Question 32616: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(-16)x+(3). For a=1≤b=2 and N=16 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)=(-16)x+(3). For a=1≤b=2 and N=16 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 = 1

Left sum ORIGINAL width over positive cell count := div(1, 16)

Left sum ORIGINAL width over positive cell count = 1/16

Left sum ORIGINAL sample0 index times width := mul(0, 1/16)

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(-16, 1)

Left sum ORIGINAL sample0 slope times x = -16

Left sum ORIGINAL sample0 affine value plus c := sum(-16, 3)

Left sum ORIGINAL sample0 affine value plus c = -13

Left sum ORIGINAL sample0 value times cell width := mul(-13, 1/16)

Left sum ORIGINAL sample0 value times cell width = -13/16

Left sum ORIGINAL accumulated term0 := sum(0, -13/16)

Left sum ORIGINAL accumulated term0 = -13/16

source roles = [0, 16, -16, 3, 1, 2, 1/16]

derived = [1/1, -13/1, -13/16, -13/16]

Left sum ORIGINAL sample1 index times width := mul(1, 1/16)

Left sum ORIGINAL sample1 index times width = 1/16

Left sum ORIGINAL sample1 lower plus offset := sum(1, 1/16)

Left sum ORIGINAL sample1 lower plus offset = 17/16

Left sum ORIGINAL sample1 slope times x := mul(-16, 17/16)

Left sum ORIGINAL sample1 slope times x = -17

Left sum ORIGINAL sample1 affine value plus c := sum(-17, 3)

Left sum ORIGINAL sample1 affine value plus c = -14

Left sum ORIGINAL sample1 value times cell width := mul(-14, 1/16)

Left sum ORIGINAL sample1 value times cell width = -7/8

Left sum ORIGINAL accumulated term1 := sum(-13/16, -7/8)

Left sum ORIGINAL accumulated term1 = -27/16

source roles = [1, 16, -16, 3, 1, 2, 1/16]

derived = [17/16, -14/1, -7/8, -27/16]

Left sum ORIGINAL sample2 index times width := mul(2, 1/16)

Left sum ORIGINAL sample2 index times width = 1/8

Left sum ORIGINAL sample2 lower plus offset := sum(1, 1/8)

Left sum ORIGINAL sample2 lower plus offset = 9/8

Left sum ORIGINAL sample2 slope times x := mul(-16, 9/8)

Left sum ORIGINAL sample2 slope times x = -18

Left sum ORIGINAL sample2 affine value plus c := sum(-18, 3)

Left sum ORIGINAL sample2 affine value plus c = -15

Left sum ORIGINAL sample2 value times cell width := mul(-15, 1/16)

Left sum ORIGINAL sample2 value times cell width = -15/16

Left sum ORIGINAL accumulated term2 := sum(-27/16, -15/16)

Left sum ORIGINAL accumulated term2 = -21/8

source roles = [2, 16, -16, 3, 1, 2, 1/16]

derived = [9/8, -15/1, -15/16, -21/8]

Left sum ORIGINAL sample3 index times width := mul(3, 1/16)

Left sum ORIGINAL sample3 index times width = 3/16

Left sum ORIGINAL sample3 lower plus offset := sum(1, 3/16)

Left sum ORIGINAL sample3 lower plus offset = 19/16

Left sum ORIGINAL sample3 slope times x := mul(-16, 19/16)

Left sum ORIGINAL sample3 slope times x = -19

Left sum ORIGINAL sample3 affine value plus c := sum(-19, 3)

Left sum ORIGINAL sample3 affine value plus c = -16

Left sum ORIGINAL sample3 value times cell width := mul(-16, 1/16)

Left sum ORIGINAL sample3 value times cell width = -1

Left sum ORIGINAL accumulated term3 := sum(-21/8, -1)

Left sum ORIGINAL accumulated term3 = -29/8

source roles = [3, 16, -16, 3, 1, 2, 1/16]

derived = [19/16, -16/1, -1/1, -29/8]

Left sum ORIGINAL sample4 index times width := mul(4, 1/16)

Left sum ORIGINAL sample4 index times width = 1/4

Left sum ORIGINAL sample4 lower plus offset := sum(1, 1/4)

Left sum ORIGINAL sample4 lower plus offset = 5/4

Left sum ORIGINAL sample4 slope times x := mul(-16, 5/4)

Left sum ORIGINAL sample4 slope times x = -20

Left sum ORIGINAL sample4 affine value plus c := sum(-20, 3)

Left sum ORIGINAL sample4 affine value plus c = -17

Left sum ORIGINAL sample4 value times cell width := mul(-17, 1/16)

Left sum ORIGINAL sample4 value times cell width = -17/16

Left sum ORIGINAL accumulated term4 := sum(-29/8, -17/16)

Left sum ORIGINAL accumulated term4 = -75/16

source roles = [4, 16, -16, 3, 1, 2, 1/16]

derived = [5/4, -17/1, -17/16, -75/16]

Left sum ORIGINAL sample5 index times width := mul(5, 1/16)

Left sum ORIGINAL sample5 index times width = 5/16

Left sum ORIGINAL sample5 lower plus offset := sum(1, 5/16)

Left sum ORIGINAL sample5 lower plus offset = 21/16

Left sum ORIGINAL sample5 slope times x := mul(-16, 21/16)

Left sum ORIGINAL sample5 slope times x = -21

Left sum ORIGINAL sample5 affine value plus c := sum(-21, 3)

Left sum ORIGINAL sample5 affine value plus c = -18

Left sum ORIGINAL sample5 value times cell width := mul(-18, 1/16)

Left sum ORIGINAL sample5 value times cell width = -9/8

Left sum ORIGINAL accumulated term5 := sum(-75/16, -9/8)

Left sum ORIGINAL accumulated term5 = -93/16

source roles = [5, 16, -16, 3, 1, 2, 1/16]

derived = [21/16, -18/1, -9/8, -93/16]

Left sum ORIGINAL sample6 index times width := mul(6, 1/16)

Left sum ORIGINAL sample6 index times width = 3/8

Left sum ORIGINAL sample6 lower plus offset := sum(1, 3/8)

Left sum ORIGINAL sample6 lower plus offset = 11/8

Left sum ORIGINAL sample6 slope times x := mul(-16, 11/8)

Left sum ORIGINAL sample6 slope times x = -22

Left sum ORIGINAL sample6 affine value plus c := sum(-22, 3)

Left sum ORIGINAL sample6 affine value plus c = -19

Left sum ORIGINAL sample6 value times cell width := mul(-19, 1/16)

Left sum ORIGINAL sample6 value times cell width = -19/16

Left sum ORIGINAL accumulated term6 := sum(-93/16, -19/16)

Left sum ORIGINAL accumulated term6 = -7

source roles = [6, 16, -16, 3, 1, 2, 1/16]

derived = [11/8, -19/1, -19/16, -7/1]

Left sum ORIGINAL sample7 index times width := mul(7, 1/16)

Left sum ORIGINAL sample7 index times width = 7/16

Left sum ORIGINAL sample7 lower plus offset := sum(1, 7/16)

Left sum ORIGINAL sample7 lower plus offset = 23/16

Left sum ORIGINAL sample7 slope times x := mul(-16, 23/16)

Left sum ORIGINAL sample7 slope times x = -23

Left sum ORIGINAL sample7 affine value plus c := sum(-23, 3)

Left sum ORIGINAL sample7 affine value plus c = -20

Left sum ORIGINAL sample7 value times cell width := mul(-20, 1/16)

Left sum ORIGINAL sample7 value times cell width = -5/4

Left sum ORIGINAL accumulated term7 := sum(-7, -5/4)

Left sum ORIGINAL accumulated term7 = -33/4

source roles = [7, 16, -16, 3, 1, 2, 1/16]

derived = [23/16, -20/1, -5/4, -33/4]

Left sum ORIGINAL sample8 index times width := mul(8, 1/16)

Left sum ORIGINAL sample8 index times width = 1/2

Left sum ORIGINAL sample8 lower plus offset := sum(1, 1/2)

Left sum ORIGINAL sample8 lower plus offset = 3/2

Left sum ORIGINAL sample8 slope times x := mul(-16, 3/2)

Left sum ORIGINAL sample8 slope times x = -24

Left sum ORIGINAL sample8 affine value plus c := sum(-24, 3)

Left sum ORIGINAL sample8 affine value plus c = -21

Left sum ORIGINAL sample8 value times cell width := mul(-21, 1/16)

Left sum ORIGINAL sample8 value times cell width = -21/16

Left sum ORIGINAL accumulated term8 := sum(-33/4, -21/16)

Left sum ORIGINAL accumulated term8 = -153/16

source roles = [8, 16, -16, 3, 1, 2, 1/16]

derived = [3/2, -21/1, -21/16, -153/16]

Left sum ORIGINAL sample9 index times width := mul(9, 1/16)

Left sum ORIGINAL sample9 index times width = 9/16

Left sum ORIGINAL sample9 lower plus offset := sum(1, 9/16)

Left sum ORIGINAL sample9 lower plus offset = 25/16

Left sum ORIGINAL sample9 slope times x := mul(-16, 25/16)

Left sum ORIGINAL sample9 slope times x = -25

Left sum ORIGINAL sample9 affine value plus c := sum(-25, 3)

Left sum ORIGINAL sample9 affine value plus c = -22

Left sum ORIGINAL sample9 value times cell width := mul(-22, 1/16)

Left sum ORIGINAL sample9 value times cell width = -11/8

Left sum ORIGINAL accumulated term9 := sum(-153/16, -11/8)

Left sum ORIGINAL accumulated term9 = -175/16

source roles = [9, 16, -16, 3, 1, 2, 1/16]

derived = [25/16, -22/1, -11/8, -175/16]

Left sum ORIGINAL sample10 index times width := mul(10, 1/16)

Left sum ORIGINAL sample10 index times width = 5/8

Left sum ORIGINAL sample10 lower plus offset := sum(1, 5/8)

Left sum ORIGINAL sample10 lower plus offset = 13/8

Left sum ORIGINAL sample10 slope times x := mul(-16, 13/8)

Left sum ORIGINAL sample10 slope times x = -26

Left sum ORIGINAL sample10 affine value plus c := sum(-26, 3)

Left sum ORIGINAL sample10 affine value plus c = -23

Left sum ORIGINAL sample10 value times cell width := mul(-23, 1/16)

Left sum ORIGINAL sample10 value times cell width = -23/16

Left sum ORIGINAL accumulated term10 := sum(-175/16, -23/16)

Left sum ORIGINAL accumulated term10 = -99/8

source roles = [10, 16, -16, 3, 1, 2, 1/16]

derived = [13/8, -23/1, -23/16, -99/8]

Left sum ORIGINAL sample11 index times width := mul(11, 1/16)

Left sum ORIGINAL sample11 index times width = 11/16

Left sum ORIGINAL sample11 lower plus offset := sum(1, 11/16)

Left sum ORIGINAL sample11 lower plus offset = 27/16

Left sum ORIGINAL sample11 slope times x := mul(-16, 27/16)

Left sum ORIGINAL sample11 slope times x = -27

Left sum ORIGINAL sample11 affine value plus c := sum(-27, 3)

Left sum ORIGINAL sample11 affine value plus c = -24

Left sum ORIGINAL sample11 value times cell width := mul(-24, 1/16)

Left sum ORIGINAL sample11 value times cell width = -3/2

Left sum ORIGINAL accumulated term11 := sum(-99/8, -3/2)

Left sum ORIGINAL accumulated term11 = -111/8

source roles = [11, 16, -16, 3, 1, 2, 1/16]

derived = [27/16, -24/1, -3/2, -111/8]

Left sum ORIGINAL sample12 index times width := mul(12, 1/16)

Left sum ORIGINAL sample12 index times width = 3/4

Left sum ORIGINAL sample12 lower plus offset := sum(1, 3/4)

Left sum ORIGINAL sample12 lower plus offset = 7/4

Left sum ORIGINAL sample12 slope times x := mul(-16, 7/4)

Left sum ORIGINAL sample12 slope times x = -28

Left sum ORIGINAL sample12 affine value plus c := sum(-28, 3)

Left sum ORIGINAL sample12 affine value plus c = -25

Left sum ORIGINAL sample12 value times cell width := mul(-25, 1/16)

Left sum ORIGINAL sample12 value times cell width = -25/16

Left sum ORIGINAL accumulated term12 := sum(-111/8, -25/16)

Left sum ORIGINAL accumulated term12 = -247/16

source roles = [12, 16, -16, 3, 1, 2, 1/16]

derived = [7/4, -25/1, -25/16, -247/16]

Left sum ORIGINAL sample13 index times width := mul(13, 1/16)

Left sum ORIGINAL sample13 index times width = 13/16

Left sum ORIGINAL sample13 lower plus offset := sum(1, 13/16)

Left sum ORIGINAL sample13 lower plus offset = 29/16

Left sum ORIGINAL sample13 slope times x := mul(-16, 29/16)

Left sum ORIGINAL sample13 slope times x = -29

Left sum ORIGINAL sample13 affine value plus c := sum(-29, 3)

Left sum ORIGINAL sample13 affine value plus c = -26

Left sum ORIGINAL sample13 value times cell width := mul(-26, 1/16)

Left sum ORIGINAL sample13 value times cell width = -13/8

Left sum ORIGINAL accumulated term13 := sum(-247/16, -13/8)

Left sum ORIGINAL accumulated term13 = -273/16

source roles = [13, 16, -16, 3, 1, 2, 1/16]

derived = [29/16, -26/1, -13/8, -273/16]

Left sum ORIGINAL sample14 index times width := mul(14, 1/16)

Left sum ORIGINAL sample14 index times width = 7/8

Left sum ORIGINAL sample14 lower plus offset := sum(1, 7/8)

Left sum ORIGINAL sample14 lower plus offset = 15/8

Left sum ORIGINAL sample14 slope times x := mul(-16, 15/8)

Left sum ORIGINAL sample14 slope times x = -30

Left sum ORIGINAL sample14 affine value plus c := sum(-30, 3)

Left sum ORIGINAL sample14 affine value plus c = -27

Left sum ORIGINAL sample14 value times cell width := mul(-27, 1/16)

Left sum ORIGINAL sample14 value times cell width = -27/16

Left sum ORIGINAL accumulated term14 := sum(-273/16, -27/16)

Left sum ORIGINAL accumulated term14 = -75/4

source roles = [14, 16, -16, 3, 1, 2, 1/16]

derived = [15/8, -27/1, -27/16, -75/4]

Left sum ORIGINAL sample15 index times width := mul(15, 1/16)

Left sum ORIGINAL sample15 index times width = 15/16

Left sum ORIGINAL sample15 lower plus offset := sum(1, 15/16)

Left sum ORIGINAL sample15 lower plus offset = 31/16

Left sum ORIGINAL sample15 slope times x := mul(-16, 31/16)

Left sum ORIGINAL sample15 slope times x = -31

Left sum ORIGINAL sample15 affine value plus c := sum(-31, 3)

Left sum ORIGINAL sample15 affine value plus c = -28

Left sum ORIGINAL sample15 value times cell width := mul(-28, 1/16)

Left sum ORIGINAL sample15 value times cell width = -7/4

Left sum ORIGINAL accumulated term15 := sum(-75/4, -7/4)

Left sum ORIGINAL accumulated term15 = -41/2

source roles = [15, 16, -16, 3, 1, 2, 1/16]

derived = [31/16, -28/1, -7/4, -41/2]

source roles = [16, -16, 3, 1, 2]

derived = [1/16, -41/2]

Reverse checks

• S116 · 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.
• 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

-41/2

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.