Question 32612: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(-12)x+(-1). For a=1≤b=3 and N=12 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)=(-12)x+(-1). For a=1≤b=3 and N=12 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(3, 1)

Left sum ORIGINAL upper minus lower endpoint = 2

Left sum ORIGINAL width over positive cell count := div(2, 12)

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(-12, 1)

Left sum ORIGINAL sample0 slope times x = -12

Left sum ORIGINAL sample0 affine value plus c := sum(-12, -1)

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, 12, -12, -1, 1, 3, 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(-12, 7/6)

Left sum ORIGINAL sample1 slope times x = -14

Left sum ORIGINAL sample1 affine value plus c := sum(-14, -1)

Left sum ORIGINAL sample1 affine value plus c = -15

Left sum ORIGINAL sample1 value times cell width := mul(-15, 1/6)

Left sum ORIGINAL sample1 value times cell width = -5/2

Left sum ORIGINAL accumulated term1 := sum(-13/6, -5/2)

Left sum ORIGINAL accumulated term1 = -14/3

source roles = [1, 12, -12, -1, 1, 3, 1/6]

derived = [7/6, -15/1, -5/2, -14/3]

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(-12, 4/3)

Left sum ORIGINAL sample2 slope times x = -16

Left sum ORIGINAL sample2 affine value plus c := sum(-16, -1)

Left sum ORIGINAL sample2 affine value plus c = -17

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

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

Left sum ORIGINAL accumulated term2 := sum(-14/3, -17/6)

Left sum ORIGINAL accumulated term2 = -15/2

source roles = [2, 12, -12, -1, 1, 3, 1/6]

derived = [4/3, -17/1, -17/6, -15/2]

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(-12, 3/2)

Left sum ORIGINAL sample3 slope times x = -18

Left sum ORIGINAL sample3 affine value plus c := sum(-18, -1)

Left sum ORIGINAL sample3 affine value plus c = -19

Left sum ORIGINAL sample3 value times cell width := mul(-19, 1/6)

Left sum ORIGINAL sample3 value times cell width = -19/6

Left sum ORIGINAL accumulated term3 := sum(-15/2, -19/6)

Left sum ORIGINAL accumulated term3 = -32/3

source roles = [3, 12, -12, -1, 1, 3, 1/6]

derived = [3/2, -19/1, -19/6, -32/3]

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(-12, 5/3)

Left sum ORIGINAL sample4 slope times x = -20

Left sum ORIGINAL sample4 affine value plus c := sum(-20, -1)

Left sum ORIGINAL sample4 affine value plus c = -21

Left sum ORIGINAL sample4 value times cell width := mul(-21, 1/6)

Left sum ORIGINAL sample4 value times cell width = -7/2

Left sum ORIGINAL accumulated term4 := sum(-32/3, -7/2)

Left sum ORIGINAL accumulated term4 = -85/6

source roles = [4, 12, -12, -1, 1, 3, 1/6]

derived = [5/3, -21/1, -7/2, -85/6]

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

Left sum ORIGINAL sample5 index times width = 5/6

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

Left sum ORIGINAL sample5 lower plus offset = 11/6

Left sum ORIGINAL sample5 slope times x := mul(-12, 11/6)

Left sum ORIGINAL sample5 slope times x = -22

Left sum ORIGINAL sample5 affine value plus c := sum(-22, -1)

Left sum ORIGINAL sample5 affine value plus c = -23

Left sum ORIGINAL sample5 value times cell width := mul(-23, 1/6)

Left sum ORIGINAL sample5 value times cell width = -23/6

Left sum ORIGINAL accumulated term5 := sum(-85/6, -23/6)

Left sum ORIGINAL accumulated term5 = -18

source roles = [5, 12, -12, -1, 1, 3, 1/6]

derived = [11/6, -23/1, -23/6, -18/1]

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(-12, 2)

Left sum ORIGINAL sample6 slope times x = -24

Left sum ORIGINAL sample6 affine value plus c := sum(-24, -1)

Left sum ORIGINAL sample6 affine value plus c = -25

Left sum ORIGINAL sample6 value times cell width := mul(-25, 1/6)

Left sum ORIGINAL sample6 value times cell width = -25/6

Left sum ORIGINAL accumulated term6 := sum(-18, -25/6)

Left sum ORIGINAL accumulated term6 = -133/6

source roles = [6, 12, -12, -1, 1, 3, 1/6]

derived = [2/1, -25/1, -25/6, -133/6]

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(-12, 13/6)

Left sum ORIGINAL sample7 slope times x = -26

Left sum ORIGINAL sample7 affine value plus c := sum(-26, -1)

Left sum ORIGINAL sample7 affine value plus c = -27

Left sum ORIGINAL sample7 value times cell width := mul(-27, 1/6)

Left sum ORIGINAL sample7 value times cell width = -9/2

Left sum ORIGINAL accumulated term7 := sum(-133/6, -9/2)

Left sum ORIGINAL accumulated term7 = -80/3

source roles = [7, 12, -12, -1, 1, 3, 1/6]

derived = [13/6, -27/1, -9/2, -80/3]

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

Left sum ORIGINAL sample8 index times width = 4/3

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

Left sum ORIGINAL sample8 lower plus offset = 7/3

Left sum ORIGINAL sample8 slope times x := mul(-12, 7/3)

Left sum ORIGINAL sample8 slope times x = -28

Left sum ORIGINAL sample8 affine value plus c := sum(-28, -1)

Left sum ORIGINAL sample8 affine value plus c = -29

Left sum ORIGINAL sample8 value times cell width := mul(-29, 1/6)

Left sum ORIGINAL sample8 value times cell width = -29/6

Left sum ORIGINAL accumulated term8 := sum(-80/3, -29/6)

Left sum ORIGINAL accumulated term8 = -63/2

source roles = [8, 12, -12, -1, 1, 3, 1/6]

derived = [7/3, -29/1, -29/6, -63/2]

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(-12, 5/2)

Left sum ORIGINAL sample9 slope times x = -30

Left sum ORIGINAL sample9 affine value plus c := sum(-30, -1)

Left sum ORIGINAL sample9 affine value plus c = -31

Left sum ORIGINAL sample9 value times cell width := mul(-31, 1/6)

Left sum ORIGINAL sample9 value times cell width = -31/6

Left sum ORIGINAL accumulated term9 := sum(-63/2, -31/6)

Left sum ORIGINAL accumulated term9 = -110/3

source roles = [9, 12, -12, -1, 1, 3, 1/6]

derived = [5/2, -31/1, -31/6, -110/3]

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(-12, 8/3)

Left sum ORIGINAL sample10 slope times x = -32

Left sum ORIGINAL sample10 affine value plus c := sum(-32, -1)

Left sum ORIGINAL sample10 affine value plus c = -33

Left sum ORIGINAL sample10 value times cell width := mul(-33, 1/6)

Left sum ORIGINAL sample10 value times cell width = -11/2

Left sum ORIGINAL accumulated term10 := sum(-110/3, -11/2)

Left sum ORIGINAL accumulated term10 = -253/6

source roles = [10, 12, -12, -1, 1, 3, 1/6]

derived = [8/3, -33/1, -11/2, -253/6]

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(-12, 17/6)

Left sum ORIGINAL sample11 slope times x = -34

Left sum ORIGINAL sample11 affine value plus c := sum(-34, -1)

Left sum ORIGINAL sample11 affine value plus c = -35

Left sum ORIGINAL sample11 value times cell width := mul(-35, 1/6)

Left sum ORIGINAL sample11 value times cell width = -35/6

Left sum ORIGINAL accumulated term11 := sum(-253/6, -35/6)

Left sum ORIGINAL accumulated term11 = -48

source roles = [11, 12, -12, -1, 1, 3, 1/6]

derived = [17/6, -35/1, -35/6, -48/1]

source roles = [12, -12, -1, 1, 3]

derived = [1/6, -48/1]

Reverse checks

• S088 · 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.
• 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

-48/1

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.