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Question 32614: Finite Left-Endpoint Sum of an Affine Function

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0064-32614 · q-20261007-advance-20261008-r07-definite-integral-0089

Original question and givens

On all R, f(x)=(-14)x+(1). For a=-1≤b=3 and N=14 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)=(-14)x+(1). For a=-1≤b=3 and N=14 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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-integral-affine-left-sum, affine_left_sum, reals, []]

After

derived = [[finite_left_sum_fraction_RETAIN1]]

Conditions: none

Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status.

Step identifiers and dependencies

Rule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001

S002 · calculation

Before

Left sum ORIGINAL upper minus lower endpoint := sub(3, -1)

After

Left sum ORIGINAL upper minus lower endpoint = 4

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Left sum ORIGINAL width over positive cell count := div(4, 14)

After

Left sum ORIGINAL width over positive cell count = 2/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Left sum ORIGINAL sample0 index times width := mul(0, 2/7)

After

Left sum ORIGINAL sample0 index times width = 0

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

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

After

Left sum ORIGINAL sample0 lower plus offset = -1

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Left sum ORIGINAL sample0 slope times x := mul(-14, -1)

After

Left sum ORIGINAL sample0 slope times x = 14

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Left sum ORIGINAL sample0 affine value plus c := sum(14, 1)

After

Left sum ORIGINAL sample0 affine value plus c = 15

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Left sum ORIGINAL sample0 value times cell width := mul(15, 2/7)

After

Left sum ORIGINAL sample0 value times cell width = 30/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Left sum ORIGINAL accumulated term0 := sum(0, 30/7)

After

Left sum ORIGINAL accumulated term0 = 30/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [0, 14, -14, 1, -1, 3, 2/7]

After

derived = [-1/1, 15/1, 30/7, 30/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S009

S011 · calculation

Before

Left sum ORIGINAL sample1 index times width := mul(1, 2/7)

After

Left sum ORIGINAL sample1 index times width = 2/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Left sum ORIGINAL sample1 lower plus offset := sum(-1, 2/7)

After

Left sum ORIGINAL sample1 lower plus offset = -5/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Left sum ORIGINAL sample1 slope times x := mul(-14, -5/7)

After

Left sum ORIGINAL sample1 slope times x = 10

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Left sum ORIGINAL sample1 affine value plus c := sum(10, 1)

After

Left sum ORIGINAL sample1 affine value plus c = 11

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Left sum ORIGINAL sample1 value times cell width := mul(11, 2/7)

After

Left sum ORIGINAL sample1 value times cell width = 22/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Left sum ORIGINAL accumulated term1 := sum(30/7, 22/7)

After

Left sum ORIGINAL accumulated term1 = 52/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [1, 14, -14, 1, -1, 3, 2/7]

After

derived = [-5/7, 11/1, 22/7, 52/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S016

S018 · calculation

Before

Left sum ORIGINAL sample2 index times width := mul(2, 2/7)

After

Left sum ORIGINAL sample2 index times width = 4/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Left sum ORIGINAL sample2 lower plus offset := sum(-1, 4/7)

After

Left sum ORIGINAL sample2 lower plus offset = -3/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Left sum ORIGINAL sample2 slope times x := mul(-14, -3/7)

After

Left sum ORIGINAL sample2 slope times x = 6

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Left sum ORIGINAL sample2 affine value plus c := sum(6, 1)

After

Left sum ORIGINAL sample2 affine value plus c = 7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Left sum ORIGINAL sample2 value times cell width := mul(7, 2/7)

After

Left sum ORIGINAL sample2 value times cell width = 2

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Left sum ORIGINAL accumulated term2 := sum(52/7, 2)

After

Left sum ORIGINAL accumulated term2 = 66/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

source roles = [2, 14, -14, 1, -1, 3, 2/7]

After

derived = [-3/7, 7/1, 2/1, 66/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S023

S025 · calculation

Before

Left sum ORIGINAL sample3 index times width := mul(3, 2/7)

After

Left sum ORIGINAL sample3 index times width = 6/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Left sum ORIGINAL sample3 lower plus offset := sum(-1, 6/7)

After

Left sum ORIGINAL sample3 lower plus offset = -1/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Left sum ORIGINAL sample3 slope times x := mul(-14, -1/7)

After

Left sum ORIGINAL sample3 slope times x = 2

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Left sum ORIGINAL sample3 affine value plus c := sum(2, 1)

After

Left sum ORIGINAL sample3 affine value plus c = 3

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Left sum ORIGINAL sample3 value times cell width := mul(3, 2/7)

After

Left sum ORIGINAL sample3 value times cell width = 6/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Left sum ORIGINAL accumulated term3 := sum(66/7, 6/7)

After

Left sum ORIGINAL accumulated term3 = 72/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

source roles = [3, 14, -14, 1, -1, 3, 2/7]

After

derived = [-1/7, 3/1, 6/7, 72/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S030

S032 · calculation

Before

Left sum ORIGINAL sample4 index times width := mul(4, 2/7)

After

Left sum ORIGINAL sample4 index times width = 8/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Left sum ORIGINAL sample4 lower plus offset := sum(-1, 8/7)

After

Left sum ORIGINAL sample4 lower plus offset = 1/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Left sum ORIGINAL sample4 slope times x := mul(-14, 1/7)

After

Left sum ORIGINAL sample4 slope times x = -2

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Left sum ORIGINAL sample4 affine value plus c := sum(-2, 1)

After

Left sum ORIGINAL sample4 affine value plus c = -1

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Left sum ORIGINAL sample4 value times cell width := mul(-1, 2/7)

After

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

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Left sum ORIGINAL accumulated term4 := sum(72/7, -2/7)

After

Left sum ORIGINAL accumulated term4 = 10

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

source roles = [4, 14, -14, 1, -1, 3, 2/7]

After

derived = [1/7, -1/1, -2/7, 10/1]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S037

S039 · calculation

Before

Left sum ORIGINAL sample5 index times width := mul(5, 2/7)

After

Left sum ORIGINAL sample5 index times width = 10/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Left sum ORIGINAL sample5 lower plus offset := sum(-1, 10/7)

After

Left sum ORIGINAL sample5 lower plus offset = 3/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Left sum ORIGINAL sample5 slope times x := mul(-14, 3/7)

After

Left sum ORIGINAL sample5 slope times x = -6

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

Left sum ORIGINAL sample5 affine value plus c := sum(-6, 1)

After

Left sum ORIGINAL sample5 affine value plus c = -5

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Left sum ORIGINAL sample5 value times cell width := mul(-5, 2/7)

After

Left sum ORIGINAL sample5 value times cell width = -10/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

Left sum ORIGINAL accumulated term5 := sum(10, -10/7)

After

Left sum ORIGINAL accumulated term5 = 60/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

source roles = [5, 14, -14, 1, -1, 3, 2/7]

After

derived = [3/7, -5/1, -10/7, 60/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S044

S046 · calculation

Before

Left sum ORIGINAL sample6 index times width := mul(6, 2/7)

After

Left sum ORIGINAL sample6 index times width = 12/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

Left sum ORIGINAL sample6 lower plus offset := sum(-1, 12/7)

After

Left sum ORIGINAL sample6 lower plus offset = 5/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

Left sum ORIGINAL sample6 slope times x := mul(-14, 5/7)

After

Left sum ORIGINAL sample6 slope times x = -10

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Left sum ORIGINAL sample6 affine value plus c := sum(-10, 1)

After

Left sum ORIGINAL sample6 affine value plus c = -9

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

Left sum ORIGINAL sample6 value times cell width := mul(-9, 2/7)

After

Left sum ORIGINAL sample6 value times cell width = -18/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

Left sum ORIGINAL accumulated term6 := sum(60/7, -18/7)

After

Left sum ORIGINAL accumulated term6 = 6

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

source roles = [6, 14, -14, 1, -1, 3, 2/7]

After

derived = [5/7, -9/1, -18/7, 6/1]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S051

S053 · calculation

Before

Left sum ORIGINAL sample7 index times width := mul(7, 2/7)

After

Left sum ORIGINAL sample7 index times width = 2

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

Left sum ORIGINAL sample7 lower plus offset := sum(-1, 2)

After

Left sum ORIGINAL sample7 lower plus offset = 1

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

Left sum ORIGINAL sample7 slope times x := mul(-14, 1)

After

Left sum ORIGINAL sample7 slope times x = -14

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

Left sum ORIGINAL sample7 affine value plus c := sum(-14, 1)

After

Left sum ORIGINAL sample7 affine value plus c = -13

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

Left sum ORIGINAL sample7 value times cell width := mul(-13, 2/7)

After

Left sum ORIGINAL sample7 value times cell width = -26/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

Left sum ORIGINAL accumulated term7 := sum(6, -26/7)

After

Left sum ORIGINAL accumulated term7 = 16/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

source roles = [7, 14, -14, 1, -1, 3, 2/7]

After

derived = [1/1, -13/1, -26/7, 16/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S058

S060 · calculation

Before

Left sum ORIGINAL sample8 index times width := mul(8, 2/7)

After

Left sum ORIGINAL sample8 index times width = 16/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

Left sum ORIGINAL sample8 lower plus offset := sum(-1, 16/7)

After

Left sum ORIGINAL sample8 lower plus offset = 9/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

Left sum ORIGINAL sample8 slope times x := mul(-14, 9/7)

After

Left sum ORIGINAL sample8 slope times x = -18

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

Left sum ORIGINAL sample8 affine value plus c := sum(-18, 1)

After

Left sum ORIGINAL sample8 affine value plus c = -17

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

Left sum ORIGINAL sample8 value times cell width := mul(-17, 2/7)

After

Left sum ORIGINAL sample8 value times cell width = -34/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

Left sum ORIGINAL accumulated term8 := sum(16/7, -34/7)

After

Left sum ORIGINAL accumulated term8 = -18/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

source roles = [8, 14, -14, 1, -1, 3, 2/7]

After

derived = [9/7, -17/1, -34/7, -18/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S065

S067 · calculation

Before

Left sum ORIGINAL sample9 index times width := mul(9, 2/7)

After

Left sum ORIGINAL sample9 index times width = 18/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

Left sum ORIGINAL sample9 lower plus offset := sum(-1, 18/7)

After

Left sum ORIGINAL sample9 lower plus offset = 11/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

Left sum ORIGINAL sample9 slope times x := mul(-14, 11/7)

After

Left sum ORIGINAL sample9 slope times x = -22

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

Left sum ORIGINAL sample9 affine value plus c := sum(-22, 1)

After

Left sum ORIGINAL sample9 affine value plus c = -21

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · calculation

Before

Left sum ORIGINAL sample9 value times cell width := mul(-21, 2/7)

After

Left sum ORIGINAL sample9 value times cell width = -6

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

Left sum ORIGINAL accumulated term9 := sum(-18/7, -6)

After

Left sum ORIGINAL accumulated term9 = -60/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · implies

Before

source roles = [9, 14, -14, 1, -1, 3, 2/7]

After

derived = [11/7, -21/1, -6/1, -60/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S072

S074 · calculation

Before

Left sum ORIGINAL sample10 index times width := mul(10, 2/7)

After

Left sum ORIGINAL sample10 index times width = 20/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · calculation

Before

Left sum ORIGINAL sample10 lower plus offset := sum(-1, 20/7)

After

Left sum ORIGINAL sample10 lower plus offset = 13/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

Left sum ORIGINAL sample10 slope times x := mul(-14, 13/7)

After

Left sum ORIGINAL sample10 slope times x = -26

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · calculation

Before

Left sum ORIGINAL sample10 affine value plus c := sum(-26, 1)

After

Left sum ORIGINAL sample10 affine value plus c = -25

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · calculation

Before

Left sum ORIGINAL sample10 value times cell width := mul(-25, 2/7)

After

Left sum ORIGINAL sample10 value times cell width = -50/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · calculation

Before

Left sum ORIGINAL accumulated term10 := sum(-60/7, -50/7)

After

Left sum ORIGINAL accumulated term10 = -110/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · implies

Before

source roles = [10, 14, -14, 1, -1, 3, 2/7]

After

derived = [13/7, -25/1, -50/7, -110/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S079

S081 · calculation

Before

Left sum ORIGINAL sample11 index times width := mul(11, 2/7)

After

Left sum ORIGINAL sample11 index times width = 22/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · calculation

Before

Left sum ORIGINAL sample11 lower plus offset := sum(-1, 22/7)

After

Left sum ORIGINAL sample11 lower plus offset = 15/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S081

S083 · calculation

Before

Left sum ORIGINAL sample11 slope times x := mul(-14, 15/7)

After

Left sum ORIGINAL sample11 slope times x = -30

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · calculation

Before

Left sum ORIGINAL sample11 affine value plus c := sum(-30, 1)

After

Left sum ORIGINAL sample11 affine value plus c = -29

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · calculation

Before

Left sum ORIGINAL sample11 value times cell width := mul(-29, 2/7)

After

Left sum ORIGINAL sample11 value times cell width = -58/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S084

S086 · calculation

Before

Left sum ORIGINAL accumulated term11 := sum(-110/7, -58/7)

After

Left sum ORIGINAL accumulated term11 = -24

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S085

S087 · implies

Before

source roles = [11, 14, -14, 1, -1, 3, 2/7]

After

derived = [15/7, -29/1, -58/7, -24/1]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S086

S088 · calculation

Before

Left sum ORIGINAL sample12 index times width := mul(12, 2/7)

After

Left sum ORIGINAL sample12 index times width = 24/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S087

S089 · calculation

Before

Left sum ORIGINAL sample12 lower plus offset := sum(-1, 24/7)

After

Left sum ORIGINAL sample12 lower plus offset = 17/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · calculation

Before

Left sum ORIGINAL sample12 slope times x := mul(-14, 17/7)

After

Left sum ORIGINAL sample12 slope times x = -34

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S089

S091 · calculation

Before

Left sum ORIGINAL sample12 affine value plus c := sum(-34, 1)

After

Left sum ORIGINAL sample12 affine value plus c = -33

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S090

S092 · calculation

Before

Left sum ORIGINAL sample12 value times cell width := mul(-33, 2/7)

After

Left sum ORIGINAL sample12 value times cell width = -66/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S091

S093 · calculation

Before

Left sum ORIGINAL accumulated term12 := sum(-24, -66/7)

After

Left sum ORIGINAL accumulated term12 = -234/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · implies

Before

source roles = [12, 14, -14, 1, -1, 3, 2/7]

After

derived = [17/7, -33/1, -66/7, -234/7]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S093

S095 · calculation

Before

Left sum ORIGINAL sample13 index times width := mul(13, 2/7)

After

Left sum ORIGINAL sample13 index times width = 26/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S094

S096 · calculation

Before

Left sum ORIGINAL sample13 lower plus offset := sum(-1, 26/7)

After

Left sum ORIGINAL sample13 lower plus offset = 19/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S095

S097 · calculation

Before

Left sum ORIGINAL sample13 slope times x := mul(-14, 19/7)

After

Left sum ORIGINAL sample13 slope times x = -38

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · calculation

Before

Left sum ORIGINAL sample13 affine value plus c := sum(-38, 1)

After

Left sum ORIGINAL sample13 affine value plus c = -37

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S097

S099 · calculation

Before

Left sum ORIGINAL sample13 value times cell width := mul(-37, 2/7)

After

Left sum ORIGINAL sample13 value times cell width = -74/7

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · calculation

Before

Left sum ORIGINAL accumulated term13 := sum(-234/7, -74/7)

After

Left sum ORIGINAL accumulated term13 = -44

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S099

S101 · implies

Before

source roles = [13, 14, -14, 1, -1, 3, 2/7]

After

derived = [19/7, -37/1, -74/7, -44/1]

Conditions: none

For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field.

Step identifiers and dependencies

Rule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S100

S102 · implies

Before

source roles = [14, -14, 1, -1, 3]

After

derived = [2/7, -44/1]

Conditions: none

This is a FIXED-N finite left sum,generally not the integral.Constants/zero-width cases may agree,without proving arbitrary integrability or a limit as N→∞.

Step identifiers and dependencies

Rule: LeftSum_FIXED_finite_partition_NOT_integral_or_limit_claim · Uses: G001, S101

Reverse checks

  • S102 · 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.
  • 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

-44/1

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 0501f764fdaf6c3df40cd5e4fd5baf89e13e6770d4db08bb1de8c8bbd6ec89f7
  • candidate record SHA-256: 0132b63631532f34fb7ad94b6ab557798a068ef63f1db016a27ee27e0dd5ca19
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON