Question 32613: Finite Left-Endpoint Sum of an Affine Function

Original question and givens

On all R, f(x)=(13)x+(0). For a=-2≤b=1 and N=13 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)=(13)x+(0). For a=-2≤b=1 and N=13 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(1, -2)

Left sum ORIGINAL upper minus lower endpoint = 3

Left sum ORIGINAL width over positive cell count := div(3, 13)

Left sum ORIGINAL width over positive cell count = 3/13

Left sum ORIGINAL sample0 index times width := mul(0, 3/13)

Left sum ORIGINAL sample0 index times width = 0

Left sum ORIGINAL sample0 lower plus offset := sum(-2, 0)

Left sum ORIGINAL sample0 lower plus offset = -2

Left sum ORIGINAL sample0 slope times x := mul(13, -2)

Left sum ORIGINAL sample0 slope times x = -26

Left sum ORIGINAL sample0 affine value plus c := sum(-26, 0)

Left sum ORIGINAL sample0 affine value plus c = -26

Left sum ORIGINAL sample0 value times cell width := mul(-26, 3/13)

Left sum ORIGINAL sample0 value times cell width = -6

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

Left sum ORIGINAL accumulated term0 = -6

source roles = [0, 13, 13, 0, -2, 1, 3/13]

derived = [-2/1, -26/1, -6/1, -6/1]

Left sum ORIGINAL sample1 index times width := mul(1, 3/13)

Left sum ORIGINAL sample1 index times width = 3/13

Left sum ORIGINAL sample1 lower plus offset := sum(-2, 3/13)

Left sum ORIGINAL sample1 lower plus offset = -23/13

Left sum ORIGINAL sample1 slope times x := mul(13, -23/13)

Left sum ORIGINAL sample1 slope times x = -23

Left sum ORIGINAL sample1 affine value plus c := sum(-23, 0)

Left sum ORIGINAL sample1 affine value plus c = -23

Left sum ORIGINAL sample1 value times cell width := mul(-23, 3/13)

Left sum ORIGINAL sample1 value times cell width = -69/13

Left sum ORIGINAL accumulated term1 := sum(-6, -69/13)

Left sum ORIGINAL accumulated term1 = -147/13

source roles = [1, 13, 13, 0, -2, 1, 3/13]

derived = [-23/13, -23/1, -69/13, -147/13]

Left sum ORIGINAL sample2 index times width := mul(2, 3/13)

Left sum ORIGINAL sample2 index times width = 6/13

Left sum ORIGINAL sample2 lower plus offset := sum(-2, 6/13)

Left sum ORIGINAL sample2 lower plus offset = -20/13

Left sum ORIGINAL sample2 slope times x := mul(13, -20/13)

Left sum ORIGINAL sample2 slope times x = -20

Left sum ORIGINAL sample2 affine value plus c := sum(-20, 0)

Left sum ORIGINAL sample2 affine value plus c = -20

Left sum ORIGINAL sample2 value times cell width := mul(-20, 3/13)

Left sum ORIGINAL sample2 value times cell width = -60/13

Left sum ORIGINAL accumulated term2 := sum(-147/13, -60/13)

Left sum ORIGINAL accumulated term2 = -207/13

source roles = [2, 13, 13, 0, -2, 1, 3/13]

derived = [-20/13, -20/1, -60/13, -207/13]

Left sum ORIGINAL sample3 index times width := mul(3, 3/13)

Left sum ORIGINAL sample3 index times width = 9/13

Left sum ORIGINAL sample3 lower plus offset := sum(-2, 9/13)

Left sum ORIGINAL sample3 lower plus offset = -17/13

Left sum ORIGINAL sample3 slope times x := mul(13, -17/13)

Left sum ORIGINAL sample3 slope times x = -17

Left sum ORIGINAL sample3 affine value plus c := sum(-17, 0)

Left sum ORIGINAL sample3 affine value plus c = -17

Left sum ORIGINAL sample3 value times cell width := mul(-17, 3/13)

Left sum ORIGINAL sample3 value times cell width = -51/13

Left sum ORIGINAL accumulated term3 := sum(-207/13, -51/13)

Left sum ORIGINAL accumulated term3 = -258/13

source roles = [3, 13, 13, 0, -2, 1, 3/13]

derived = [-17/13, -17/1, -51/13, -258/13]

Left sum ORIGINAL sample4 index times width := mul(4, 3/13)

Left sum ORIGINAL sample4 index times width = 12/13

Left sum ORIGINAL sample4 lower plus offset := sum(-2, 12/13)

Left sum ORIGINAL sample4 lower plus offset = -14/13

Left sum ORIGINAL sample4 slope times x := mul(13, -14/13)

Left sum ORIGINAL sample4 slope times x = -14

Left sum ORIGINAL sample4 affine value plus c := sum(-14, 0)

Left sum ORIGINAL sample4 affine value plus c = -14

Left sum ORIGINAL sample4 value times cell width := mul(-14, 3/13)

Left sum ORIGINAL sample4 value times cell width = -42/13

Left sum ORIGINAL accumulated term4 := sum(-258/13, -42/13)

Left sum ORIGINAL accumulated term4 = -300/13

source roles = [4, 13, 13, 0, -2, 1, 3/13]

derived = [-14/13, -14/1, -42/13, -300/13]

Left sum ORIGINAL sample5 index times width := mul(5, 3/13)

Left sum ORIGINAL sample5 index times width = 15/13

Left sum ORIGINAL sample5 lower plus offset := sum(-2, 15/13)

Left sum ORIGINAL sample5 lower plus offset = -11/13

Left sum ORIGINAL sample5 slope times x := mul(13, -11/13)

Left sum ORIGINAL sample5 slope times x = -11

Left sum ORIGINAL sample5 affine value plus c := sum(-11, 0)

Left sum ORIGINAL sample5 affine value plus c = -11

Left sum ORIGINAL sample5 value times cell width := mul(-11, 3/13)

Left sum ORIGINAL sample5 value times cell width = -33/13

Left sum ORIGINAL accumulated term5 := sum(-300/13, -33/13)

Left sum ORIGINAL accumulated term5 = -333/13

source roles = [5, 13, 13, 0, -2, 1, 3/13]

derived = [-11/13, -11/1, -33/13, -333/13]

Left sum ORIGINAL sample6 index times width := mul(6, 3/13)

Left sum ORIGINAL sample6 index times width = 18/13

Left sum ORIGINAL sample6 lower plus offset := sum(-2, 18/13)

Left sum ORIGINAL sample6 lower plus offset = -8/13

Left sum ORIGINAL sample6 slope times x := mul(13, -8/13)

Left sum ORIGINAL sample6 slope times x = -8

Left sum ORIGINAL sample6 affine value plus c := sum(-8, 0)

Left sum ORIGINAL sample6 affine value plus c = -8

Left sum ORIGINAL sample6 value times cell width := mul(-8, 3/13)

Left sum ORIGINAL sample6 value times cell width = -24/13

Left sum ORIGINAL accumulated term6 := sum(-333/13, -24/13)

Left sum ORIGINAL accumulated term6 = -357/13

source roles = [6, 13, 13, 0, -2, 1, 3/13]

derived = [-8/13, -8/1, -24/13, -357/13]

Left sum ORIGINAL sample7 index times width := mul(7, 3/13)

Left sum ORIGINAL sample7 index times width = 21/13

Left sum ORIGINAL sample7 lower plus offset := sum(-2, 21/13)

Left sum ORIGINAL sample7 lower plus offset = -5/13

Left sum ORIGINAL sample7 slope times x := mul(13, -5/13)

Left sum ORIGINAL sample7 slope times x = -5

Left sum ORIGINAL sample7 affine value plus c := sum(-5, 0)

Left sum ORIGINAL sample7 affine value plus c = -5

Left sum ORIGINAL sample7 value times cell width := mul(-5, 3/13)

Left sum ORIGINAL sample7 value times cell width = -15/13

Left sum ORIGINAL accumulated term7 := sum(-357/13, -15/13)

Left sum ORIGINAL accumulated term7 = -372/13

source roles = [7, 13, 13, 0, -2, 1, 3/13]

derived = [-5/13, -5/1, -15/13, -372/13]

Left sum ORIGINAL sample8 index times width := mul(8, 3/13)

Left sum ORIGINAL sample8 index times width = 24/13

Left sum ORIGINAL sample8 lower plus offset := sum(-2, 24/13)

Left sum ORIGINAL sample8 lower plus offset = -2/13

Left sum ORIGINAL sample8 slope times x := mul(13, -2/13)

Left sum ORIGINAL sample8 slope times x = -2

Left sum ORIGINAL sample8 affine value plus c := sum(-2, 0)

Left sum ORIGINAL sample8 affine value plus c = -2

Left sum ORIGINAL sample8 value times cell width := mul(-2, 3/13)

Left sum ORIGINAL sample8 value times cell width = -6/13

Left sum ORIGINAL accumulated term8 := sum(-372/13, -6/13)

Left sum ORIGINAL accumulated term8 = -378/13

source roles = [8, 13, 13, 0, -2, 1, 3/13]

derived = [-2/13, -2/1, -6/13, -378/13]

Left sum ORIGINAL sample9 index times width := mul(9, 3/13)

Left sum ORIGINAL sample9 index times width = 27/13

Left sum ORIGINAL sample9 lower plus offset := sum(-2, 27/13)

Left sum ORIGINAL sample9 lower plus offset = 1/13

Left sum ORIGINAL sample9 slope times x := mul(13, 1/13)

Left sum ORIGINAL sample9 slope times x = 1

Left sum ORIGINAL sample9 affine value plus c := sum(1, 0)

Left sum ORIGINAL sample9 affine value plus c = 1

Left sum ORIGINAL sample9 value times cell width := mul(1, 3/13)

Left sum ORIGINAL sample9 value times cell width = 3/13

Left sum ORIGINAL accumulated term9 := sum(-378/13, 3/13)

Left sum ORIGINAL accumulated term9 = -375/13

source roles = [9, 13, 13, 0, -2, 1, 3/13]

derived = [1/13, 1/1, 3/13, -375/13]

Left sum ORIGINAL sample10 index times width := mul(10, 3/13)

Left sum ORIGINAL sample10 index times width = 30/13

Left sum ORIGINAL sample10 lower plus offset := sum(-2, 30/13)

Left sum ORIGINAL sample10 lower plus offset = 4/13

Left sum ORIGINAL sample10 slope times x := mul(13, 4/13)

Left sum ORIGINAL sample10 slope times x = 4

Left sum ORIGINAL sample10 affine value plus c := sum(4, 0)

Left sum ORIGINAL sample10 affine value plus c = 4

Left sum ORIGINAL sample10 value times cell width := mul(4, 3/13)

Left sum ORIGINAL sample10 value times cell width = 12/13

Left sum ORIGINAL accumulated term10 := sum(-375/13, 12/13)

Left sum ORIGINAL accumulated term10 = -363/13

source roles = [10, 13, 13, 0, -2, 1, 3/13]

derived = [4/13, 4/1, 12/13, -363/13]

Left sum ORIGINAL sample11 index times width := mul(11, 3/13)

Left sum ORIGINAL sample11 index times width = 33/13

Left sum ORIGINAL sample11 lower plus offset := sum(-2, 33/13)

Left sum ORIGINAL sample11 lower plus offset = 7/13

Left sum ORIGINAL sample11 slope times x := mul(13, 7/13)

Left sum ORIGINAL sample11 slope times x = 7

Left sum ORIGINAL sample11 affine value plus c := sum(7, 0)

Left sum ORIGINAL sample11 affine value plus c = 7

Left sum ORIGINAL sample11 value times cell width := mul(7, 3/13)

Left sum ORIGINAL sample11 value times cell width = 21/13

Left sum ORIGINAL accumulated term11 := sum(-363/13, 21/13)

Left sum ORIGINAL accumulated term11 = -342/13

source roles = [11, 13, 13, 0, -2, 1, 3/13]

derived = [7/13, 7/1, 21/13, -342/13]

Left sum ORIGINAL sample12 index times width := mul(12, 3/13)

Left sum ORIGINAL sample12 index times width = 36/13

Left sum ORIGINAL sample12 lower plus offset := sum(-2, 36/13)

Left sum ORIGINAL sample12 lower plus offset = 10/13

Left sum ORIGINAL sample12 slope times x := mul(13, 10/13)

Left sum ORIGINAL sample12 slope times x = 10

Left sum ORIGINAL sample12 affine value plus c := sum(10, 0)

Left sum ORIGINAL sample12 affine value plus c = 10

Left sum ORIGINAL sample12 value times cell width := mul(10, 3/13)

Left sum ORIGINAL sample12 value times cell width = 30/13

Left sum ORIGINAL accumulated term12 := sum(-342/13, 30/13)

Left sum ORIGINAL accumulated term12 = -24

source roles = [12, 13, 13, 0, -2, 1, 3/13]

derived = [10/13, 10/1, 30/13, -24/1]

source roles = [13, 13, 0, -2, 1]

derived = [3/13, -24/1]

Reverse checks

• S095 · 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.
• 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

-24/1

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.