Question 32619: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(19)x+(6). For a=0≤b=4 and N=19 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)=(19)x+(6). For a=0≤b=4 and N=19 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(4, 0)

Left sum ORIGINAL upper minus lower endpoint = 4

Left sum ORIGINAL width over positive cell count := div(4, 19)

Left sum ORIGINAL width over positive cell count = 4/19

Left sum ORIGINAL sample0 index times width := mul(0, 4/19)

Left sum ORIGINAL sample0 index times width = 0

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

Left sum ORIGINAL sample0 lower plus offset = 0

Left sum ORIGINAL sample0 slope times x := mul(19, 0)

Left sum ORIGINAL sample0 slope times x = 0

Left sum ORIGINAL sample0 affine value plus c := sum(0, 6)

Left sum ORIGINAL sample0 affine value plus c = 6

Left sum ORIGINAL sample0 value times cell width := mul(6, 4/19)

Left sum ORIGINAL sample0 value times cell width = 24/19

Left sum ORIGINAL accumulated term0 := sum(0, 24/19)

Left sum ORIGINAL accumulated term0 = 24/19

source roles = [0, 19, 19, 6, 0, 4, 4/19]

derived = [0/1, 6/1, 24/19, 24/19]

Left sum ORIGINAL sample1 index times width := mul(1, 4/19)

Left sum ORIGINAL sample1 index times width = 4/19

Left sum ORIGINAL sample1 lower plus offset := sum(0, 4/19)

Left sum ORIGINAL sample1 lower plus offset = 4/19

Left sum ORIGINAL sample1 slope times x := mul(19, 4/19)

Left sum ORIGINAL sample1 slope times x = 4

Left sum ORIGINAL sample1 affine value plus c := sum(4, 6)

Left sum ORIGINAL sample1 affine value plus c = 10

Left sum ORIGINAL sample1 value times cell width := mul(10, 4/19)

Left sum ORIGINAL sample1 value times cell width = 40/19

Left sum ORIGINAL accumulated term1 := sum(24/19, 40/19)

Left sum ORIGINAL accumulated term1 = 64/19

source roles = [1, 19, 19, 6, 0, 4, 4/19]

derived = [4/19, 10/1, 40/19, 64/19]

Left sum ORIGINAL sample2 index times width := mul(2, 4/19)

Left sum ORIGINAL sample2 index times width = 8/19

Left sum ORIGINAL sample2 lower plus offset := sum(0, 8/19)

Left sum ORIGINAL sample2 lower plus offset = 8/19

Left sum ORIGINAL sample2 slope times x := mul(19, 8/19)

Left sum ORIGINAL sample2 slope times x = 8

Left sum ORIGINAL sample2 affine value plus c := sum(8, 6)

Left sum ORIGINAL sample2 affine value plus c = 14

Left sum ORIGINAL sample2 value times cell width := mul(14, 4/19)

Left sum ORIGINAL sample2 value times cell width = 56/19

Left sum ORIGINAL accumulated term2 := sum(64/19, 56/19)

Left sum ORIGINAL accumulated term2 = 120/19

source roles = [2, 19, 19, 6, 0, 4, 4/19]

derived = [8/19, 14/1, 56/19, 120/19]

Left sum ORIGINAL sample3 index times width := mul(3, 4/19)

Left sum ORIGINAL sample3 index times width = 12/19

Left sum ORIGINAL sample3 lower plus offset := sum(0, 12/19)

Left sum ORIGINAL sample3 lower plus offset = 12/19

Left sum ORIGINAL sample3 slope times x := mul(19, 12/19)

Left sum ORIGINAL sample3 slope times x = 12

Left sum ORIGINAL sample3 affine value plus c := sum(12, 6)

Left sum ORIGINAL sample3 affine value plus c = 18

Left sum ORIGINAL sample3 value times cell width := mul(18, 4/19)

Left sum ORIGINAL sample3 value times cell width = 72/19

Left sum ORIGINAL accumulated term3 := sum(120/19, 72/19)

Left sum ORIGINAL accumulated term3 = 192/19

source roles = [3, 19, 19, 6, 0, 4, 4/19]

derived = [12/19, 18/1, 72/19, 192/19]

Left sum ORIGINAL sample4 index times width := mul(4, 4/19)

Left sum ORIGINAL sample4 index times width = 16/19

Left sum ORIGINAL sample4 lower plus offset := sum(0, 16/19)

Left sum ORIGINAL sample4 lower plus offset = 16/19

Left sum ORIGINAL sample4 slope times x := mul(19, 16/19)

Left sum ORIGINAL sample4 slope times x = 16

Left sum ORIGINAL sample4 affine value plus c := sum(16, 6)

Left sum ORIGINAL sample4 affine value plus c = 22

Left sum ORIGINAL sample4 value times cell width := mul(22, 4/19)

Left sum ORIGINAL sample4 value times cell width = 88/19

Left sum ORIGINAL accumulated term4 := sum(192/19, 88/19)

Left sum ORIGINAL accumulated term4 = 280/19

source roles = [4, 19, 19, 6, 0, 4, 4/19]

derived = [16/19, 22/1, 88/19, 280/19]

Left sum ORIGINAL sample5 index times width := mul(5, 4/19)

Left sum ORIGINAL sample5 index times width = 20/19

Left sum ORIGINAL sample5 lower plus offset := sum(0, 20/19)

Left sum ORIGINAL sample5 lower plus offset = 20/19

Left sum ORIGINAL sample5 slope times x := mul(19, 20/19)

Left sum ORIGINAL sample5 slope times x = 20

Left sum ORIGINAL sample5 affine value plus c := sum(20, 6)

Left sum ORIGINAL sample5 affine value plus c = 26

Left sum ORIGINAL sample5 value times cell width := mul(26, 4/19)

Left sum ORIGINAL sample5 value times cell width = 104/19

Left sum ORIGINAL accumulated term5 := sum(280/19, 104/19)

Left sum ORIGINAL accumulated term5 = 384/19

source roles = [5, 19, 19, 6, 0, 4, 4/19]

derived = [20/19, 26/1, 104/19, 384/19]

Left sum ORIGINAL sample6 index times width := mul(6, 4/19)

Left sum ORIGINAL sample6 index times width = 24/19

Left sum ORIGINAL sample6 lower plus offset := sum(0, 24/19)

Left sum ORIGINAL sample6 lower plus offset = 24/19

Left sum ORIGINAL sample6 slope times x := mul(19, 24/19)

Left sum ORIGINAL sample6 slope times x = 24

Left sum ORIGINAL sample6 affine value plus c := sum(24, 6)

Left sum ORIGINAL sample6 affine value plus c = 30

Left sum ORIGINAL sample6 value times cell width := mul(30, 4/19)

Left sum ORIGINAL sample6 value times cell width = 120/19

Left sum ORIGINAL accumulated term6 := sum(384/19, 120/19)

Left sum ORIGINAL accumulated term6 = 504/19

source roles = [6, 19, 19, 6, 0, 4, 4/19]

derived = [24/19, 30/1, 120/19, 504/19]

Left sum ORIGINAL sample7 index times width := mul(7, 4/19)

Left sum ORIGINAL sample7 index times width = 28/19

Left sum ORIGINAL sample7 lower plus offset := sum(0, 28/19)

Left sum ORIGINAL sample7 lower plus offset = 28/19

Left sum ORIGINAL sample7 slope times x := mul(19, 28/19)

Left sum ORIGINAL sample7 slope times x = 28

Left sum ORIGINAL sample7 affine value plus c := sum(28, 6)

Left sum ORIGINAL sample7 affine value plus c = 34

Left sum ORIGINAL sample7 value times cell width := mul(34, 4/19)

Left sum ORIGINAL sample7 value times cell width = 136/19

Left sum ORIGINAL accumulated term7 := sum(504/19, 136/19)

Left sum ORIGINAL accumulated term7 = 640/19

source roles = [7, 19, 19, 6, 0, 4, 4/19]

derived = [28/19, 34/1, 136/19, 640/19]

Left sum ORIGINAL sample8 index times width := mul(8, 4/19)

Left sum ORIGINAL sample8 index times width = 32/19

Left sum ORIGINAL sample8 lower plus offset := sum(0, 32/19)

Left sum ORIGINAL sample8 lower plus offset = 32/19

Left sum ORIGINAL sample8 slope times x := mul(19, 32/19)

Left sum ORIGINAL sample8 slope times x = 32

Left sum ORIGINAL sample8 affine value plus c := sum(32, 6)

Left sum ORIGINAL sample8 affine value plus c = 38

Left sum ORIGINAL sample8 value times cell width := mul(38, 4/19)

Left sum ORIGINAL sample8 value times cell width = 8

Left sum ORIGINAL accumulated term8 := sum(640/19, 8)

Left sum ORIGINAL accumulated term8 = 792/19

source roles = [8, 19, 19, 6, 0, 4, 4/19]

derived = [32/19, 38/1, 8/1, 792/19]

Left sum ORIGINAL sample9 index times width := mul(9, 4/19)

Left sum ORIGINAL sample9 index times width = 36/19

Left sum ORIGINAL sample9 lower plus offset := sum(0, 36/19)

Left sum ORIGINAL sample9 lower plus offset = 36/19

Left sum ORIGINAL sample9 slope times x := mul(19, 36/19)

Left sum ORIGINAL sample9 slope times x = 36

Left sum ORIGINAL sample9 affine value plus c := sum(36, 6)

Left sum ORIGINAL sample9 affine value plus c = 42

Left sum ORIGINAL sample9 value times cell width := mul(42, 4/19)

Left sum ORIGINAL sample9 value times cell width = 168/19

Left sum ORIGINAL accumulated term9 := sum(792/19, 168/19)

Left sum ORIGINAL accumulated term9 = 960/19

source roles = [9, 19, 19, 6, 0, 4, 4/19]

derived = [36/19, 42/1, 168/19, 960/19]

Left sum ORIGINAL sample10 index times width := mul(10, 4/19)

Left sum ORIGINAL sample10 index times width = 40/19

Left sum ORIGINAL sample10 lower plus offset := sum(0, 40/19)

Left sum ORIGINAL sample10 lower plus offset = 40/19

Left sum ORIGINAL sample10 slope times x := mul(19, 40/19)

Left sum ORIGINAL sample10 slope times x = 40

Left sum ORIGINAL sample10 affine value plus c := sum(40, 6)

Left sum ORIGINAL sample10 affine value plus c = 46

Left sum ORIGINAL sample10 value times cell width := mul(46, 4/19)

Left sum ORIGINAL sample10 value times cell width = 184/19

Left sum ORIGINAL accumulated term10 := sum(960/19, 184/19)

Left sum ORIGINAL accumulated term10 = 1144/19

source roles = [10, 19, 19, 6, 0, 4, 4/19]

derived = [40/19, 46/1, 184/19, 1144/19]

Left sum ORIGINAL sample11 index times width := mul(11, 4/19)

Left sum ORIGINAL sample11 index times width = 44/19

Left sum ORIGINAL sample11 lower plus offset := sum(0, 44/19)

Left sum ORIGINAL sample11 lower plus offset = 44/19

Left sum ORIGINAL sample11 slope times x := mul(19, 44/19)

Left sum ORIGINAL sample11 slope times x = 44

Left sum ORIGINAL sample11 affine value plus c := sum(44, 6)

Left sum ORIGINAL sample11 affine value plus c = 50

Left sum ORIGINAL sample11 value times cell width := mul(50, 4/19)

Left sum ORIGINAL sample11 value times cell width = 200/19

Left sum ORIGINAL accumulated term11 := sum(1144/19, 200/19)

Left sum ORIGINAL accumulated term11 = 1344/19

source roles = [11, 19, 19, 6, 0, 4, 4/19]

derived = [44/19, 50/1, 200/19, 1344/19]

Left sum ORIGINAL sample12 index times width := mul(12, 4/19)

Left sum ORIGINAL sample12 index times width = 48/19

Left sum ORIGINAL sample12 lower plus offset := sum(0, 48/19)

Left sum ORIGINAL sample12 lower plus offset = 48/19

Left sum ORIGINAL sample12 slope times x := mul(19, 48/19)

Left sum ORIGINAL sample12 slope times x = 48

Left sum ORIGINAL sample12 affine value plus c := sum(48, 6)

Left sum ORIGINAL sample12 affine value plus c = 54

Left sum ORIGINAL sample12 value times cell width := mul(54, 4/19)

Left sum ORIGINAL sample12 value times cell width = 216/19

Left sum ORIGINAL accumulated term12 := sum(1344/19, 216/19)

Left sum ORIGINAL accumulated term12 = 1560/19

source roles = [12, 19, 19, 6, 0, 4, 4/19]

derived = [48/19, 54/1, 216/19, 1560/19]

Left sum ORIGINAL sample13 index times width := mul(13, 4/19)

Left sum ORIGINAL sample13 index times width = 52/19

Left sum ORIGINAL sample13 lower plus offset := sum(0, 52/19)

Left sum ORIGINAL sample13 lower plus offset = 52/19

Left sum ORIGINAL sample13 slope times x := mul(19, 52/19)

Left sum ORIGINAL sample13 slope times x = 52

Left sum ORIGINAL sample13 affine value plus c := sum(52, 6)

Left sum ORIGINAL sample13 affine value plus c = 58

Left sum ORIGINAL sample13 value times cell width := mul(58, 4/19)

Left sum ORIGINAL sample13 value times cell width = 232/19

Left sum ORIGINAL accumulated term13 := sum(1560/19, 232/19)

Left sum ORIGINAL accumulated term13 = 1792/19

source roles = [13, 19, 19, 6, 0, 4, 4/19]

derived = [52/19, 58/1, 232/19, 1792/19]

Left sum ORIGINAL sample14 index times width := mul(14, 4/19)

Left sum ORIGINAL sample14 index times width = 56/19

Left sum ORIGINAL sample14 lower plus offset := sum(0, 56/19)

Left sum ORIGINAL sample14 lower plus offset = 56/19

Left sum ORIGINAL sample14 slope times x := mul(19, 56/19)

Left sum ORIGINAL sample14 slope times x = 56

Left sum ORIGINAL sample14 affine value plus c := sum(56, 6)

Left sum ORIGINAL sample14 affine value plus c = 62

Left sum ORIGINAL sample14 value times cell width := mul(62, 4/19)

Left sum ORIGINAL sample14 value times cell width = 248/19

Left sum ORIGINAL accumulated term14 := sum(1792/19, 248/19)

Left sum ORIGINAL accumulated term14 = 2040/19

source roles = [14, 19, 19, 6, 0, 4, 4/19]

derived = [56/19, 62/1, 248/19, 2040/19]

Left sum ORIGINAL sample15 index times width := mul(15, 4/19)

Left sum ORIGINAL sample15 index times width = 60/19

Left sum ORIGINAL sample15 lower plus offset := sum(0, 60/19)

Left sum ORIGINAL sample15 lower plus offset = 60/19

Left sum ORIGINAL sample15 slope times x := mul(19, 60/19)

Left sum ORIGINAL sample15 slope times x = 60

Left sum ORIGINAL sample15 affine value plus c := sum(60, 6)

Left sum ORIGINAL sample15 affine value plus c = 66

Left sum ORIGINAL sample15 value times cell width := mul(66, 4/19)

Left sum ORIGINAL sample15 value times cell width = 264/19

Left sum ORIGINAL accumulated term15 := sum(2040/19, 264/19)

Left sum ORIGINAL accumulated term15 = 2304/19

source roles = [15, 19, 19, 6, 0, 4, 4/19]

derived = [60/19, 66/1, 264/19, 2304/19]

Left sum ORIGINAL sample16 index times width := mul(16, 4/19)

Left sum ORIGINAL sample16 index times width = 64/19

Left sum ORIGINAL sample16 lower plus offset := sum(0, 64/19)

Left sum ORIGINAL sample16 lower plus offset = 64/19

Left sum ORIGINAL sample16 slope times x := mul(19, 64/19)

Left sum ORIGINAL sample16 slope times x = 64

Left sum ORIGINAL sample16 affine value plus c := sum(64, 6)

Left sum ORIGINAL sample16 affine value plus c = 70

Left sum ORIGINAL sample16 value times cell width := mul(70, 4/19)

Left sum ORIGINAL sample16 value times cell width = 280/19

Left sum ORIGINAL accumulated term16 := sum(2304/19, 280/19)

Left sum ORIGINAL accumulated term16 = 136

source roles = [16, 19, 19, 6, 0, 4, 4/19]

derived = [64/19, 70/1, 280/19, 136/1]

Left sum ORIGINAL sample17 index times width := mul(17, 4/19)

Left sum ORIGINAL sample17 index times width = 68/19

Left sum ORIGINAL sample17 lower plus offset := sum(0, 68/19)

Left sum ORIGINAL sample17 lower plus offset = 68/19

Left sum ORIGINAL sample17 slope times x := mul(19, 68/19)

Left sum ORIGINAL sample17 slope times x = 68

Left sum ORIGINAL sample17 affine value plus c := sum(68, 6)

Left sum ORIGINAL sample17 affine value plus c = 74

Left sum ORIGINAL sample17 value times cell width := mul(74, 4/19)

Left sum ORIGINAL sample17 value times cell width = 296/19

Left sum ORIGINAL accumulated term17 := sum(136, 296/19)

Left sum ORIGINAL accumulated term17 = 2880/19

source roles = [17, 19, 19, 6, 0, 4, 4/19]

derived = [68/19, 74/1, 296/19, 2880/19]

Left sum ORIGINAL sample18 index times width := mul(18, 4/19)

Left sum ORIGINAL sample18 index times width = 72/19

Left sum ORIGINAL sample18 lower plus offset := sum(0, 72/19)

Left sum ORIGINAL sample18 lower plus offset = 72/19

Left sum ORIGINAL sample18 slope times x := mul(19, 72/19)

Left sum ORIGINAL sample18 slope times x = 72

Left sum ORIGINAL sample18 affine value plus c := sum(72, 6)

Left sum ORIGINAL sample18 affine value plus c = 78

Left sum ORIGINAL sample18 value times cell width := mul(78, 4/19)

Left sum ORIGINAL sample18 value times cell width = 312/19

Left sum ORIGINAL accumulated term18 := sum(2880/19, 312/19)

Left sum ORIGINAL accumulated term18 = 168

source roles = [18, 19, 19, 6, 0, 4, 4/19]

derived = [72/19, 78/1, 312/19, 168/1]

source roles = [19, 19, 6, 0, 4]

derived = [4/19, 168/1]

Reverse checks

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

Result

168/1

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.