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Question 32624: 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

R0064-32624 · q-20261007-advance-20261008-r07-definite-integral-0099

Original question and givens

On all R, f(x)=(-24)x+(11). For a=1≤b=5 and N=24 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)=(-24)x+(11). For a=1≤b=5 and N=24 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(5, 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, 24)

After

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

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, S002

S004 · calculation

Before

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

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(-24, 1)

After

Left sum ORIGINAL sample0 slope times x = -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, S005

S007 · calculation

Before

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

After

Left sum ORIGINAL sample0 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, S006

S008 · calculation

Before

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

After

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

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, S007

S009 · calculation

Before

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

After

Left sum ORIGINAL accumulated term0 = -13/6

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, S008

S010 · implies

Before

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

After

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

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, 1/6)

After

Left sum ORIGINAL sample1 index times width = 1/6

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, S010

S012 · calculation

Before

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

After

Left sum ORIGINAL sample1 lower plus offset = 7/6

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, S011

S013 · calculation

Before

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

After

Left sum ORIGINAL sample1 slope times x = -28

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(-28, 11)

After

Left sum ORIGINAL sample1 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, S013

S015 · calculation

Before

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

After

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

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, S014

S016 · calculation

Before

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

After

Left sum ORIGINAL accumulated term1 = -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, S015

S017 · implies

Before

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

After

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

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, S016

S018 · calculation

Before

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

After

Left sum ORIGINAL sample2 index times width = 1/3

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, S017

S019 · calculation

Before

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

After

Left sum ORIGINAL sample2 lower plus offset = 4/3

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, S018

S020 · calculation

Before

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

After

Left sum ORIGINAL sample2 slope times x = -32

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(-32, 11)

After

Left sum ORIGINAL sample2 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, S020

S022 · calculation

Before

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

After

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

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(-5, -7/2)

After

Left sum ORIGINAL accumulated term2 = -17/2

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, S022

S024 · implies

Before

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

After

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

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, 1/6)

After

Left sum ORIGINAL sample3 index times width = 1/2

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, S024

S026 · calculation

Before

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

After

Left sum ORIGINAL sample3 lower plus offset = 3/2

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, S025

S027 · calculation

Before

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

After

Left sum ORIGINAL sample3 slope times x = -36

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(-36, 11)

After

Left sum ORIGINAL sample3 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, S027

S029 · calculation

Before

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

After

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

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, S028

S030 · calculation

Before

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

After

Left sum ORIGINAL accumulated term3 = -38/3

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, S029

S031 · implies

Before

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

After

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

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, 1/6)

After

Left sum ORIGINAL sample4 index times width = 2/3

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, S031

S033 · calculation

Before

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

After

Left sum ORIGINAL sample4 lower plus offset = 5/3

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, S032

S034 · calculation

Before

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

After

Left sum ORIGINAL sample4 slope times x = -40

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(-40, 11)

After

Left sum ORIGINAL sample4 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, S034

S036 · calculation

Before

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

After

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

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, S035

S037 · calculation

Before

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

After

Left sum ORIGINAL accumulated term4 = -35/2

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, S036

S038 · implies

Before

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

After

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

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, 1/6)

After

Left sum ORIGINAL sample5 index times width = 5/6

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, S038

S040 · calculation

Before

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

After

Left sum ORIGINAL sample5 lower plus offset = 11/6

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, S039

S041 · calculation

Before

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

After

Left sum ORIGINAL sample5 slope times x = -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, S040

S042 · calculation

Before

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

After

Left sum ORIGINAL sample5 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, S041

S043 · calculation

Before

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

After

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

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, S042

S044 · calculation

Before

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

After

Left sum ORIGINAL accumulated term5 = -23

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, 24, -24, 11, 1, 5, 1/6]

After

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

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, S044

S046 · calculation

Before

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

After

Left sum ORIGINAL sample6 index times width = 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, S045

S047 · calculation

Before

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

After

Left sum ORIGINAL sample6 lower plus offset = 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, S046

S048 · calculation

Before

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

After

Left sum ORIGINAL sample6 slope times x = -48

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(-48, 11)

After

Left sum ORIGINAL sample6 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, S048

S050 · calculation

Before

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

After

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

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, S049

S051 · calculation

Before

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

After

Left sum ORIGINAL accumulated term6 = -175/6

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, 24, -24, 11, 1, 5, 1/6]

After

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

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, 1/6)

After

Left sum ORIGINAL sample7 index times width = 7/6

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, S052

S054 · calculation

Before

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

After

Left sum ORIGINAL sample7 lower plus offset = 13/6

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, S053

S055 · calculation

Before

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

After

Left sum ORIGINAL sample7 slope times x = -52

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(-52, 11)

After

Left sum ORIGINAL sample7 affine value plus c = -41

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(-41, 1/6)

After

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

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, S056

S058 · calculation

Before

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

After

Left sum ORIGINAL accumulated term7 = -36

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, 24, -24, 11, 1, 5, 1/6]

After

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

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, S058

S060 · calculation

Before

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

After

Left sum ORIGINAL sample8 index times width = 4/3

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, S059

S061 · calculation

Before

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

After

Left sum ORIGINAL sample8 lower plus offset = 7/3

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, S060

S062 · calculation

Before

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

After

Left sum ORIGINAL sample8 slope times x = -56

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(-56, 11)

After

Left sum ORIGINAL sample8 affine value plus c = -45

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(-45, 1/6)

After

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

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, S063

S065 · calculation

Before

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

After

Left sum ORIGINAL accumulated term8 = -87/2

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, S064

S066 · implies

Before

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

After

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

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, 1/6)

After

Left sum ORIGINAL sample9 index times width = 3/2

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, S066

S068 · calculation

Before

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

After

Left sum ORIGINAL sample9 lower plus offset = 5/2

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, S067

S069 · calculation

Before

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

After

Left sum ORIGINAL sample9 slope times x = -60

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(-60, 11)

After

Left sum ORIGINAL sample9 affine value plus c = -49

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(-49, 1/6)

After

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

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(-87/2, -49/6)

After

Left sum ORIGINAL accumulated term9 = -155/3

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, S071

S073 · implies

Before

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

After

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

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, 1/6)

After

Left sum ORIGINAL sample10 index times width = 5/3

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, S073

S075 · calculation

Before

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

After

Left sum ORIGINAL sample10 lower plus offset = 8/3

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, S074

S076 · calculation

Before

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

After

Left sum ORIGINAL sample10 slope times x = -64

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(-64, 11)

After

Left sum ORIGINAL sample10 affine value plus c = -53

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(-53, 1/6)

After

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

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, S077

S079 · calculation

Before

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

After

Left sum ORIGINAL accumulated term10 = -121/2

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, S078

S080 · implies

Before

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

After

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

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, 1/6)

After

Left sum ORIGINAL sample11 index times width = 11/6

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, S080

S082 · calculation

Before

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

After

Left sum ORIGINAL sample11 lower plus offset = 17/6

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, S081

S083 · calculation

Before

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

After

Left sum ORIGINAL sample11 slope times x = -68

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(-68, 11)

After

Left sum ORIGINAL sample11 affine value plus c = -57

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(-57, 1/6)

After

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

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, S084

S086 · calculation

Before

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

After

Left sum ORIGINAL accumulated term11 = -70

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, 24, -24, 11, 1, 5, 1/6]

After

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

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, 1/6)

After

Left sum ORIGINAL sample12 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, S087

S089 · calculation

Before

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

After

Left sum ORIGINAL sample12 lower plus offset = 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, S088

S090 · calculation

Before

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

After

Left sum ORIGINAL sample12 slope times x = -72

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(-72, 11)

After

Left sum ORIGINAL sample12 affine value plus c = -61

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(-61, 1/6)

After

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

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, S091

S093 · calculation

Before

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

After

Left sum ORIGINAL accumulated term12 = -481/6

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, S092

S094 · implies

Before

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

After

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

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, 1/6)

After

Left sum ORIGINAL sample13 index times width = 13/6

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, S094

S096 · calculation

Before

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

After

Left sum ORIGINAL sample13 lower plus offset = 19/6

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, S095

S097 · calculation

Before

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

After

Left sum ORIGINAL sample13 slope times x = -76

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(-76, 11)

After

Left sum ORIGINAL sample13 affine value plus c = -65

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(-65, 1/6)

After

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

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, S098

S100 · calculation

Before

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

After

Left sum ORIGINAL accumulated term13 = -91

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, 24, -24, 11, 1, 5, 1/6]

After

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

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 · calculation

Before

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

After

Left sum ORIGINAL sample14 index times width = 7/3

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, S101

S103 · calculation

Before

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

After

Left sum ORIGINAL sample14 lower plus offset = 10/3

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, S102

S104 · calculation

Before

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

After

Left sum ORIGINAL sample14 slope times x = -80

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, S103

S105 · calculation

Before

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

After

Left sum ORIGINAL sample14 affine value plus c = -69

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, S104

S106 · calculation

Before

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

After

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

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, S105

S107 · calculation

Before

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

After

Left sum ORIGINAL accumulated term14 = -205/2

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, S106

S108 · implies

Before

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

After

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

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, S107

S109 · calculation

Before

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

After

Left sum ORIGINAL sample15 index times width = 5/2

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, S108

S110 · calculation

Before

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

After

Left sum ORIGINAL sample15 lower plus offset = 7/2

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, S109

S111 · calculation

Before

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

After

Left sum ORIGINAL sample15 slope times x = -84

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, S110

S112 · calculation

Before

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

After

Left sum ORIGINAL sample15 affine value plus c = -73

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, S111

S113 · calculation

Before

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

After

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

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, S112

S114 · calculation

Before

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

After

Left sum ORIGINAL accumulated term15 = -344/3

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, S113

S115 · implies

Before

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

After

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

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, S114

S116 · calculation

Before

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

After

Left sum ORIGINAL sample16 index times width = 8/3

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, S115

S117 · calculation

Before

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

After

Left sum ORIGINAL sample16 lower plus offset = 11/3

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, S116

S118 · calculation

Before

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

After

Left sum ORIGINAL sample16 slope times x = -88

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, S117

S119 · calculation

Before

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

After

Left sum ORIGINAL sample16 affine value plus c = -77

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, S118

S120 · calculation

Before

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

After

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

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, S119

S121 · calculation

Before

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

After

Left sum ORIGINAL accumulated term16 = -255/2

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, S120

S122 · implies

Before

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

After

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

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, S121

S123 · calculation

Before

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

After

Left sum ORIGINAL sample17 index times width = 17/6

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, S122

S124 · calculation

Before

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

After

Left sum ORIGINAL sample17 lower plus offset = 23/6

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, S123

S125 · calculation

Before

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

After

Left sum ORIGINAL sample17 slope times x = -92

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, S124

S126 · calculation

Before

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

After

Left sum ORIGINAL sample17 affine value plus c = -81

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, S125

S127 · calculation

Before

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

After

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

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, S126

S128 · calculation

Before

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

After

Left sum ORIGINAL accumulated term17 = -141

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, S127

S129 · implies

Before

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

After

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

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, S128

S130 · calculation

Before

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

After

Left sum ORIGINAL sample18 index times width = 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, S129

S131 · calculation

Before

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

After

Left sum ORIGINAL sample18 lower plus offset = 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, S130

S132 · calculation

Before

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

After

Left sum ORIGINAL sample18 slope times x = -96

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, S131

S133 · calculation

Before

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

After

Left sum ORIGINAL sample18 affine value plus c = -85

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, S132

S134 · calculation

Before

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

After

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

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, S133

S135 · calculation

Before

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

After

Left sum ORIGINAL accumulated term18 = -931/6

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, S134

S136 · implies

Before

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

After

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

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, S135

S137 · calculation

Before

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

After

Left sum ORIGINAL sample19 index times width = 19/6

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, S136

S138 · calculation

Before

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

After

Left sum ORIGINAL sample19 lower plus offset = 25/6

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, S137

S139 · calculation

Before

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

After

Left sum ORIGINAL sample19 slope times x = -100

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, S138

S140 · calculation

Before

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

After

Left sum ORIGINAL sample19 affine value plus c = -89

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, S139

S141 · calculation

Before

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

After

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

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, S140

S142 · calculation

Before

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

After

Left sum ORIGINAL accumulated term19 = -170

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, S141

S143 · implies

Before

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

After

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

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, S142

S144 · calculation

Before

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

After

Left sum ORIGINAL sample20 index times width = 10/3

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, S143

S145 · calculation

Before

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

After

Left sum ORIGINAL sample20 lower plus offset = 13/3

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, S144

S146 · calculation

Before

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

After

Left sum ORIGINAL sample20 slope times x = -104

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, S145

S147 · calculation

Before

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

After

Left sum ORIGINAL sample20 affine value plus c = -93

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, S146

S148 · calculation

Before

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

After

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

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, S147

S149 · calculation

Before

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

After

Left sum ORIGINAL accumulated term20 = -371/2

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, S148

S150 · implies

Before

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

After

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

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, S149

S151 · calculation

Before

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

After

Left sum ORIGINAL sample21 index times width = 7/2

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, S150

S152 · calculation

Before

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

After

Left sum ORIGINAL sample21 lower plus offset = 9/2

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, S151

S153 · calculation

Before

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

After

Left sum ORIGINAL sample21 slope times x = -108

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, S152

S154 · calculation

Before

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

After

Left sum ORIGINAL sample21 affine value plus c = -97

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, S153

S155 · calculation

Before

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

After

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

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, S154

S156 · calculation

Before

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

After

Left sum ORIGINAL accumulated term21 = -605/3

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, S155

S157 · implies

Before

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

After

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

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, S156

S158 · calculation

Before

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

After

Left sum ORIGINAL sample22 index times width = 11/3

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, S157

S159 · calculation

Before

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

After

Left sum ORIGINAL sample22 lower plus offset = 14/3

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, S158

S160 · calculation

Before

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

After

Left sum ORIGINAL sample22 slope times x = -112

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, S159

S161 · calculation

Before

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

After

Left sum ORIGINAL sample22 affine value plus c = -101

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, S160

S162 · calculation

Before

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

After

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

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, S161

S163 · calculation

Before

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

After

Left sum ORIGINAL accumulated term22 = -437/2

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, S162

S164 · implies

Before

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

After

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

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, S163

S165 · calculation

Before

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

After

Left sum ORIGINAL sample23 index times width = 23/6

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, S164

S166 · calculation

Before

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

After

Left sum ORIGINAL sample23 lower plus offset = 29/6

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, S165

S167 · calculation

Before

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

After

Left sum ORIGINAL sample23 slope times x = -116

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, S166

S168 · calculation

Before

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

After

Left sum ORIGINAL sample23 affine value plus c = -105

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, S167

S169 · calculation

Before

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

After

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

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, S168

S170 · calculation

Before

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

After

Left sum ORIGINAL accumulated term23 = -236

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, S169

S171 · implies

Before

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

After

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

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, S170

S172 · implies

Before

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

After

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

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, S171

Reverse checks

  • S172 · 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.
  • S171 · 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.
  • S170 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term23 with exact rational arithmetic, not rounded decimals.
  • S169 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 value times cell width with exact rational arithmetic, not rounded decimals.
  • S168 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S167 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 slope times x with exact rational arithmetic, not rounded decimals.
  • S166 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S165 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 index times width with exact rational arithmetic, not rounded decimals.
  • S164 · 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.
  • S163 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term22 with exact rational arithmetic, not rounded decimals.
  • S162 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 value times cell width with exact rational arithmetic, not rounded decimals.
  • S161 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S160 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 slope times x with exact rational arithmetic, not rounded decimals.
  • S159 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S158 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 index times width with exact rational arithmetic, not rounded decimals.
  • S157 · 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.
  • S156 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term21 with exact rational arithmetic, not rounded decimals.
  • S155 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 value times cell width with exact rational arithmetic, not rounded decimals.
  • S154 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S153 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 slope times x with exact rational arithmetic, not rounded decimals.
  • S152 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S151 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 index times width with exact rational arithmetic, not rounded decimals.
  • S150 · 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.
  • S149 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term20 with exact rational arithmetic, not rounded decimals.
  • S148 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 value times cell width with exact rational arithmetic, not rounded decimals.
  • S147 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S146 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 slope times x with exact rational arithmetic, not rounded decimals.
  • S145 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S144 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 index times width with exact rational arithmetic, not rounded decimals.
  • S143 · 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.
  • S142 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term19 with exact rational arithmetic, not rounded decimals.
  • S141 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 value times cell width with exact rational arithmetic, not rounded decimals.
  • S140 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S139 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 slope times x with exact rational arithmetic, not rounded decimals.
  • S138 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S137 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 index times width with exact rational arithmetic, not rounded decimals.
  • S136 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S135 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term18 with exact rational arithmetic, not rounded decimals.
  • S134 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 value times cell width with exact rational arithmetic, not rounded decimals.
  • S133 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S132 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 slope times x with exact rational arithmetic, not rounded decimals.
  • S131 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S130 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 index times width with exact rational arithmetic, not rounded decimals.
  • S129 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S128 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term17 with exact rational arithmetic, not rounded decimals.
  • S127 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 value times cell width with exact rational arithmetic, not rounded decimals.
  • S126 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S125 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 slope times x with exact rational arithmetic, not rounded decimals.
  • S124 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S123 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 index times width with exact rational arithmetic, not rounded decimals.
  • S122 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S121 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term16 with exact rational arithmetic, not rounded decimals.
  • S120 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 value times cell width with exact rational arithmetic, not rounded decimals.
  • S119 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S118 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 slope times x with exact rational arithmetic, not rounded decimals.
  • S117 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S116 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 index times width with exact rational arithmetic, not rounded decimals.
  • S115 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S114 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term15 with exact rational arithmetic, not rounded decimals.
  • S113 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 value times cell width with exact rational arithmetic, not rounded decimals.
  • S112 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S111 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 slope times x with exact rational arithmetic, not rounded decimals.
  • S110 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S109 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 index times width with exact rational arithmetic, not rounded decimals.
  • S108 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S107 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term14 with exact rational arithmetic, not rounded decimals.
  • S106 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 value times cell width with exact rational arithmetic, not rounded decimals.
  • S105 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S104 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 slope times x with exact rational arithmetic, not rounded decimals.
  • S103 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S102 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 index times width with exact rational arithmetic, not rounded decimals.
  • S101 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S100 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term13 with exact rational arithmetic, not rounded decimals.
  • S099 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 value times cell width with exact rational arithmetic, not rounded decimals.
  • S098 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S097 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 slope times x with exact rational arithmetic, not rounded decimals.
  • S096 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S095 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 index times width with exact rational arithmetic, not rounded decimals.
  • S094 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S093 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term12 with exact rational arithmetic, not rounded decimals.
  • S092 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 value times cell width with exact rational arithmetic, not rounded decimals.
  • S091 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S090 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 slope times x with exact rational arithmetic, not rounded decimals.
  • S089 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S088 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 index times width with exact rational arithmetic, not rounded decimals.
  • S087 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S086 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term11 with exact rational arithmetic, not rounded decimals.
  • S085 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 value times cell width with exact rational arithmetic, not rounded decimals.
  • S084 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S083 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 slope times x with exact rational arithmetic, not rounded decimals.
  • S082 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S081 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 index times width with exact rational arithmetic, not rounded decimals.
  • S080 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S079 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term10 with exact rational arithmetic, not rounded decimals.
  • S078 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 value times cell width with exact rational arithmetic, not rounded decimals.
  • S077 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S076 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 slope times x with exact rational arithmetic, not rounded decimals.
  • S075 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S074 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 index times width with exact rational arithmetic, not rounded decimals.
  • S073 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S072 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term9 with exact rational arithmetic, not rounded decimals.
  • S071 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 value times cell width with exact rational arithmetic, not rounded decimals.
  • S070 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S069 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 slope times x with exact rational arithmetic, not rounded decimals.
  • S068 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S067 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 index times width with exact rational arithmetic, not rounded decimals.
  • S066 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S065 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term8 with exact rational arithmetic, not rounded decimals.
  • S064 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 value times cell width with exact rational arithmetic, not rounded decimals.
  • S063 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S062 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 slope times x with exact rational arithmetic, not rounded decimals.
  • S061 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S060 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 index times width with exact rational arithmetic, not rounded decimals.
  • S059 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S058 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term7 with exact rational arithmetic, not rounded decimals.
  • S057 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 value times cell width with exact rational arithmetic, not rounded decimals.
  • S056 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S055 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 slope times x with exact rational arithmetic, not rounded decimals.
  • S054 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S053 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 index times width with exact rational arithmetic, not rounded decimals.
  • S052 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S051 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term6 with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 value times cell width with exact rational arithmetic, not rounded decimals.
  • S049 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S048 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 slope times x with exact rational arithmetic, not rounded decimals.
  • S047 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 index times width with exact rational arithmetic, not rounded decimals.
  • S045 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S044 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term5 with exact rational arithmetic, not rounded decimals.
  • S043 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 value times cell width with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 slope times x with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 index times width with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S037 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term4 with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 value times cell width with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 slope times x with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 index times width with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S030 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term3 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 value times cell width with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 slope times x with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 index times width with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S023 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term2 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 value times cell width with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 slope times x with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 index times width with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S016 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 value times cell width with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 slope times x with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 index times width with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 value times cell width with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 affine value plus c with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 slope times x with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 lower plus offset with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 index times width with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Left sum ORIGINAL width over positive cell count with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Left sum ORIGINAL upper minus lower endpoint with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

-236/1

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: 277bd8619d17823b71393f4770726c7b7b74536e60b6d6b02420b9d49c772df8
  • candidate record SHA-256: 166b756025214e8fffaf49801cd4ad74ba3cdca7dc8e72c86ed871df21d107f7
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON