Question 32617: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(17)x+(4). For a=-2≤b=0 and N=17 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)=(17)x+(4). For a=-2≤b=0 and N=17 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(0, -2)

Left sum ORIGINAL upper minus lower endpoint = 2

Left sum ORIGINAL width over positive cell count := div(2, 17)

Left sum ORIGINAL width over positive cell count = 2/17

Left sum ORIGINAL sample0 index times width := mul(0, 2/17)

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(17, -2)

Left sum ORIGINAL sample0 slope times x = -34

Left sum ORIGINAL sample0 affine value plus c := sum(-34, 4)

Left sum ORIGINAL sample0 affine value plus c = -30

Left sum ORIGINAL sample0 value times cell width := mul(-30, 2/17)

Left sum ORIGINAL sample0 value times cell width = -60/17

Left sum ORIGINAL accumulated term0 := sum(0, -60/17)

Left sum ORIGINAL accumulated term0 = -60/17

source roles = [0, 17, 17, 4, -2, 0, 2/17]

derived = [-2/1, -30/1, -60/17, -60/17]

Left sum ORIGINAL sample1 index times width := mul(1, 2/17)

Left sum ORIGINAL sample1 index times width = 2/17

Left sum ORIGINAL sample1 lower plus offset := sum(-2, 2/17)

Left sum ORIGINAL sample1 lower plus offset = -32/17

Left sum ORIGINAL sample1 slope times x := mul(17, -32/17)

Left sum ORIGINAL sample1 slope times x = -32

Left sum ORIGINAL sample1 affine value plus c := sum(-32, 4)

Left sum ORIGINAL sample1 affine value plus c = -28

Left sum ORIGINAL sample1 value times cell width := mul(-28, 2/17)

Left sum ORIGINAL sample1 value times cell width = -56/17

Left sum ORIGINAL accumulated term1 := sum(-60/17, -56/17)

Left sum ORIGINAL accumulated term1 = -116/17

source roles = [1, 17, 17, 4, -2, 0, 2/17]

derived = [-32/17, -28/1, -56/17, -116/17]

Left sum ORIGINAL sample2 index times width := mul(2, 2/17)

Left sum ORIGINAL sample2 index times width = 4/17

Left sum ORIGINAL sample2 lower plus offset := sum(-2, 4/17)

Left sum ORIGINAL sample2 lower plus offset = -30/17

Left sum ORIGINAL sample2 slope times x := mul(17, -30/17)

Left sum ORIGINAL sample2 slope times x = -30

Left sum ORIGINAL sample2 affine value plus c := sum(-30, 4)

Left sum ORIGINAL sample2 affine value plus c = -26

Left sum ORIGINAL sample2 value times cell width := mul(-26, 2/17)

Left sum ORIGINAL sample2 value times cell width = -52/17

Left sum ORIGINAL accumulated term2 := sum(-116/17, -52/17)

Left sum ORIGINAL accumulated term2 = -168/17

source roles = [2, 17, 17, 4, -2, 0, 2/17]

derived = [-30/17, -26/1, -52/17, -168/17]

Left sum ORIGINAL sample3 index times width := mul(3, 2/17)

Left sum ORIGINAL sample3 index times width = 6/17

Left sum ORIGINAL sample3 lower plus offset := sum(-2, 6/17)

Left sum ORIGINAL sample3 lower plus offset = -28/17

Left sum ORIGINAL sample3 slope times x := mul(17, -28/17)

Left sum ORIGINAL sample3 slope times x = -28

Left sum ORIGINAL sample3 affine value plus c := sum(-28, 4)

Left sum ORIGINAL sample3 affine value plus c = -24

Left sum ORIGINAL sample3 value times cell width := mul(-24, 2/17)

Left sum ORIGINAL sample3 value times cell width = -48/17

Left sum ORIGINAL accumulated term3 := sum(-168/17, -48/17)

Left sum ORIGINAL accumulated term3 = -216/17

source roles = [3, 17, 17, 4, -2, 0, 2/17]

derived = [-28/17, -24/1, -48/17, -216/17]

Left sum ORIGINAL sample4 index times width := mul(4, 2/17)

Left sum ORIGINAL sample4 index times width = 8/17

Left sum ORIGINAL sample4 lower plus offset := sum(-2, 8/17)

Left sum ORIGINAL sample4 lower plus offset = -26/17

Left sum ORIGINAL sample4 slope times x := mul(17, -26/17)

Left sum ORIGINAL sample4 slope times x = -26

Left sum ORIGINAL sample4 affine value plus c := sum(-26, 4)

Left sum ORIGINAL sample4 affine value plus c = -22

Left sum ORIGINAL sample4 value times cell width := mul(-22, 2/17)

Left sum ORIGINAL sample4 value times cell width = -44/17

Left sum ORIGINAL accumulated term4 := sum(-216/17, -44/17)

Left sum ORIGINAL accumulated term4 = -260/17

source roles = [4, 17, 17, 4, -2, 0, 2/17]

derived = [-26/17, -22/1, -44/17, -260/17]

Left sum ORIGINAL sample5 index times width := mul(5, 2/17)

Left sum ORIGINAL sample5 index times width = 10/17

Left sum ORIGINAL sample5 lower plus offset := sum(-2, 10/17)

Left sum ORIGINAL sample5 lower plus offset = -24/17

Left sum ORIGINAL sample5 slope times x := mul(17, -24/17)

Left sum ORIGINAL sample5 slope times x = -24

Left sum ORIGINAL sample5 affine value plus c := sum(-24, 4)

Left sum ORIGINAL sample5 affine value plus c = -20

Left sum ORIGINAL sample5 value times cell width := mul(-20, 2/17)

Left sum ORIGINAL sample5 value times cell width = -40/17

Left sum ORIGINAL accumulated term5 := sum(-260/17, -40/17)

Left sum ORIGINAL accumulated term5 = -300/17

source roles = [5, 17, 17, 4, -2, 0, 2/17]

derived = [-24/17, -20/1, -40/17, -300/17]

Left sum ORIGINAL sample6 index times width := mul(6, 2/17)

Left sum ORIGINAL sample6 index times width = 12/17

Left sum ORIGINAL sample6 lower plus offset := sum(-2, 12/17)

Left sum ORIGINAL sample6 lower plus offset = -22/17

Left sum ORIGINAL sample6 slope times x := mul(17, -22/17)

Left sum ORIGINAL sample6 slope times x = -22

Left sum ORIGINAL sample6 affine value plus c := sum(-22, 4)

Left sum ORIGINAL sample6 affine value plus c = -18

Left sum ORIGINAL sample6 value times cell width := mul(-18, 2/17)

Left sum ORIGINAL sample6 value times cell width = -36/17

Left sum ORIGINAL accumulated term6 := sum(-300/17, -36/17)

Left sum ORIGINAL accumulated term6 = -336/17

source roles = [6, 17, 17, 4, -2, 0, 2/17]

derived = [-22/17, -18/1, -36/17, -336/17]

Left sum ORIGINAL sample7 index times width := mul(7, 2/17)

Left sum ORIGINAL sample7 index times width = 14/17

Left sum ORIGINAL sample7 lower plus offset := sum(-2, 14/17)

Left sum ORIGINAL sample7 lower plus offset = -20/17

Left sum ORIGINAL sample7 slope times x := mul(17, -20/17)

Left sum ORIGINAL sample7 slope times x = -20

Left sum ORIGINAL sample7 affine value plus c := sum(-20, 4)

Left sum ORIGINAL sample7 affine value plus c = -16

Left sum ORIGINAL sample7 value times cell width := mul(-16, 2/17)

Left sum ORIGINAL sample7 value times cell width = -32/17

Left sum ORIGINAL accumulated term7 := sum(-336/17, -32/17)

Left sum ORIGINAL accumulated term7 = -368/17

source roles = [7, 17, 17, 4, -2, 0, 2/17]

derived = [-20/17, -16/1, -32/17, -368/17]

Left sum ORIGINAL sample8 index times width := mul(8, 2/17)

Left sum ORIGINAL sample8 index times width = 16/17

Left sum ORIGINAL sample8 lower plus offset := sum(-2, 16/17)

Left sum ORIGINAL sample8 lower plus offset = -18/17

Left sum ORIGINAL sample8 slope times x := mul(17, -18/17)

Left sum ORIGINAL sample8 slope times x = -18

Left sum ORIGINAL sample8 affine value plus c := sum(-18, 4)

Left sum ORIGINAL sample8 affine value plus c = -14

Left sum ORIGINAL sample8 value times cell width := mul(-14, 2/17)

Left sum ORIGINAL sample8 value times cell width = -28/17

Left sum ORIGINAL accumulated term8 := sum(-368/17, -28/17)

Left sum ORIGINAL accumulated term8 = -396/17

source roles = [8, 17, 17, 4, -2, 0, 2/17]

derived = [-18/17, -14/1, -28/17, -396/17]

Left sum ORIGINAL sample9 index times width := mul(9, 2/17)

Left sum ORIGINAL sample9 index times width = 18/17

Left sum ORIGINAL sample9 lower plus offset := sum(-2, 18/17)

Left sum ORIGINAL sample9 lower plus offset = -16/17

Left sum ORIGINAL sample9 slope times x := mul(17, -16/17)

Left sum ORIGINAL sample9 slope times x = -16

Left sum ORIGINAL sample9 affine value plus c := sum(-16, 4)

Left sum ORIGINAL sample9 affine value plus c = -12

Left sum ORIGINAL sample9 value times cell width := mul(-12, 2/17)

Left sum ORIGINAL sample9 value times cell width = -24/17

Left sum ORIGINAL accumulated term9 := sum(-396/17, -24/17)

Left sum ORIGINAL accumulated term9 = -420/17

source roles = [9, 17, 17, 4, -2, 0, 2/17]

derived = [-16/17, -12/1, -24/17, -420/17]

Left sum ORIGINAL sample10 index times width := mul(10, 2/17)

Left sum ORIGINAL sample10 index times width = 20/17

Left sum ORIGINAL sample10 lower plus offset := sum(-2, 20/17)

Left sum ORIGINAL sample10 lower plus offset = -14/17

Left sum ORIGINAL sample10 slope times x := mul(17, -14/17)

Left sum ORIGINAL sample10 slope times x = -14

Left sum ORIGINAL sample10 affine value plus c := sum(-14, 4)

Left sum ORIGINAL sample10 affine value plus c = -10

Left sum ORIGINAL sample10 value times cell width := mul(-10, 2/17)

Left sum ORIGINAL sample10 value times cell width = -20/17

Left sum ORIGINAL accumulated term10 := sum(-420/17, -20/17)

Left sum ORIGINAL accumulated term10 = -440/17

source roles = [10, 17, 17, 4, -2, 0, 2/17]

derived = [-14/17, -10/1, -20/17, -440/17]

Left sum ORIGINAL sample11 index times width := mul(11, 2/17)

Left sum ORIGINAL sample11 index times width = 22/17

Left sum ORIGINAL sample11 lower plus offset := sum(-2, 22/17)

Left sum ORIGINAL sample11 lower plus offset = -12/17

Left sum ORIGINAL sample11 slope times x := mul(17, -12/17)

Left sum ORIGINAL sample11 slope times x = -12

Left sum ORIGINAL sample11 affine value plus c := sum(-12, 4)

Left sum ORIGINAL sample11 affine value plus c = -8

Left sum ORIGINAL sample11 value times cell width := mul(-8, 2/17)

Left sum ORIGINAL sample11 value times cell width = -16/17

Left sum ORIGINAL accumulated term11 := sum(-440/17, -16/17)

Left sum ORIGINAL accumulated term11 = -456/17

source roles = [11, 17, 17, 4, -2, 0, 2/17]

derived = [-12/17, -8/1, -16/17, -456/17]

Left sum ORIGINAL sample12 index times width := mul(12, 2/17)

Left sum ORIGINAL sample12 index times width = 24/17

Left sum ORIGINAL sample12 lower plus offset := sum(-2, 24/17)

Left sum ORIGINAL sample12 lower plus offset = -10/17

Left sum ORIGINAL sample12 slope times x := mul(17, -10/17)

Left sum ORIGINAL sample12 slope times x = -10

Left sum ORIGINAL sample12 affine value plus c := sum(-10, 4)

Left sum ORIGINAL sample12 affine value plus c = -6

Left sum ORIGINAL sample12 value times cell width := mul(-6, 2/17)

Left sum ORIGINAL sample12 value times cell width = -12/17

Left sum ORIGINAL accumulated term12 := sum(-456/17, -12/17)

Left sum ORIGINAL accumulated term12 = -468/17

source roles = [12, 17, 17, 4, -2, 0, 2/17]

derived = [-10/17, -6/1, -12/17, -468/17]

Left sum ORIGINAL sample13 index times width := mul(13, 2/17)

Left sum ORIGINAL sample13 index times width = 26/17

Left sum ORIGINAL sample13 lower plus offset := sum(-2, 26/17)

Left sum ORIGINAL sample13 lower plus offset = -8/17

Left sum ORIGINAL sample13 slope times x := mul(17, -8/17)

Left sum ORIGINAL sample13 slope times x = -8

Left sum ORIGINAL sample13 affine value plus c := sum(-8, 4)

Left sum ORIGINAL sample13 affine value plus c = -4

Left sum ORIGINAL sample13 value times cell width := mul(-4, 2/17)

Left sum ORIGINAL sample13 value times cell width = -8/17

Left sum ORIGINAL accumulated term13 := sum(-468/17, -8/17)

Left sum ORIGINAL accumulated term13 = -28

source roles = [13, 17, 17, 4, -2, 0, 2/17]

derived = [-8/17, -4/1, -8/17, -28/1]

Left sum ORIGINAL sample14 index times width := mul(14, 2/17)

Left sum ORIGINAL sample14 index times width = 28/17

Left sum ORIGINAL sample14 lower plus offset := sum(-2, 28/17)

Left sum ORIGINAL sample14 lower plus offset = -6/17

Left sum ORIGINAL sample14 slope times x := mul(17, -6/17)

Left sum ORIGINAL sample14 slope times x = -6

Left sum ORIGINAL sample14 affine value plus c := sum(-6, 4)

Left sum ORIGINAL sample14 affine value plus c = -2

Left sum ORIGINAL sample14 value times cell width := mul(-2, 2/17)

Left sum ORIGINAL sample14 value times cell width = -4/17

Left sum ORIGINAL accumulated term14 := sum(-28, -4/17)

Left sum ORIGINAL accumulated term14 = -480/17

source roles = [14, 17, 17, 4, -2, 0, 2/17]

derived = [-6/17, -2/1, -4/17, -480/17]

Left sum ORIGINAL sample15 index times width := mul(15, 2/17)

Left sum ORIGINAL sample15 index times width = 30/17

Left sum ORIGINAL sample15 lower plus offset := sum(-2, 30/17)

Left sum ORIGINAL sample15 lower plus offset = -4/17

Left sum ORIGINAL sample15 slope times x := mul(17, -4/17)

Left sum ORIGINAL sample15 slope times x = -4

Left sum ORIGINAL sample15 affine value plus c := sum(-4, 4)

Left sum ORIGINAL sample15 affine value plus c = 0

Left sum ORIGINAL sample15 value times cell width := mul(0, 2/17)

Left sum ORIGINAL sample15 value times cell width = 0

Left sum ORIGINAL accumulated term15 := sum(-480/17, 0)

Left sum ORIGINAL accumulated term15 = -480/17

source roles = [15, 17, 17, 4, -2, 0, 2/17]

derived = [-4/17, 0/1, 0/1, -480/17]

Left sum ORIGINAL sample16 index times width := mul(16, 2/17)

Left sum ORIGINAL sample16 index times width = 32/17

Left sum ORIGINAL sample16 lower plus offset := sum(-2, 32/17)

Left sum ORIGINAL sample16 lower plus offset = -2/17

Left sum ORIGINAL sample16 slope times x := mul(17, -2/17)

Left sum ORIGINAL sample16 slope times x = -2

Left sum ORIGINAL sample16 affine value plus c := sum(-2, 4)

Left sum ORIGINAL sample16 affine value plus c = 2

Left sum ORIGINAL sample16 value times cell width := mul(2, 2/17)

Left sum ORIGINAL sample16 value times cell width = 4/17

Left sum ORIGINAL accumulated term16 := sum(-480/17, 4/17)

Left sum ORIGINAL accumulated term16 = -28

source roles = [16, 17, 17, 4, -2, 0, 2/17]

derived = [-2/17, 2/1, 4/17, -28/1]

source roles = [17, 17, 4, -2, 0]

derived = [2/17, -28/1]

Reverse checks

• S123 · 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.
• 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

-28/1

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.