Question 32618: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(-18)x+(5). For a=-1≤b=2 and N=18 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)=(-18)x+(5). For a=-1≤b=2 and N=18 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(2, -1)

Left sum ORIGINAL upper minus lower endpoint = 3

Left sum ORIGINAL width over positive cell count := div(3, 18)

Left sum ORIGINAL width over positive cell count = 1/6

Left sum ORIGINAL sample0 index times width := mul(0, 1/6)

Left sum ORIGINAL sample0 index times width = 0

Left sum ORIGINAL sample0 lower plus offset := sum(-1, 0)

Left sum ORIGINAL sample0 lower plus offset = -1

Left sum ORIGINAL sample0 slope times x := mul(-18, -1)

Left sum ORIGINAL sample0 slope times x = 18

Left sum ORIGINAL sample0 affine value plus c := sum(18, 5)

Left sum ORIGINAL sample0 affine value plus c = 23

Left sum ORIGINAL sample0 value times cell width := mul(23, 1/6)

Left sum ORIGINAL sample0 value times cell width = 23/6

Left sum ORIGINAL accumulated term0 := sum(0, 23/6)

Left sum ORIGINAL accumulated term0 = 23/6

source roles = [0, 18, -18, 5, -1, 2, 1/6]

derived = [-1/1, 23/1, 23/6, 23/6]

Left sum ORIGINAL sample1 index times width := mul(1, 1/6)

Left sum ORIGINAL sample1 index times width = 1/6

Left sum ORIGINAL sample1 lower plus offset := sum(-1, 1/6)

Left sum ORIGINAL sample1 lower plus offset = -5/6

Left sum ORIGINAL sample1 slope times x := mul(-18, -5/6)

Left sum ORIGINAL sample1 slope times x = 15

Left sum ORIGINAL sample1 affine value plus c := sum(15, 5)

Left sum ORIGINAL sample1 affine value plus c = 20

Left sum ORIGINAL sample1 value times cell width := mul(20, 1/6)

Left sum ORIGINAL sample1 value times cell width = 10/3

Left sum ORIGINAL accumulated term1 := sum(23/6, 10/3)

Left sum ORIGINAL accumulated term1 = 43/6

source roles = [1, 18, -18, 5, -1, 2, 1/6]

derived = [-5/6, 20/1, 10/3, 43/6]

Left sum ORIGINAL sample2 index times width := mul(2, 1/6)

Left sum ORIGINAL sample2 index times width = 1/3

Left sum ORIGINAL sample2 lower plus offset := sum(-1, 1/3)

Left sum ORIGINAL sample2 lower plus offset = -2/3

Left sum ORIGINAL sample2 slope times x := mul(-18, -2/3)

Left sum ORIGINAL sample2 slope times x = 12

Left sum ORIGINAL sample2 affine value plus c := sum(12, 5)

Left sum ORIGINAL sample2 affine value plus c = 17

Left sum ORIGINAL sample2 value times cell width := mul(17, 1/6)

Left sum ORIGINAL sample2 value times cell width = 17/6

Left sum ORIGINAL accumulated term2 := sum(43/6, 17/6)

Left sum ORIGINAL accumulated term2 = 10

source roles = [2, 18, -18, 5, -1, 2, 1/6]

derived = [-2/3, 17/1, 17/6, 10/1]

Left sum ORIGINAL sample3 index times width := mul(3, 1/6)

Left sum ORIGINAL sample3 index times width = 1/2

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

Left sum ORIGINAL sample3 lower plus offset = -1/2

Left sum ORIGINAL sample3 slope times x := mul(-18, -1/2)

Left sum ORIGINAL sample3 slope times x = 9

Left sum ORIGINAL sample3 affine value plus c := sum(9, 5)

Left sum ORIGINAL sample3 affine value plus c = 14

Left sum ORIGINAL sample3 value times cell width := mul(14, 1/6)

Left sum ORIGINAL sample3 value times cell width = 7/3

Left sum ORIGINAL accumulated term3 := sum(10, 7/3)

Left sum ORIGINAL accumulated term3 = 37/3

source roles = [3, 18, -18, 5, -1, 2, 1/6]

derived = [-1/2, 14/1, 7/3, 37/3]

Left sum ORIGINAL sample4 index times width := mul(4, 1/6)

Left sum ORIGINAL sample4 index times width = 2/3

Left sum ORIGINAL sample4 lower plus offset := sum(-1, 2/3)

Left sum ORIGINAL sample4 lower plus offset = -1/3

Left sum ORIGINAL sample4 slope times x := mul(-18, -1/3)

Left sum ORIGINAL sample4 slope times x = 6

Left sum ORIGINAL sample4 affine value plus c := sum(6, 5)

Left sum ORIGINAL sample4 affine value plus c = 11

Left sum ORIGINAL sample4 value times cell width := mul(11, 1/6)

Left sum ORIGINAL sample4 value times cell width = 11/6

Left sum ORIGINAL accumulated term4 := sum(37/3, 11/6)

Left sum ORIGINAL accumulated term4 = 85/6

source roles = [4, 18, -18, 5, -1, 2, 1/6]

derived = [-1/3, 11/1, 11/6, 85/6]

Left sum ORIGINAL sample5 index times width := mul(5, 1/6)

Left sum ORIGINAL sample5 index times width = 5/6

Left sum ORIGINAL sample5 lower plus offset := sum(-1, 5/6)

Left sum ORIGINAL sample5 lower plus offset = -1/6

Left sum ORIGINAL sample5 slope times x := mul(-18, -1/6)

Left sum ORIGINAL sample5 slope times x = 3

Left sum ORIGINAL sample5 affine value plus c := sum(3, 5)

Left sum ORIGINAL sample5 affine value plus c = 8

Left sum ORIGINAL sample5 value times cell width := mul(8, 1/6)

Left sum ORIGINAL sample5 value times cell width = 4/3

Left sum ORIGINAL accumulated term5 := sum(85/6, 4/3)

Left sum ORIGINAL accumulated term5 = 31/2

source roles = [5, 18, -18, 5, -1, 2, 1/6]

derived = [-1/6, 8/1, 4/3, 31/2]

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

Left sum ORIGINAL sample6 index times width = 1

Left sum ORIGINAL sample6 lower plus offset := sum(-1, 1)

Left sum ORIGINAL sample6 lower plus offset = 0

Left sum ORIGINAL sample6 slope times x := mul(-18, 0)

Left sum ORIGINAL sample6 slope times x = 0

Left sum ORIGINAL sample6 affine value plus c := sum(0, 5)

Left sum ORIGINAL sample6 affine value plus c = 5

Left sum ORIGINAL sample6 value times cell width := mul(5, 1/6)

Left sum ORIGINAL sample6 value times cell width = 5/6

Left sum ORIGINAL accumulated term6 := sum(31/2, 5/6)

Left sum ORIGINAL accumulated term6 = 49/3

source roles = [6, 18, -18, 5, -1, 2, 1/6]

derived = [0/1, 5/1, 5/6, 49/3]

Left sum ORIGINAL sample7 index times width := mul(7, 1/6)

Left sum ORIGINAL sample7 index times width = 7/6

Left sum ORIGINAL sample7 lower plus offset := sum(-1, 7/6)

Left sum ORIGINAL sample7 lower plus offset = 1/6

Left sum ORIGINAL sample7 slope times x := mul(-18, 1/6)

Left sum ORIGINAL sample7 slope times x = -3

Left sum ORIGINAL sample7 affine value plus c := sum(-3, 5)

Left sum ORIGINAL sample7 affine value plus c = 2

Left sum ORIGINAL sample7 value times cell width := mul(2, 1/6)

Left sum ORIGINAL sample7 value times cell width = 1/3

Left sum ORIGINAL accumulated term7 := sum(49/3, 1/3)

Left sum ORIGINAL accumulated term7 = 50/3

source roles = [7, 18, -18, 5, -1, 2, 1/6]

derived = [1/6, 2/1, 1/3, 50/3]

Left sum ORIGINAL sample8 index times width := mul(8, 1/6)

Left sum ORIGINAL sample8 index times width = 4/3

Left sum ORIGINAL sample8 lower plus offset := sum(-1, 4/3)

Left sum ORIGINAL sample8 lower plus offset = 1/3

Left sum ORIGINAL sample8 slope times x := mul(-18, 1/3)

Left sum ORIGINAL sample8 slope times x = -6

Left sum ORIGINAL sample8 affine value plus c := sum(-6, 5)

Left sum ORIGINAL sample8 affine value plus c = -1

Left sum ORIGINAL sample8 value times cell width := mul(-1, 1/6)

Left sum ORIGINAL sample8 value times cell width = -1/6

Left sum ORIGINAL accumulated term8 := sum(50/3, -1/6)

Left sum ORIGINAL accumulated term8 = 33/2

source roles = [8, 18, -18, 5, -1, 2, 1/6]

derived = [1/3, -1/1, -1/6, 33/2]

Left sum ORIGINAL sample9 index times width := mul(9, 1/6)

Left sum ORIGINAL sample9 index times width = 3/2

Left sum ORIGINAL sample9 lower plus offset := sum(-1, 3/2)

Left sum ORIGINAL sample9 lower plus offset = 1/2

Left sum ORIGINAL sample9 slope times x := mul(-18, 1/2)

Left sum ORIGINAL sample9 slope times x = -9

Left sum ORIGINAL sample9 affine value plus c := sum(-9, 5)

Left sum ORIGINAL sample9 affine value plus c = -4

Left sum ORIGINAL sample9 value times cell width := mul(-4, 1/6)

Left sum ORIGINAL sample9 value times cell width = -2/3

Left sum ORIGINAL accumulated term9 := sum(33/2, -2/3)

Left sum ORIGINAL accumulated term9 = 95/6

source roles = [9, 18, -18, 5, -1, 2, 1/6]

derived = [1/2, -4/1, -2/3, 95/6]

Left sum ORIGINAL sample10 index times width := mul(10, 1/6)

Left sum ORIGINAL sample10 index times width = 5/3

Left sum ORIGINAL sample10 lower plus offset := sum(-1, 5/3)

Left sum ORIGINAL sample10 lower plus offset = 2/3

Left sum ORIGINAL sample10 slope times x := mul(-18, 2/3)

Left sum ORIGINAL sample10 slope times x = -12

Left sum ORIGINAL sample10 affine value plus c := sum(-12, 5)

Left sum ORIGINAL sample10 affine value plus c = -7

Left sum ORIGINAL sample10 value times cell width := mul(-7, 1/6)

Left sum ORIGINAL sample10 value times cell width = -7/6

Left sum ORIGINAL accumulated term10 := sum(95/6, -7/6)

Left sum ORIGINAL accumulated term10 = 44/3

source roles = [10, 18, -18, 5, -1, 2, 1/6]

derived = [2/3, -7/1, -7/6, 44/3]

Left sum ORIGINAL sample11 index times width := mul(11, 1/6)

Left sum ORIGINAL sample11 index times width = 11/6

Left sum ORIGINAL sample11 lower plus offset := sum(-1, 11/6)

Left sum ORIGINAL sample11 lower plus offset = 5/6

Left sum ORIGINAL sample11 slope times x := mul(-18, 5/6)

Left sum ORIGINAL sample11 slope times x = -15

Left sum ORIGINAL sample11 affine value plus c := sum(-15, 5)

Left sum ORIGINAL sample11 affine value plus c = -10

Left sum ORIGINAL sample11 value times cell width := mul(-10, 1/6)

Left sum ORIGINAL sample11 value times cell width = -5/3

Left sum ORIGINAL accumulated term11 := sum(44/3, -5/3)

Left sum ORIGINAL accumulated term11 = 13

source roles = [11, 18, -18, 5, -1, 2, 1/6]

derived = [5/6, -10/1, -5/3, 13/1]

Left sum ORIGINAL sample12 index times width := mul(12, 1/6)

Left sum ORIGINAL sample12 index times width = 2

Left sum ORIGINAL sample12 lower plus offset := sum(-1, 2)

Left sum ORIGINAL sample12 lower plus offset = 1

Left sum ORIGINAL sample12 slope times x := mul(-18, 1)

Left sum ORIGINAL sample12 slope times x = -18

Left sum ORIGINAL sample12 affine value plus c := sum(-18, 5)

Left sum ORIGINAL sample12 affine value plus c = -13

Left sum ORIGINAL sample12 value times cell width := mul(-13, 1/6)

Left sum ORIGINAL sample12 value times cell width = -13/6

Left sum ORIGINAL accumulated term12 := sum(13, -13/6)

Left sum ORIGINAL accumulated term12 = 65/6

source roles = [12, 18, -18, 5, -1, 2, 1/6]

derived = [1/1, -13/1, -13/6, 65/6]

Left sum ORIGINAL sample13 index times width := mul(13, 1/6)

Left sum ORIGINAL sample13 index times width = 13/6

Left sum ORIGINAL sample13 lower plus offset := sum(-1, 13/6)

Left sum ORIGINAL sample13 lower plus offset = 7/6

Left sum ORIGINAL sample13 slope times x := mul(-18, 7/6)

Left sum ORIGINAL sample13 slope times x = -21

Left sum ORIGINAL sample13 affine value plus c := sum(-21, 5)

Left sum ORIGINAL sample13 affine value plus c = -16

Left sum ORIGINAL sample13 value times cell width := mul(-16, 1/6)

Left sum ORIGINAL sample13 value times cell width = -8/3

Left sum ORIGINAL accumulated term13 := sum(65/6, -8/3)

Left sum ORIGINAL accumulated term13 = 49/6

source roles = [13, 18, -18, 5, -1, 2, 1/6]

derived = [7/6, -16/1, -8/3, 49/6]

Left sum ORIGINAL sample14 index times width := mul(14, 1/6)

Left sum ORIGINAL sample14 index times width = 7/3

Left sum ORIGINAL sample14 lower plus offset := sum(-1, 7/3)

Left sum ORIGINAL sample14 lower plus offset = 4/3

Left sum ORIGINAL sample14 slope times x := mul(-18, 4/3)

Left sum ORIGINAL sample14 slope times x = -24

Left sum ORIGINAL sample14 affine value plus c := sum(-24, 5)

Left sum ORIGINAL sample14 affine value plus c = -19

Left sum ORIGINAL sample14 value times cell width := mul(-19, 1/6)

Left sum ORIGINAL sample14 value times cell width = -19/6

Left sum ORIGINAL accumulated term14 := sum(49/6, -19/6)

Left sum ORIGINAL accumulated term14 = 5

source roles = [14, 18, -18, 5, -1, 2, 1/6]

derived = [4/3, -19/1, -19/6, 5/1]

Left sum ORIGINAL sample15 index times width := mul(15, 1/6)

Left sum ORIGINAL sample15 index times width = 5/2

Left sum ORIGINAL sample15 lower plus offset := sum(-1, 5/2)

Left sum ORIGINAL sample15 lower plus offset = 3/2

Left sum ORIGINAL sample15 slope times x := mul(-18, 3/2)

Left sum ORIGINAL sample15 slope times x = -27

Left sum ORIGINAL sample15 affine value plus c := sum(-27, 5)

Left sum ORIGINAL sample15 affine value plus c = -22

Left sum ORIGINAL sample15 value times cell width := mul(-22, 1/6)

Left sum ORIGINAL sample15 value times cell width = -11/3

Left sum ORIGINAL accumulated term15 := sum(5, -11/3)

Left sum ORIGINAL accumulated term15 = 4/3

source roles = [15, 18, -18, 5, -1, 2, 1/6]

derived = [3/2, -22/1, -11/3, 4/3]

Left sum ORIGINAL sample16 index times width := mul(16, 1/6)

Left sum ORIGINAL sample16 index times width = 8/3

Left sum ORIGINAL sample16 lower plus offset := sum(-1, 8/3)

Left sum ORIGINAL sample16 lower plus offset = 5/3

Left sum ORIGINAL sample16 slope times x := mul(-18, 5/3)

Left sum ORIGINAL sample16 slope times x = -30

Left sum ORIGINAL sample16 affine value plus c := sum(-30, 5)

Left sum ORIGINAL sample16 affine value plus c = -25

Left sum ORIGINAL sample16 value times cell width := mul(-25, 1/6)

Left sum ORIGINAL sample16 value times cell width = -25/6

Left sum ORIGINAL accumulated term16 := sum(4/3, -25/6)

Left sum ORIGINAL accumulated term16 = -17/6

source roles = [16, 18, -18, 5, -1, 2, 1/6]

derived = [5/3, -25/1, -25/6, -17/6]

Left sum ORIGINAL sample17 index times width := mul(17, 1/6)

Left sum ORIGINAL sample17 index times width = 17/6

Left sum ORIGINAL sample17 lower plus offset := sum(-1, 17/6)

Left sum ORIGINAL sample17 lower plus offset = 11/6

Left sum ORIGINAL sample17 slope times x := mul(-18, 11/6)

Left sum ORIGINAL sample17 slope times x = -33

Left sum ORIGINAL sample17 affine value plus c := sum(-33, 5)

Left sum ORIGINAL sample17 affine value plus c = -28

Left sum ORIGINAL sample17 value times cell width := mul(-28, 1/6)

Left sum ORIGINAL sample17 value times cell width = -14/3

Left sum ORIGINAL accumulated term17 := sum(-17/6, -14/3)

Left sum ORIGINAL accumulated term17 = -15/2

source roles = [17, 18, -18, 5, -1, 2, 1/6]

derived = [11/6, -28/1, -14/3, -15/2]

source roles = [18, -18, 5, -1, 2]

derived = [1/6, -15/2]

Reverse checks

• S130 · 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.
• 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

-15/2

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.