第 32624 題：仿射式的左端有限和

原題與已知條件

全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-24x+(11)，區間 [1,5] 分為 N=24 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。

• G001 — 全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-24x+(11)，區間 [1,5] 分為 N=24 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-definite-integral

逐步推導

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(5, 1)

Left sum ORIGINAL upper minus lower endpoint = 4

Left sum ORIGINAL width over positive cell count := div(4, 24)

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

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

Left sum ORIGINAL sample0 index times width = 0

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

Left sum ORIGINAL sample0 lower plus offset = 1

Left sum ORIGINAL sample0 slope times x := mul(-24, 1)

Left sum ORIGINAL sample0 slope times x = -24

Left sum ORIGINAL sample0 affine value plus c := sum(-24, 11)

Left sum ORIGINAL sample0 affine value plus c = -13

Left sum ORIGINAL sample0 value times cell width := mul(-13, 1/6)

Left sum ORIGINAL sample0 value times cell width = -13/6

Left sum ORIGINAL accumulated term0 := sum(0, -13/6)

Left sum ORIGINAL accumulated term0 = -13/6

source roles = [0, 24, -24, 11, 1, 5, 1/6]

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

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

Left sum ORIGINAL sample1 index times width = 1/6

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

Left sum ORIGINAL sample1 lower plus offset = 7/6

Left sum ORIGINAL sample1 slope times x := mul(-24, 7/6)

Left sum ORIGINAL sample1 slope times x = -28

Left sum ORIGINAL sample1 affine value plus c := sum(-28, 11)

Left sum ORIGINAL sample1 affine value plus c = -17

Left sum ORIGINAL sample1 value times cell width := mul(-17, 1/6)

Left sum ORIGINAL sample1 value times cell width = -17/6

Left sum ORIGINAL accumulated term1 := sum(-13/6, -17/6)

Left sum ORIGINAL accumulated term1 = -5

source roles = [1, 24, -24, 11, 1, 5, 1/6]

derived = [7/6, -17/1, -17/6, -5/1]

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

Left sum ORIGINAL sample2 index times width = 1/3

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

Left sum ORIGINAL sample2 lower plus offset = 4/3

Left sum ORIGINAL sample2 slope times x := mul(-24, 4/3)

Left sum ORIGINAL sample2 slope times x = -32

Left sum ORIGINAL sample2 affine value plus c := sum(-32, 11)

Left sum ORIGINAL sample2 affine value plus c = -21

Left sum ORIGINAL sample2 value times cell width := mul(-21, 1/6)

Left sum ORIGINAL sample2 value times cell width = -7/2

Left sum ORIGINAL accumulated term2 := sum(-5, -7/2)

Left sum ORIGINAL accumulated term2 = -17/2

source roles = [2, 24, -24, 11, 1, 5, 1/6]

derived = [4/3, -21/1, -7/2, -17/2]

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

Left sum ORIGINAL sample3 index times width = 1/2

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

Left sum ORIGINAL sample3 lower plus offset = 3/2

Left sum ORIGINAL sample3 slope times x := mul(-24, 3/2)

Left sum ORIGINAL sample3 slope times x = -36

Left sum ORIGINAL sample3 affine value plus c := sum(-36, 11)

Left sum ORIGINAL sample3 affine value plus c = -25

Left sum ORIGINAL sample3 value times cell width := mul(-25, 1/6)

Left sum ORIGINAL sample3 value times cell width = -25/6

Left sum ORIGINAL accumulated term3 := sum(-17/2, -25/6)

Left sum ORIGINAL accumulated term3 = -38/3

source roles = [3, 24, -24, 11, 1, 5, 1/6]

derived = [3/2, -25/1, -25/6, -38/3]

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

Left sum ORIGINAL sample4 index times width = 2/3

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

Left sum ORIGINAL sample4 lower plus offset = 5/3

Left sum ORIGINAL sample4 slope times x := mul(-24, 5/3)

Left sum ORIGINAL sample4 slope times x = -40

Left sum ORIGINAL sample4 affine value plus c := sum(-40, 11)

Left sum ORIGINAL sample4 affine value plus c = -29

Left sum ORIGINAL sample4 value times cell width := mul(-29, 1/6)

Left sum ORIGINAL sample4 value times cell width = -29/6

Left sum ORIGINAL accumulated term4 := sum(-38/3, -29/6)

Left sum ORIGINAL accumulated term4 = -35/2

source roles = [4, 24, -24, 11, 1, 5, 1/6]

derived = [5/3, -29/1, -29/6, -35/2]

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

Left sum ORIGINAL sample5 index times width = 5/6

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

Left sum ORIGINAL sample5 lower plus offset = 11/6

Left sum ORIGINAL sample5 slope times x := mul(-24, 11/6)

Left sum ORIGINAL sample5 slope times x = -44

Left sum ORIGINAL sample5 affine value plus c := sum(-44, 11)

Left sum ORIGINAL sample5 affine value plus c = -33

Left sum ORIGINAL sample5 value times cell width := mul(-33, 1/6)

Left sum ORIGINAL sample5 value times cell width = -11/2

Left sum ORIGINAL accumulated term5 := sum(-35/2, -11/2)

Left sum ORIGINAL accumulated term5 = -23

source roles = [5, 24, -24, 11, 1, 5, 1/6]

derived = [11/6, -33/1, -11/2, -23/1]

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

Left sum ORIGINAL sample6 index times width = 1

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

Left sum ORIGINAL sample6 lower plus offset = 2

Left sum ORIGINAL sample6 slope times x := mul(-24, 2)

Left sum ORIGINAL sample6 slope times x = -48

Left sum ORIGINAL sample6 affine value plus c := sum(-48, 11)

Left sum ORIGINAL sample6 affine value plus c = -37

Left sum ORIGINAL sample6 value times cell width := mul(-37, 1/6)

Left sum ORIGINAL sample6 value times cell width = -37/6

Left sum ORIGINAL accumulated term6 := sum(-23, -37/6)

Left sum ORIGINAL accumulated term6 = -175/6

source roles = [6, 24, -24, 11, 1, 5, 1/6]

derived = [2/1, -37/1, -37/6, -175/6]

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

Left sum ORIGINAL sample7 index times width = 7/6

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

Left sum ORIGINAL sample7 lower plus offset = 13/6

Left sum ORIGINAL sample7 slope times x := mul(-24, 13/6)

Left sum ORIGINAL sample7 slope times x = -52

Left sum ORIGINAL sample7 affine value plus c := sum(-52, 11)

Left sum ORIGINAL sample7 affine value plus c = -41

Left sum ORIGINAL sample7 value times cell width := mul(-41, 1/6)

Left sum ORIGINAL sample7 value times cell width = -41/6

Left sum ORIGINAL accumulated term7 := sum(-175/6, -41/6)

Left sum ORIGINAL accumulated term7 = -36

source roles = [7, 24, -24, 11, 1, 5, 1/6]

derived = [13/6, -41/1, -41/6, -36/1]

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

Left sum ORIGINAL sample8 index times width = 4/3

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

Left sum ORIGINAL sample8 lower plus offset = 7/3

Left sum ORIGINAL sample8 slope times x := mul(-24, 7/3)

Left sum ORIGINAL sample8 slope times x = -56

Left sum ORIGINAL sample8 affine value plus c := sum(-56, 11)

Left sum ORIGINAL sample8 affine value plus c = -45

Left sum ORIGINAL sample8 value times cell width := mul(-45, 1/6)

Left sum ORIGINAL sample8 value times cell width = -15/2

Left sum ORIGINAL accumulated term8 := sum(-36, -15/2)

Left sum ORIGINAL accumulated term8 = -87/2

source roles = [8, 24, -24, 11, 1, 5, 1/6]

derived = [7/3, -45/1, -15/2, -87/2]

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

Left sum ORIGINAL sample9 index times width = 3/2

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

Left sum ORIGINAL sample9 lower plus offset = 5/2

Left sum ORIGINAL sample9 slope times x := mul(-24, 5/2)

Left sum ORIGINAL sample9 slope times x = -60

Left sum ORIGINAL sample9 affine value plus c := sum(-60, 11)

Left sum ORIGINAL sample9 affine value plus c = -49

Left sum ORIGINAL sample9 value times cell width := mul(-49, 1/6)

Left sum ORIGINAL sample9 value times cell width = -49/6

Left sum ORIGINAL accumulated term9 := sum(-87/2, -49/6)

Left sum ORIGINAL accumulated term9 = -155/3

source roles = [9, 24, -24, 11, 1, 5, 1/6]

derived = [5/2, -49/1, -49/6, -155/3]

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

Left sum ORIGINAL sample10 index times width = 5/3

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

Left sum ORIGINAL sample10 lower plus offset = 8/3

Left sum ORIGINAL sample10 slope times x := mul(-24, 8/3)

Left sum ORIGINAL sample10 slope times x = -64

Left sum ORIGINAL sample10 affine value plus c := sum(-64, 11)

Left sum ORIGINAL sample10 affine value plus c = -53

Left sum ORIGINAL sample10 value times cell width := mul(-53, 1/6)

Left sum ORIGINAL sample10 value times cell width = -53/6

Left sum ORIGINAL accumulated term10 := sum(-155/3, -53/6)

Left sum ORIGINAL accumulated term10 = -121/2

source roles = [10, 24, -24, 11, 1, 5, 1/6]

derived = [8/3, -53/1, -53/6, -121/2]

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

Left sum ORIGINAL sample11 index times width = 11/6

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

Left sum ORIGINAL sample11 lower plus offset = 17/6

Left sum ORIGINAL sample11 slope times x := mul(-24, 17/6)

Left sum ORIGINAL sample11 slope times x = -68

Left sum ORIGINAL sample11 affine value plus c := sum(-68, 11)

Left sum ORIGINAL sample11 affine value plus c = -57

Left sum ORIGINAL sample11 value times cell width := mul(-57, 1/6)

Left sum ORIGINAL sample11 value times cell width = -19/2

Left sum ORIGINAL accumulated term11 := sum(-121/2, -19/2)

Left sum ORIGINAL accumulated term11 = -70

source roles = [11, 24, -24, 11, 1, 5, 1/6]

derived = [17/6, -57/1, -19/2, -70/1]

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

Left sum ORIGINAL sample12 index times width = 2

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

Left sum ORIGINAL sample12 lower plus offset = 3

Left sum ORIGINAL sample12 slope times x := mul(-24, 3)

Left sum ORIGINAL sample12 slope times x = -72

Left sum ORIGINAL sample12 affine value plus c := sum(-72, 11)

Left sum ORIGINAL sample12 affine value plus c = -61

Left sum ORIGINAL sample12 value times cell width := mul(-61, 1/6)

Left sum ORIGINAL sample12 value times cell width = -61/6

Left sum ORIGINAL accumulated term12 := sum(-70, -61/6)

Left sum ORIGINAL accumulated term12 = -481/6

source roles = [12, 24, -24, 11, 1, 5, 1/6]

derived = [3/1, -61/1, -61/6, -481/6]

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

Left sum ORIGINAL sample13 index times width = 13/6

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

Left sum ORIGINAL sample13 lower plus offset = 19/6

Left sum ORIGINAL sample13 slope times x := mul(-24, 19/6)

Left sum ORIGINAL sample13 slope times x = -76

Left sum ORIGINAL sample13 affine value plus c := sum(-76, 11)

Left sum ORIGINAL sample13 affine value plus c = -65

Left sum ORIGINAL sample13 value times cell width := mul(-65, 1/6)

Left sum ORIGINAL sample13 value times cell width = -65/6

Left sum ORIGINAL accumulated term13 := sum(-481/6, -65/6)

Left sum ORIGINAL accumulated term13 = -91

source roles = [13, 24, -24, 11, 1, 5, 1/6]

derived = [19/6, -65/1, -65/6, -91/1]

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

Left sum ORIGINAL sample14 index times width = 7/3

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

Left sum ORIGINAL sample14 lower plus offset = 10/3

Left sum ORIGINAL sample14 slope times x := mul(-24, 10/3)

Left sum ORIGINAL sample14 slope times x = -80

Left sum ORIGINAL sample14 affine value plus c := sum(-80, 11)

Left sum ORIGINAL sample14 affine value plus c = -69

Left sum ORIGINAL sample14 value times cell width := mul(-69, 1/6)

Left sum ORIGINAL sample14 value times cell width = -23/2

Left sum ORIGINAL accumulated term14 := sum(-91, -23/2)

Left sum ORIGINAL accumulated term14 = -205/2

source roles = [14, 24, -24, 11, 1, 5, 1/6]

derived = [10/3, -69/1, -23/2, -205/2]

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

Left sum ORIGINAL sample15 index times width = 5/2

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

Left sum ORIGINAL sample15 lower plus offset = 7/2

Left sum ORIGINAL sample15 slope times x := mul(-24, 7/2)

Left sum ORIGINAL sample15 slope times x = -84

Left sum ORIGINAL sample15 affine value plus c := sum(-84, 11)

Left sum ORIGINAL sample15 affine value plus c = -73

Left sum ORIGINAL sample15 value times cell width := mul(-73, 1/6)

Left sum ORIGINAL sample15 value times cell width = -73/6

Left sum ORIGINAL accumulated term15 := sum(-205/2, -73/6)

Left sum ORIGINAL accumulated term15 = -344/3

source roles = [15, 24, -24, 11, 1, 5, 1/6]

derived = [7/2, -73/1, -73/6, -344/3]

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

Left sum ORIGINAL sample16 index times width = 8/3

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

Left sum ORIGINAL sample16 lower plus offset = 11/3

Left sum ORIGINAL sample16 slope times x := mul(-24, 11/3)

Left sum ORIGINAL sample16 slope times x = -88

Left sum ORIGINAL sample16 affine value plus c := sum(-88, 11)

Left sum ORIGINAL sample16 affine value plus c = -77

Left sum ORIGINAL sample16 value times cell width := mul(-77, 1/6)

Left sum ORIGINAL sample16 value times cell width = -77/6

Left sum ORIGINAL accumulated term16 := sum(-344/3, -77/6)

Left sum ORIGINAL accumulated term16 = -255/2

source roles = [16, 24, -24, 11, 1, 5, 1/6]

derived = [11/3, -77/1, -77/6, -255/2]

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

Left sum ORIGINAL sample17 index times width = 17/6

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

Left sum ORIGINAL sample17 lower plus offset = 23/6

Left sum ORIGINAL sample17 slope times x := mul(-24, 23/6)

Left sum ORIGINAL sample17 slope times x = -92

Left sum ORIGINAL sample17 affine value plus c := sum(-92, 11)

Left sum ORIGINAL sample17 affine value plus c = -81

Left sum ORIGINAL sample17 value times cell width := mul(-81, 1/6)

Left sum ORIGINAL sample17 value times cell width = -27/2

Left sum ORIGINAL accumulated term17 := sum(-255/2, -27/2)

Left sum ORIGINAL accumulated term17 = -141

source roles = [17, 24, -24, 11, 1, 5, 1/6]

derived = [23/6, -81/1, -27/2, -141/1]

Left sum ORIGINAL sample18 index times width := mul(18, 1/6)

Left sum ORIGINAL sample18 index times width = 3

Left sum ORIGINAL sample18 lower plus offset := sum(1, 3)

Left sum ORIGINAL sample18 lower plus offset = 4

Left sum ORIGINAL sample18 slope times x := mul(-24, 4)

Left sum ORIGINAL sample18 slope times x = -96

Left sum ORIGINAL sample18 affine value plus c := sum(-96, 11)

Left sum ORIGINAL sample18 affine value plus c = -85

Left sum ORIGINAL sample18 value times cell width := mul(-85, 1/6)

Left sum ORIGINAL sample18 value times cell width = -85/6

Left sum ORIGINAL accumulated term18 := sum(-141, -85/6)

Left sum ORIGINAL accumulated term18 = -931/6

source roles = [18, 24, -24, 11, 1, 5, 1/6]

derived = [4/1, -85/1, -85/6, -931/6]

Left sum ORIGINAL sample19 index times width := mul(19, 1/6)

Left sum ORIGINAL sample19 index times width = 19/6

Left sum ORIGINAL sample19 lower plus offset := sum(1, 19/6)

Left sum ORIGINAL sample19 lower plus offset = 25/6

Left sum ORIGINAL sample19 slope times x := mul(-24, 25/6)

Left sum ORIGINAL sample19 slope times x = -100

Left sum ORIGINAL sample19 affine value plus c := sum(-100, 11)

Left sum ORIGINAL sample19 affine value plus c = -89

Left sum ORIGINAL sample19 value times cell width := mul(-89, 1/6)

Left sum ORIGINAL sample19 value times cell width = -89/6

Left sum ORIGINAL accumulated term19 := sum(-931/6, -89/6)

Left sum ORIGINAL accumulated term19 = -170

source roles = [19, 24, -24, 11, 1, 5, 1/6]

derived = [25/6, -89/1, -89/6, -170/1]

Left sum ORIGINAL sample20 index times width := mul(20, 1/6)

Left sum ORIGINAL sample20 index times width = 10/3

Left sum ORIGINAL sample20 lower plus offset := sum(1, 10/3)

Left sum ORIGINAL sample20 lower plus offset = 13/3

Left sum ORIGINAL sample20 slope times x := mul(-24, 13/3)

Left sum ORIGINAL sample20 slope times x = -104

Left sum ORIGINAL sample20 affine value plus c := sum(-104, 11)

Left sum ORIGINAL sample20 affine value plus c = -93

Left sum ORIGINAL sample20 value times cell width := mul(-93, 1/6)

Left sum ORIGINAL sample20 value times cell width = -31/2

Left sum ORIGINAL accumulated term20 := sum(-170, -31/2)

Left sum ORIGINAL accumulated term20 = -371/2

source roles = [20, 24, -24, 11, 1, 5, 1/6]

derived = [13/3, -93/1, -31/2, -371/2]

Left sum ORIGINAL sample21 index times width := mul(21, 1/6)

Left sum ORIGINAL sample21 index times width = 7/2

Left sum ORIGINAL sample21 lower plus offset := sum(1, 7/2)

Left sum ORIGINAL sample21 lower plus offset = 9/2

Left sum ORIGINAL sample21 slope times x := mul(-24, 9/2)

Left sum ORIGINAL sample21 slope times x = -108

Left sum ORIGINAL sample21 affine value plus c := sum(-108, 11)

Left sum ORIGINAL sample21 affine value plus c = -97

Left sum ORIGINAL sample21 value times cell width := mul(-97, 1/6)

Left sum ORIGINAL sample21 value times cell width = -97/6

Left sum ORIGINAL accumulated term21 := sum(-371/2, -97/6)

Left sum ORIGINAL accumulated term21 = -605/3

source roles = [21, 24, -24, 11, 1, 5, 1/6]

derived = [9/2, -97/1, -97/6, -605/3]

Left sum ORIGINAL sample22 index times width := mul(22, 1/6)

Left sum ORIGINAL sample22 index times width = 11/3

Left sum ORIGINAL sample22 lower plus offset := sum(1, 11/3)

Left sum ORIGINAL sample22 lower plus offset = 14/3

Left sum ORIGINAL sample22 slope times x := mul(-24, 14/3)

Left sum ORIGINAL sample22 slope times x = -112

Left sum ORIGINAL sample22 affine value plus c := sum(-112, 11)

Left sum ORIGINAL sample22 affine value plus c = -101

Left sum ORIGINAL sample22 value times cell width := mul(-101, 1/6)

Left sum ORIGINAL sample22 value times cell width = -101/6

Left sum ORIGINAL accumulated term22 := sum(-605/3, -101/6)

Left sum ORIGINAL accumulated term22 = -437/2

source roles = [22, 24, -24, 11, 1, 5, 1/6]

derived = [14/3, -101/1, -101/6, -437/2]

Left sum ORIGINAL sample23 index times width := mul(23, 1/6)

Left sum ORIGINAL sample23 index times width = 23/6

Left sum ORIGINAL sample23 lower plus offset := sum(1, 23/6)

Left sum ORIGINAL sample23 lower plus offset = 29/6

Left sum ORIGINAL sample23 slope times x := mul(-24, 29/6)

Left sum ORIGINAL sample23 slope times x = -116

Left sum ORIGINAL sample23 affine value plus c := sum(-116, 11)

Left sum ORIGINAL sample23 affine value plus c = -105

Left sum ORIGINAL sample23 value times cell width := mul(-105, 1/6)

Left sum ORIGINAL sample23 value times cell width = -35/2

Left sum ORIGINAL accumulated term23 := sum(-437/2, -35/2)

Left sum ORIGINAL accumulated term23 = -236

source roles = [23, 24, -24, 11, 1, 5, 1/6]

derived = [29/6, -105/1, -35/2, -236/1]

source roles = [24, -24, 11, 1, 5]

derived = [1/6, -236/1]

逐步反向核對

• S172 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S171 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S170 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term23，不以小數近似替代。
• S169 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample23 value times cell width，不以小數近似替代。
• S168 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample23 affine value plus c，不以小數近似替代。
• S167 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample23 slope times x，不以小數近似替代。
• S166 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample23 lower plus offset，不以小數近似替代。
• S165 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample23 index times width，不以小數近似替代。
• S164 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S163 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term22，不以小數近似替代。
• S162 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample22 value times cell width，不以小數近似替代。
• S161 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample22 affine value plus c，不以小數近似替代。
• S160 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample22 slope times x，不以小數近似替代。
• S159 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample22 lower plus offset，不以小數近似替代。
• S158 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample22 index times width，不以小數近似替代。
• S157 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S156 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term21，不以小數近似替代。
• S155 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 value times cell width，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 affine value plus c，不以小數近似替代。
• S153 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 slope times x，不以小數近似替代。
• S152 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 lower plus offset，不以小數近似替代。
• S151 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 index times width，不以小數近似替代。
• S150 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S149 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term20，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 value times cell width，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 affine value plus c，不以小數近似替代。
• S146 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 slope times x，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 lower plus offset，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 index times width，不以小數近似替代。
• S143 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S142 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term19，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 value times cell width，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 affine value plus c，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 slope times x，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 lower plus offset，不以小數近似替代。
• S137 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 index times width，不以小數近似替代。
• S136 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S135 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term18，不以小數近似替代。
• S134 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 value times cell width，不以小數近似替代。
• S133 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 affine value plus c，不以小數近似替代。
• S132 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 slope times x，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 lower plus offset，不以小數近似替代。
• S130 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 index times width，不以小數近似替代。
• S129 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S128 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term17，不以小數近似替代。
• S127 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 value times cell width，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 affine value plus c，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 slope times x，不以小數近似替代。
• S124 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 lower plus offset，不以小數近似替代。
• S123 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 index times width，不以小數近似替代。
• S122 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S121 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term16，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 value times cell width，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 affine value plus c，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 slope times x，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 lower plus offset，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 index times width，不以小數近似替代。
• S115 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S114 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term15，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 value times cell width，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 affine value plus c，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 slope times x，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 lower plus offset，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 index times width，不以小數近似替代。
• S108 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S107 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term14，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 value times cell width，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 affine value plus c，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 slope times x，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 lower plus offset，不以小數近似替代。
• S102 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 index times width，不以小數近似替代。
• S101 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S100 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term13，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 value times cell width，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 affine value plus c，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 slope times x，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 lower plus offset，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 index times width，不以小數近似替代。
• S094 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S093 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term12，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 value times cell width，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 affine value plus c，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 slope times x，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 lower plus offset，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 index times width，不以小數近似替代。
• S087 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S086 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term11，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 value times cell width，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 affine value plus c，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 slope times x，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 lower plus offset，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 index times width，不以小數近似替代。
• S080 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S079 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term10，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 value times cell width，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 affine value plus c，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 slope times x，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 lower plus offset，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 index times width，不以小數近似替代。
• S073 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S072 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term9，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 value times cell width，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 affine value plus c，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 slope times x，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 lower plus offset，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 index times width，不以小數近似替代。
• S066 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S065 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term8，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 value times cell width，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 affine value plus c，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 slope times x，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 lower plus offset，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 index times width，不以小數近似替代。
• S059 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S058 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term7，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 value times cell width，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 affine value plus c，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 slope times x，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 lower plus offset，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 index times width，不以小數近似替代。
• S052 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S051 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term6，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 value times cell width，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 affine value plus c，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 slope times x，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 lower plus offset，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 index times width，不以小數近似替代。
• S045 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S044 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term5，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 value times cell width，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 affine value plus c，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 slope times x，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 lower plus offset，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 index times width，不以小數近似替代。
• S038 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S037 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term4，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 value times cell width，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 affine value plus c，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 slope times x，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 lower plus offset，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 index times width，不以小數近似替代。
• S031 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S030 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term3，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 value times cell width，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 affine value plus c，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 slope times x，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 lower plus offset，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 index times width，不以小數近似替代。
• S024 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S023 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term2，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 value times cell width，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 affine value plus c，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 slope times x，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 lower plus offset，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 index times width，不以小數近似替代。
• S017 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S016 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 value times cell width，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 affine value plus c，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 slope times x，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 lower plus offset，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 index times width，不以小數近似替代。
• S010 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S009 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 value times cell width，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 affine value plus c，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 slope times x，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 lower plus offset，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 index times width，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL width over positive cell count，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL upper minus lower endpoint，不以小數近似替代。
• S001 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。

結論

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依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。