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Question 32622: 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

R0064-32622 · q-20261007-advance-20261008-r07-definite-integral-0097

Original question and givens

On all R, f(x)=(-22)x+(9). For a=-1≤b=1 and N=22 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)=(-22)x+(9). For a=-1≤b=1 and N=22 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(1, -1)

After

Left sum ORIGINAL upper minus lower endpoint = 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, S001

S003 · calculation

Before

Left sum ORIGINAL width over positive cell count := div(2, 22)

After

Left sum ORIGINAL width over positive cell count = 1/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Left sum ORIGINAL sample0 index times width := mul(0, 1/11)

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(-22, -1)

After

Left sum ORIGINAL sample0 slope times x = 22

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Left sum ORIGINAL sample0 affine value plus c := sum(22, 9)

After

Left sum ORIGINAL sample0 affine value plus c = 31

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(31, 1/11)

After

Left sum ORIGINAL sample0 value times cell width = 31/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Left sum ORIGINAL accumulated term0 := sum(0, 31/11)

After

Left sum ORIGINAL accumulated term0 = 31/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [0, 22, -22, 9, -1, 1, 1/11]

After

derived = [-1/1, 31/1, 31/11, 31/11]

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, 1/11)

After

Left sum ORIGINAL sample1 index times width = 1/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Left sum ORIGINAL sample1 lower plus offset := sum(-1, 1/11)

After

Left sum ORIGINAL sample1 lower plus offset = -10/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Left sum ORIGINAL sample1 slope times x := mul(-22, -10/11)

After

Left sum ORIGINAL sample1 slope times x = 20

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(20, 9)

After

Left sum ORIGINAL sample1 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, S013

S015 · calculation

Before

Left sum ORIGINAL sample1 value times cell width := mul(29, 1/11)

After

Left sum ORIGINAL sample1 value times cell width = 29/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Left sum ORIGINAL accumulated term1 := sum(31/11, 29/11)

After

Left sum ORIGINAL accumulated term1 = 60/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [1, 22, -22, 9, -1, 1, 1/11]

After

derived = [-10/11, 29/1, 29/11, 60/11]

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, 1/11)

After

Left sum ORIGINAL sample2 index times width = 2/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Left sum ORIGINAL sample2 lower plus offset := sum(-1, 2/11)

After

Left sum ORIGINAL sample2 lower plus offset = -9/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Left sum ORIGINAL sample2 slope times x := mul(-22, -9/11)

After

Left sum ORIGINAL sample2 slope times x = 18

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Left sum ORIGINAL sample2 affine value plus c := sum(18, 9)

After

Left sum ORIGINAL sample2 affine value plus c = 27

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(27, 1/11)

After

Left sum ORIGINAL sample2 value times cell width = 27/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Left sum ORIGINAL accumulated term2 := sum(60/11, 27/11)

After

Left sum ORIGINAL accumulated term2 = 87/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

source roles = [2, 22, -22, 9, -1, 1, 1/11]

After

derived = [-9/11, 27/1, 27/11, 87/11]

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, 1/11)

After

Left sum ORIGINAL sample3 index times width = 3/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Left sum ORIGINAL sample3 lower plus offset := sum(-1, 3/11)

After

Left sum ORIGINAL sample3 lower plus offset = -8/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Left sum ORIGINAL sample3 slope times x := mul(-22, -8/11)

After

Left sum ORIGINAL sample3 slope times x = 16

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(16, 9)

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, 1/11)

After

Left sum ORIGINAL sample3 value times cell width = 25/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Left sum ORIGINAL accumulated term3 := sum(87/11, 25/11)

After

Left sum ORIGINAL accumulated term3 = 112/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

source roles = [3, 22, -22, 9, -1, 1, 1/11]

After

derived = [-8/11, 25/1, 25/11, 112/11]

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, 1/11)

After

Left sum ORIGINAL sample4 index times width = 4/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Left sum ORIGINAL sample4 lower plus offset := sum(-1, 4/11)

After

Left sum ORIGINAL sample4 lower plus offset = -7/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Left sum ORIGINAL sample4 slope times x := mul(-22, -7/11)

After

Left sum ORIGINAL sample4 slope times x = 14

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Left sum ORIGINAL sample4 affine value plus c := sum(14, 9)

After

Left sum ORIGINAL sample4 affine value plus c = 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, S034

S036 · calculation

Before

Left sum ORIGINAL sample4 value times cell width := mul(23, 1/11)

After

Left sum ORIGINAL sample4 value times cell width = 23/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Left sum ORIGINAL accumulated term4 := sum(112/11, 23/11)

After

Left sum ORIGINAL accumulated term4 = 135/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

source roles = [4, 22, -22, 9, -1, 1, 1/11]

After

derived = [-7/11, 23/1, 23/11, 135/11]

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, 1/11)

After

Left sum ORIGINAL sample5 index times width = 5/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Left sum ORIGINAL sample5 lower plus offset := sum(-1, 5/11)

After

Left sum ORIGINAL sample5 lower plus offset = -6/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Left sum ORIGINAL sample5 slope times x := mul(-22, -6/11)

After

Left sum ORIGINAL sample5 slope times x = 12

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(12, 9)

After

Left sum ORIGINAL sample5 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, S041

S043 · calculation

Before

Left sum ORIGINAL sample5 value times cell width := mul(21, 1/11)

After

Left sum ORIGINAL sample5 value times cell width = 21/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

Left sum ORIGINAL accumulated term5 := sum(135/11, 21/11)

After

Left sum ORIGINAL accumulated term5 = 156/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

source roles = [5, 22, -22, 9, -1, 1, 1/11]

After

derived = [-6/11, 21/1, 21/11, 156/11]

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, 1/11)

After

Left sum ORIGINAL sample6 index times width = 6/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

Left sum ORIGINAL sample6 lower plus offset := sum(-1, 6/11)

After

Left sum ORIGINAL sample6 lower plus offset = -5/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

Left sum ORIGINAL sample6 slope times x := mul(-22, -5/11)

After

Left sum ORIGINAL sample6 slope times x = 10

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Left sum ORIGINAL sample6 affine value plus c := sum(10, 9)

After

Left sum ORIGINAL sample6 affine value plus c = 19

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(19, 1/11)

After

Left sum ORIGINAL sample6 value times cell width = 19/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

Left sum ORIGINAL accumulated term6 := sum(156/11, 19/11)

After

Left sum ORIGINAL accumulated term6 = 175/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

source roles = [6, 22, -22, 9, -1, 1, 1/11]

After

derived = [-5/11, 19/1, 19/11, 175/11]

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, 1/11)

After

Left sum ORIGINAL sample7 index times width = 7/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

Left sum ORIGINAL sample7 lower plus offset := sum(-1, 7/11)

After

Left sum ORIGINAL sample7 lower plus offset = -4/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

Left sum ORIGINAL sample7 slope times x := mul(-22, -4/11)

After

Left sum ORIGINAL sample7 slope times x = 8

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(8, 9)

After

Left sum ORIGINAL sample7 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, S055

S057 · calculation

Before

Left sum ORIGINAL sample7 value times cell width := mul(17, 1/11)

After

Left sum ORIGINAL sample7 value times cell width = 17/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

Left sum ORIGINAL accumulated term7 := sum(175/11, 17/11)

After

Left sum ORIGINAL accumulated term7 = 192/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

source roles = [7, 22, -22, 9, -1, 1, 1/11]

After

derived = [-4/11, 17/1, 17/11, 192/11]

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, 1/11)

After

Left sum ORIGINAL sample8 index times width = 8/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

Left sum ORIGINAL sample8 lower plus offset := sum(-1, 8/11)

After

Left sum ORIGINAL sample8 lower plus offset = -3/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

Left sum ORIGINAL sample8 slope times x := mul(-22, -3/11)

After

Left sum ORIGINAL sample8 slope times x = 6

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

Left sum ORIGINAL sample8 affine value plus c := sum(6, 9)

After

Left sum ORIGINAL sample8 affine value plus c = 15

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

Left sum ORIGINAL sample8 value times cell width := mul(15, 1/11)

After

Left sum ORIGINAL sample8 value times cell width = 15/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

Left sum ORIGINAL accumulated term8 := sum(192/11, 15/11)

After

Left sum ORIGINAL accumulated term8 = 207/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

source roles = [8, 22, -22, 9, -1, 1, 1/11]

After

derived = [-3/11, 15/1, 15/11, 207/11]

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, 1/11)

After

Left sum ORIGINAL sample9 index times width = 9/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

Left sum ORIGINAL sample9 lower plus offset := sum(-1, 9/11)

After

Left sum ORIGINAL sample9 lower plus offset = -2/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

Left sum ORIGINAL sample9 slope times x := mul(-22, -2/11)

After

Left sum ORIGINAL sample9 slope times x = 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, S068

S070 · calculation

Before

Left sum ORIGINAL sample9 affine value plus c := sum(4, 9)

After

Left sum ORIGINAL sample9 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, S069

S071 · calculation

Before

Left sum ORIGINAL sample9 value times cell width := mul(13, 1/11)

After

Left sum ORIGINAL sample9 value times cell width = 13/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

Left sum ORIGINAL accumulated term9 := sum(207/11, 13/11)

After

Left sum ORIGINAL accumulated term9 = 20

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, 22, -22, 9, -1, 1, 1/11]

After

derived = [-2/11, 13/1, 13/11, 20/1]

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, S072

S074 · calculation

Before

Left sum ORIGINAL sample10 index times width := mul(10, 1/11)

After

Left sum ORIGINAL sample10 index times width = 10/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · calculation

Before

Left sum ORIGINAL sample10 lower plus offset := sum(-1, 10/11)

After

Left sum ORIGINAL sample10 lower plus offset = -1/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

Left sum ORIGINAL sample10 slope times x := mul(-22, -1/11)

After

Left sum ORIGINAL sample10 slope times x = 2

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · calculation

Before

Left sum ORIGINAL sample10 affine value plus c := sum(2, 9)

After

Left sum ORIGINAL sample10 affine value plus c = 11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · calculation

Before

Left sum ORIGINAL sample10 value times cell width := mul(11, 1/11)

After

Left sum ORIGINAL sample10 value times cell 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, S077

S079 · calculation

Before

Left sum ORIGINAL accumulated term10 := sum(20, 1)

After

Left sum ORIGINAL accumulated term10 = 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, S078

S080 · implies

Before

source roles = [10, 22, -22, 9, -1, 1, 1/11]

After

derived = [-1/11, 11/1, 1/1, 21/1]

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, S079

S081 · calculation

Before

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

After

Left sum ORIGINAL sample11 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, S080

S082 · calculation

Before

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

After

Left sum ORIGINAL sample11 lower plus offset = 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, S081

S083 · calculation

Before

Left sum ORIGINAL sample11 slope times x := mul(-22, 0)

After

Left sum ORIGINAL sample11 slope times x = 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, S082

S084 · calculation

Before

Left sum ORIGINAL sample11 affine value plus c := sum(0, 9)

After

Left sum ORIGINAL sample11 affine value plus c = 9

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · calculation

Before

Left sum ORIGINAL sample11 value times cell width := mul(9, 1/11)

After

Left sum ORIGINAL sample11 value times cell width = 9/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S084

S086 · calculation

Before

Left sum ORIGINAL accumulated term11 := sum(21, 9/11)

After

Left sum ORIGINAL accumulated term11 = 240/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S085

S087 · implies

Before

source roles = [11, 22, -22, 9, -1, 1, 1/11]

After

derived = [0/1, 9/1, 9/11, 240/11]

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, 1/11)

After

Left sum ORIGINAL sample12 index times width = 12/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S087

S089 · calculation

Before

Left sum ORIGINAL sample12 lower plus offset := sum(-1, 12/11)

After

Left sum ORIGINAL sample12 lower plus offset = 1/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · calculation

Before

Left sum ORIGINAL sample12 slope times x := mul(-22, 1/11)

After

Left sum ORIGINAL sample12 slope times x = -2

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S089

S091 · calculation

Before

Left sum ORIGINAL sample12 affine value plus c := sum(-2, 9)

After

Left sum ORIGINAL sample12 affine value plus c = 7

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S090

S092 · calculation

Before

Left sum ORIGINAL sample12 value times cell width := mul(7, 1/11)

After

Left sum ORIGINAL sample12 value times cell width = 7/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S091

S093 · calculation

Before

Left sum ORIGINAL accumulated term12 := sum(240/11, 7/11)

After

Left sum ORIGINAL accumulated term12 = 247/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · implies

Before

source roles = [12, 22, -22, 9, -1, 1, 1/11]

After

derived = [1/11, 7/1, 7/11, 247/11]

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, 1/11)

After

Left sum ORIGINAL sample13 index times width = 13/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S094

S096 · calculation

Before

Left sum ORIGINAL sample13 lower plus offset := sum(-1, 13/11)

After

Left sum ORIGINAL sample13 lower plus offset = 2/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S095

S097 · calculation

Before

Left sum ORIGINAL sample13 slope times x := mul(-22, 2/11)

After

Left sum ORIGINAL sample13 slope times x = -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, S096

S098 · calculation

Before

Left sum ORIGINAL sample13 affine value plus c := sum(-4, 9)

After

Left sum ORIGINAL sample13 affine value plus c = 5

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S097

S099 · calculation

Before

Left sum ORIGINAL sample13 value times cell width := mul(5, 1/11)

After

Left sum ORIGINAL sample13 value times cell width = 5/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · calculation

Before

Left sum ORIGINAL accumulated term13 := sum(247/11, 5/11)

After

Left sum ORIGINAL accumulated term13 = 252/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S099

S101 · implies

Before

source roles = [13, 22, -22, 9, -1, 1, 1/11]

After

derived = [2/11, 5/1, 5/11, 252/11]

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, 1/11)

After

Left sum ORIGINAL sample14 index times width = 14/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S101

S103 · calculation

Before

Left sum ORIGINAL sample14 lower plus offset := sum(-1, 14/11)

After

Left sum ORIGINAL sample14 lower plus offset = 3/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S102

S104 · calculation

Before

Left sum ORIGINAL sample14 slope times x := mul(-22, 3/11)

After

Left sum ORIGINAL sample14 slope times x = -6

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S103

S105 · calculation

Before

Left sum ORIGINAL sample14 affine value plus c := sum(-6, 9)

After

Left sum ORIGINAL sample14 affine value plus c = 3

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · calculation

Before

Left sum ORIGINAL sample14 value times cell width := mul(3, 1/11)

After

Left sum ORIGINAL sample14 value times cell width = 3/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S105

S107 · calculation

Before

Left sum ORIGINAL accumulated term14 := sum(252/11, 3/11)

After

Left sum ORIGINAL accumulated term14 = 255/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S106

S108 · implies

Before

source roles = [14, 22, -22, 9, -1, 1, 1/11]

After

derived = [3/11, 3/1, 3/11, 255/11]

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, 1/11)

After

Left sum ORIGINAL sample15 index times width = 15/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S108

S110 · calculation

Before

Left sum ORIGINAL sample15 lower plus offset := sum(-1, 15/11)

After

Left sum ORIGINAL sample15 lower plus offset = 4/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S109

S111 · calculation

Before

Left sum ORIGINAL sample15 slope times x := mul(-22, 4/11)

After

Left sum ORIGINAL sample15 slope times x = -8

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(-8, 9)

After

Left sum ORIGINAL sample15 affine value plus c = 1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S111

S113 · calculation

Before

Left sum ORIGINAL sample15 value times cell width := mul(1, 1/11)

After

Left sum ORIGINAL sample15 value times cell width = 1/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S112

S114 · calculation

Before

Left sum ORIGINAL accumulated term15 := sum(255/11, 1/11)

After

Left sum ORIGINAL accumulated term15 = 256/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S113

S115 · implies

Before

source roles = [15, 22, -22, 9, -1, 1, 1/11]

After

derived = [4/11, 1/1, 1/11, 256/11]

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, 1/11)

After

Left sum ORIGINAL sample16 index times width = 16/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S115

S117 · calculation

Before

Left sum ORIGINAL sample16 lower plus offset := sum(-1, 16/11)

After

Left sum ORIGINAL sample16 lower plus offset = 5/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S116

S118 · calculation

Before

Left sum ORIGINAL sample16 slope times x := mul(-22, 5/11)

After

Left sum ORIGINAL sample16 slope times x = -10

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S117

S119 · calculation

Before

Left sum ORIGINAL sample16 affine value plus c := sum(-10, 9)

After

Left sum ORIGINAL sample16 affine value plus c = -1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S118

S120 · calculation

Before

Left sum ORIGINAL sample16 value times cell width := mul(-1, 1/11)

After

Left sum ORIGINAL sample16 value times cell width = -1/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S119

S121 · calculation

Before

Left sum ORIGINAL accumulated term16 := sum(256/11, -1/11)

After

Left sum ORIGINAL accumulated term16 = 255/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S120

S122 · implies

Before

source roles = [16, 22, -22, 9, -1, 1, 1/11]

After

derived = [5/11, -1/1, -1/11, 255/11]

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, 1/11)

After

Left sum ORIGINAL sample17 index times width = 17/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S122

S124 · calculation

Before

Left sum ORIGINAL sample17 lower plus offset := sum(-1, 17/11)

After

Left sum ORIGINAL sample17 lower plus offset = 6/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S123

S125 · calculation

Before

Left sum ORIGINAL sample17 slope times x := mul(-22, 6/11)

After

Left sum ORIGINAL sample17 slope times x = -12

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(-12, 9)

After

Left sum ORIGINAL sample17 affine value plus c = -3

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S125

S127 · calculation

Before

Left sum ORIGINAL sample17 value times cell width := mul(-3, 1/11)

After

Left sum ORIGINAL sample17 value times cell width = -3/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S126

S128 · calculation

Before

Left sum ORIGINAL accumulated term17 := sum(255/11, -3/11)

After

Left sum ORIGINAL accumulated term17 = 252/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S127

S129 · implies

Before

source roles = [17, 22, -22, 9, -1, 1, 1/11]

After

derived = [6/11, -3/1, -3/11, 252/11]

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, 1/11)

After

Left sum ORIGINAL sample18 index times width = 18/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S129

S131 · calculation

Before

Left sum ORIGINAL sample18 lower plus offset := sum(-1, 18/11)

After

Left sum ORIGINAL sample18 lower plus offset = 7/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S130

S132 · calculation

Before

Left sum ORIGINAL sample18 slope times x := mul(-22, 7/11)

After

Left sum ORIGINAL sample18 slope times x = -14

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S131

S133 · calculation

Before

Left sum ORIGINAL sample18 affine value plus c := sum(-14, 9)

After

Left sum ORIGINAL sample18 affine value plus c = -5

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S132

S134 · calculation

Before

Left sum ORIGINAL sample18 value times cell width := mul(-5, 1/11)

After

Left sum ORIGINAL sample18 value times cell width = -5/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S133

S135 · calculation

Before

Left sum ORIGINAL accumulated term18 := sum(252/11, -5/11)

After

Left sum ORIGINAL accumulated term18 = 247/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S134

S136 · implies

Before

source roles = [18, 22, -22, 9, -1, 1, 1/11]

After

derived = [7/11, -5/1, -5/11, 247/11]

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, 1/11)

After

Left sum ORIGINAL sample19 index times width = 19/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S136

S138 · calculation

Before

Left sum ORIGINAL sample19 lower plus offset := sum(-1, 19/11)

After

Left sum ORIGINAL sample19 lower plus offset = 8/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S137

S139 · calculation

Before

Left sum ORIGINAL sample19 slope times x := mul(-22, 8/11)

After

Left sum ORIGINAL sample19 slope times x = -16

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(-16, 9)

After

Left sum ORIGINAL sample19 affine value plus c = -7

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S139

S141 · calculation

Before

Left sum ORIGINAL sample19 value times cell width := mul(-7, 1/11)

After

Left sum ORIGINAL sample19 value times cell width = -7/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S140

S142 · calculation

Before

Left sum ORIGINAL accumulated term19 := sum(247/11, -7/11)

After

Left sum ORIGINAL accumulated term19 = 240/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S141

S143 · implies

Before

source roles = [19, 22, -22, 9, -1, 1, 1/11]

After

derived = [8/11, -7/1, -7/11, 240/11]

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, 1/11)

After

Left sum ORIGINAL sample20 index times width = 20/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S143

S145 · calculation

Before

Left sum ORIGINAL sample20 lower plus offset := sum(-1, 20/11)

After

Left sum ORIGINAL sample20 lower plus offset = 9/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S144

S146 · calculation

Before

Left sum ORIGINAL sample20 slope times x := mul(-22, 9/11)

After

Left sum ORIGINAL sample20 slope times x = -18

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S145

S147 · calculation

Before

Left sum ORIGINAL sample20 affine value plus c := sum(-18, 9)

After

Left sum ORIGINAL sample20 affine value plus c = -9

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S146

S148 · calculation

Before

Left sum ORIGINAL sample20 value times cell width := mul(-9, 1/11)

After

Left sum ORIGINAL sample20 value times cell width = -9/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S147

S149 · calculation

Before

Left sum ORIGINAL accumulated term20 := sum(240/11, -9/11)

After

Left sum ORIGINAL accumulated term20 = 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, S148

S150 · implies

Before

source roles = [20, 22, -22, 9, -1, 1, 1/11]

After

derived = [9/11, -9/1, -9/11, 21/1]

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, S149

S151 · calculation

Before

Left sum ORIGINAL sample21 index times width := mul(21, 1/11)

After

Left sum ORIGINAL sample21 index times width = 21/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S150

S152 · calculation

Before

Left sum ORIGINAL sample21 lower plus offset := sum(-1, 21/11)

After

Left sum ORIGINAL sample21 lower plus offset = 10/11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S151

S153 · calculation

Before

Left sum ORIGINAL sample21 slope times x := mul(-22, 10/11)

After

Left sum ORIGINAL sample21 slope times x = -20

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(-20, 9)

After

Left sum ORIGINAL sample21 affine value plus c = -11

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S153

S155 · calculation

Before

Left sum ORIGINAL sample21 value times cell width := mul(-11, 1/11)

After

Left sum ORIGINAL sample21 value times cell 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, S154

S156 · calculation

Before

Left sum ORIGINAL accumulated term21 := sum(21, -1)

After

Left sum ORIGINAL accumulated term21 = 20

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, 22, -22, 9, -1, 1, 1/11]

After

derived = [10/11, -11/1, -1/1, 20/1]

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, S156

S158 · implies

Before

source roles = [22, -22, 9, -1, 1]

After

derived = [1/11, 20/1]

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, S157

Reverse checks

  • S158 · 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.
  • 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

20/1

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: 6dbf2ec2ff8e843428932c9e9c706428a4b5a4139f4631aec3784df1021f5934
  • candidate record SHA-256: dc9f46683c59d88d605d38fca0075114ee20255b0d6c9d5cd1979d85402fd483
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON