plus offset with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 index times width with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S114 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term15 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 value times cell width with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 affine value plus c with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 slope times x with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 lower plus offset with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 index times width with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S107 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term14 with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 value times cell width with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 affine value plus c with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 slope times x with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 lower plus offset with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 index times width with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S100 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term13 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 value times cell width with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 affine value plus c with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 slope times x with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 lower plus offset with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 index times width with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S093 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term12 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 value times cell width with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 affine value plus c with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 slope times x with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 lower plus offset with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 index times width with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S086 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term11 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 value times cell width with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 affine value plus c with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 slope times x with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 lower plus offset with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 index times width with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S079 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term10 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 value times cell width with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 affine value plus c with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 slope times x with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 lower plus offset with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 index times width with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S072 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term9 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 value times cell width with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 affine value plus c with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 slope times x with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 lower plus offset with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 index times width with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S065 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term8 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 value times cell width with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 affine value plus c with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 slope times x with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 lower plus offset with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 index times width with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S058 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term7 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 value times cell width with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 affine value plus c with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 slope times x with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 lower plus offset with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 index times width with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S051 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term6 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 value times cell width with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 affine value plus c with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 slope times x with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 lower plus offset with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 index times width with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S044 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term5 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 value times cell width with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 affine value plus c with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 slope times x with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 lower plus offset with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 index times width with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S037 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term4 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 value times cell width with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 affine value plus c with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 slope times x with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 lower plus offset with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 index times width with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S030 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term3 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 value times cell width with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 affine value plus c with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 slope times x with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 lower plus offset with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 index times width with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S023 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term2 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 value times cell width with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 affine value plus c with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 slope times x with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 lower plus offset with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 index times width with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S016 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 value times cell width with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 affine value plus c with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 slope times x with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 lower plus offset with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 index times width with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S009 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 value times cell width with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 affine value plus c with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 slope times x with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 lower plus offset with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 index times width with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Left sum ORIGINAL width over positive cell count with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Left sum ORIGINAL upper minus lower endpoint with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

-24/1

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.