# Question 32619: Finite Left-Endpoint Sum of an Affine Function

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0064-32619 · q-20261007-advance-20261008-r07-definite-integral-0094



## Original question and givens

On all R, f(x)=(19)x+(6). For a=0≤b=4 and N=19 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)=(19)x+(6). For a=0≤b=4 and N=19 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, retaining /1 for integers.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-definite-integral](https://ocme.evemisslab.com/en/mko/mko-definite-integral/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001





S002 · calculationBefore

Left sum ORIGINAL upper minus lower endpoint := sub(4, 0)

After

Left sum ORIGINAL upper minus lower endpoint = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Left sum ORIGINAL width over positive cell count := div(4, 19)

After

Left sum ORIGINAL width over positive cell count = 4/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

Left sum ORIGINAL sample0 index times width := mul(0, 4/19)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

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

After

Left sum ORIGINAL sample0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

Left sum ORIGINAL sample0 slope times x := mul(19, 0)

After

Left sum ORIGINAL sample0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

Left sum ORIGINAL sample0 affine value plus c := sum(0, 6)

After

Left sum ORIGINAL sample0 affine value plus c = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

Left sum ORIGINAL sample0 value times cell width := mul(6, 4/19)

After

Left sum ORIGINAL sample0 value times cell width = 24/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

Left sum ORIGINAL accumulated term0 := sum(0, 24/19)

After

Left sum ORIGINAL accumulated term0 = 24/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

source roles = [0, 19, 19, 6, 0, 4, 4/19]

After

derived = [0/1, 6/1, 24/19, 24/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S009





S011 · calculationBefore

Left sum ORIGINAL sample1 index times width := mul(1, 4/19)

After

Left sum ORIGINAL sample1 index times width = 4/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

Left sum ORIGINAL sample1 lower plus offset := sum(0, 4/19)

After

Left sum ORIGINAL sample1 lower plus offset = 4/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

Left sum ORIGINAL sample1 slope times x := mul(19, 4/19)

After

Left sum ORIGINAL sample1 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

Left sum ORIGINAL sample1 affine value plus c := sum(4, 6)

After

Left sum ORIGINAL sample1 affine value plus c = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

Left sum ORIGINAL sample1 value times cell width := mul(10, 4/19)

After

Left sum ORIGINAL sample1 value times cell width = 40/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

Left sum ORIGINAL accumulated term1 := sum(24/19, 40/19)

After

Left sum ORIGINAL accumulated term1 = 64/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · impliesBefore

source roles = [1, 19, 19, 6, 0, 4, 4/19]

After

derived = [4/19, 10/1, 40/19, 64/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S016





S018 · calculationBefore

Left sum ORIGINAL sample2 index times width := mul(2, 4/19)

After

Left sum ORIGINAL sample2 index times width = 8/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

Left sum ORIGINAL sample2 lower plus offset := sum(0, 8/19)

After

Left sum ORIGINAL sample2 lower plus offset = 8/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

Left sum ORIGINAL sample2 slope times x := mul(19, 8/19)

After

Left sum ORIGINAL sample2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

Left sum ORIGINAL sample2 affine value plus c := sum(8, 6)

After

Left sum ORIGINAL sample2 affine value plus c = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

Left sum ORIGINAL sample2 value times cell width := mul(14, 4/19)

After

Left sum ORIGINAL sample2 value times cell width = 56/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · calculationBefore

Left sum ORIGINAL accumulated term2 := sum(64/19, 56/19)

After

Left sum ORIGINAL accumulated term2 = 120/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · impliesBefore

source roles = [2, 19, 19, 6, 0, 4, 4/19]

After

derived = [8/19, 14/1, 56/19, 120/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S023





S025 · calculationBefore

Left sum ORIGINAL sample3 index times width := mul(3, 4/19)

After

Left sum ORIGINAL sample3 index times width = 12/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · calculationBefore

Left sum ORIGINAL sample3 lower plus offset := sum(0, 12/19)

After

Left sum ORIGINAL sample3 lower plus offset = 12/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · calculationBefore

Left sum ORIGINAL sample3 slope times x := mul(19, 12/19)

After

Left sum ORIGINAL sample3 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · calculationBefore

Left sum ORIGINAL sample3 affine value plus c := sum(12, 6)

After

Left sum ORIGINAL sample3 affine value plus c = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

Left sum ORIGINAL sample3 value times cell width := mul(18, 4/19)

After

Left sum ORIGINAL sample3 value times cell width = 72/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · calculationBefore

Left sum ORIGINAL accumulated term3 := sum(120/19, 72/19)

After

Left sum ORIGINAL accumulated term3 = 192/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S029





S031 · impliesBefore

source roles = [3, 19, 19, 6, 0, 4, 4/19]

After

derived = [12/19, 18/1, 72/19, 192/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S030





S032 · calculationBefore

Left sum ORIGINAL sample4 index times width := mul(4, 4/19)

After

Left sum ORIGINAL sample4 index times width = 16/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · calculationBefore

Left sum ORIGINAL sample4 lower plus offset := sum(0, 16/19)

After

Left sum ORIGINAL sample4 lower plus offset = 16/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · calculationBefore

Left sum ORIGINAL sample4 slope times x := mul(19, 16/19)

After

Left sum ORIGINAL sample4 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S033





S035 · calculationBefore

Left sum ORIGINAL sample4 affine value plus c := sum(16, 6)

After

Left sum ORIGINAL sample4 affine value plus c = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · calculationBefore

Left sum ORIGINAL sample4 value times cell width := mul(22, 4/19)

After

Left sum ORIGINAL sample4 value times cell width = 88/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · calculationBefore

Left sum ORIGINAL accumulated term4 := sum(192/19, 88/19)

After

Left sum ORIGINAL accumulated term4 = 280/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

source roles = [4, 19, 19, 6, 0, 4, 4/19]

After

derived = [16/19, 22/1, 88/19, 280/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S037





S039 · calculationBefore

Left sum ORIGINAL sample5 index times width := mul(5, 4/19)

After

Left sum ORIGINAL sample5 index times width = 20/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · calculationBefore

Left sum ORIGINAL sample5 lower plus offset := sum(0, 20/19)

After

Left sum ORIGINAL sample5 lower plus offset = 20/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · calculationBefore

Left sum ORIGINAL sample5 slope times x := mul(19, 20/19)

After

Left sum ORIGINAL sample5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · calculationBefore

Left sum ORIGINAL sample5 affine value plus c := sum(20, 6)

After

Left sum ORIGINAL sample5 affine value plus c = 26

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041





S043 · calculationBefore

Left sum ORIGINAL sample5 value times cell width := mul(26, 4/19)

After

Left sum ORIGINAL sample5 value times cell width = 104/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · calculationBefore

Left sum ORIGINAL accumulated term5 := sum(280/19, 104/19)

After

Left sum ORIGINAL accumulated term5 = 384/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S043





S045 · impliesBefore

source roles = [5, 19, 19, 6, 0, 4, 4/19]

After

derived = [20/19, 26/1, 104/19, 384/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S044





S046 · calculationBefore

Left sum ORIGINAL sample6 index times width := mul(6, 4/19)

After

Left sum ORIGINAL sample6 index times width = 24/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S045





S047 · calculationBefore

Left sum ORIGINAL sample6 lower plus offset := sum(0, 24/19)

After

Left sum ORIGINAL sample6 lower plus offset = 24/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · calculationBefore

Left sum ORIGINAL sample6 slope times x := mul(19, 24/19)

After

Left sum ORIGINAL sample6 slope times x = 24

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047





S049 · calculationBefore

Left sum ORIGINAL sample6 affine value plus c := sum(24, 6)

After

Left sum ORIGINAL sample6 affine value plus c = 30

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · calculationBefore

Left sum ORIGINAL sample6 value times cell width := mul(30, 4/19)

After

Left sum ORIGINAL sample6 value times cell width = 120/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S049





S051 · calculationBefore

Left sum ORIGINAL accumulated term6 := sum(384/19, 120/19)

After

Left sum ORIGINAL accumulated term6 = 504/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S050





S052 · impliesBefore

source roles = [6, 19, 19, 6, 0, 4, 4/19]

After

derived = [24/19, 30/1, 120/19, 504/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S051





S053 · calculationBefore

Left sum ORIGINAL sample7 index times width := mul(7, 4/19)

After

Left sum ORIGINAL sample7 index times width = 28/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S052





S054 · calculationBefore

Left sum ORIGINAL sample7 lower plus offset := sum(0, 28/19)

After

Left sum ORIGINAL sample7 lower plus offset = 28/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S053





S055 · calculationBefore

Left sum ORIGINAL sample7 slope times x := mul(19, 28/19)

After

Left sum ORIGINAL sample7 slope times x = 28

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054





S056 · calculationBefore

Left sum ORIGINAL sample7 affine value plus c := sum(28, 6)

After

Left sum ORIGINAL sample7 affine value plus c = 34

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055





S057 · calculationBefore

Left sum ORIGINAL sample7 value times cell width := mul(34, 4/19)

After

Left sum ORIGINAL sample7 value times cell width = 136/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S056





S058 · calculationBefore

Left sum ORIGINAL accumulated term7 := sum(504/19, 136/19)

After

Left sum ORIGINAL accumulated term7 = 640/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S057





S059 · impliesBefore

source roles = [7, 19, 19, 6, 0, 4, 4/19]

After

derived = [28/19, 34/1, 136/19, 640/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S058





S060 · calculationBefore

Left sum ORIGINAL sample8 index times width := mul(8, 4/19)

After

Left sum ORIGINAL sample8 index times width = 32/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S059





S061 · calculationBefore

Left sum ORIGINAL sample8 lower plus offset := sum(0, 32/19)

After

Left sum ORIGINAL sample8 lower plus offset = 32/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S060





S062 · calculationBefore

Left sum ORIGINAL sample8 slope times x := mul(19, 32/19)

After

Left sum ORIGINAL sample8 slope times x = 32

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061





S063 · calculationBefore

Left sum ORIGINAL sample8 affine value plus c := sum(32, 6)

After

Left sum ORIGINAL sample8 affine value plus c = 38

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062





S064 · calculationBefore

Left sum ORIGINAL sample8 value times cell width := mul(38, 4/19)

After

Left sum ORIGINAL sample8 value times cell width = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S063





S065 · calculationBefore

Left sum ORIGINAL accumulated term8 := sum(640/19, 8)

After

Left sum ORIGINAL accumulated term8 = 792/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S064





S066 · impliesBefore

source roles = [8, 19, 19, 6, 0, 4, 4/19]

After

derived = [32/19, 38/1, 8/1, 792/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S065





S067 · calculationBefore

Left sum ORIGINAL sample9 index times width := mul(9, 4/19)

After

Left sum ORIGINAL sample9 index times width = 36/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S066





S068 · calculationBefore

Left sum ORIGINAL sample9 lower plus offset := sum(0, 36/19)

After

Left sum ORIGINAL sample9 lower plus offset = 36/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S067





S069 · calculationBefore

Left sum ORIGINAL sample9 slope times x := mul(19, 36/19)

After

Left sum ORIGINAL sample9 slope times x = 36

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068





S070 · calculationBefore

Left sum ORIGINAL sample9 affine value plus c := sum(36, 6)

After

Left sum ORIGINAL sample9 affine value plus c = 42

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069





S071 · calculationBefore

Left sum ORIGINAL sample9 value times cell width := mul(42, 4/19)

After

Left sum ORIGINAL sample9 value times cell width = 168/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S070





S072 · calculationBefore

Left sum ORIGINAL accumulated term9 := sum(792/19, 168/19)

After

Left sum ORIGINAL accumulated term9 = 960/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S071





S073 · impliesBefore

source roles = [9, 19, 19, 6, 0, 4, 4/19]

After

derived = [36/19, 42/1, 168/19, 960/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S072





S074 · calculationBefore

Left sum ORIGINAL sample10 index times width := mul(10, 4/19)

After

Left sum ORIGINAL sample10 index times width = 40/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S073





S075 · calculationBefore

Left sum ORIGINAL sample10 lower plus offset := sum(0, 40/19)

After

Left sum ORIGINAL sample10 lower plus offset = 40/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S074





S076 · calculationBefore

Left sum ORIGINAL sample10 slope times x := mul(19, 40/19)

After

Left sum ORIGINAL sample10 slope times x = 40

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075





S077 · calculationBefore

Left sum ORIGINAL sample10 affine value plus c := sum(40, 6)

After

Left sum ORIGINAL sample10 affine value plus c = 46

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076





S078 · calculationBefore

Left sum ORIGINAL sample10 value times cell width := mul(46, 4/19)

After

Left sum ORIGINAL sample10 value times cell width = 184/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S077





S079 · calculationBefore

Left sum ORIGINAL accumulated term10 := sum(960/19, 184/19)

After

Left sum ORIGINAL accumulated term10 = 1144/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S078





S080 · impliesBefore

source roles = [10, 19, 19, 6, 0, 4, 4/19]

After

derived = [40/19, 46/1, 184/19, 1144/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S079





S081 · calculationBefore

Left sum ORIGINAL sample11 index times width := mul(11, 4/19)

After

Left sum ORIGINAL sample11 index times width = 44/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S080





S082 · calculationBefore

Left sum ORIGINAL sample11 lower plus offset := sum(0, 44/19)

After

Left sum ORIGINAL sample11 lower plus offset = 44/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S081





S083 · calculationBefore

Left sum ORIGINAL sample11 slope times x := mul(19, 44/19)

After

Left sum ORIGINAL sample11 slope times x = 44

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082





S084 · calculationBefore

Left sum ORIGINAL sample11 affine value plus c := sum(44, 6)

After

Left sum ORIGINAL sample11 affine value plus c = 50

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083





S085 · calculationBefore

Left sum ORIGINAL sample11 value times cell width := mul(50, 4/19)

After

Left sum ORIGINAL sample11 value times cell width = 200/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S084





S086 · calculationBefore

Left sum ORIGINAL accumulated term11 := sum(1144/19, 200/19)

After

Left sum ORIGINAL accumulated term11 = 1344/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S085





S087 · impliesBefore

source roles = [11, 19, 19, 6, 0, 4, 4/19]

After

derived = [44/19, 50/1, 200/19, 1344/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S086





S088 · calculationBefore

Left sum ORIGINAL sample12 index times width := mul(12, 4/19)

After

Left sum ORIGINAL sample12 index times width = 48/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S087





S089 · calculationBefore

Left sum ORIGINAL sample12 lower plus offset := sum(0, 48/19)

After

Left sum ORIGINAL sample12 lower plus offset = 48/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S088





S090 · calculationBefore

Left sum ORIGINAL sample12 slope times x := mul(19, 48/19)

After

Left sum ORIGINAL sample12 slope times x = 48

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089





S091 · calculationBefore

Left sum ORIGINAL sample12 affine value plus c := sum(48, 6)

After

Left sum ORIGINAL sample12 affine value plus c = 54

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S090





S092 · calculationBefore

Left sum ORIGINAL sample12 value times cell width := mul(54, 4/19)

After

Left sum ORIGINAL sample12 value times cell width = 216/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S091





S093 · calculationBefore

Left sum ORIGINAL accumulated term12 := sum(1344/19, 216/19)

After

Left sum ORIGINAL accumulated term12 = 1560/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S092





S094 · impliesBefore

source roles = [12, 19, 19, 6, 0, 4, 4/19]

After

derived = [48/19, 54/1, 216/19, 1560/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S093





S095 · calculationBefore

Left sum ORIGINAL sample13 index times width := mul(13, 4/19)

After

Left sum ORIGINAL sample13 index times width = 52/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S094





S096 · calculationBefore

Left sum ORIGINAL sample13 lower plus offset := sum(0, 52/19)

After

Left sum ORIGINAL sample13 lower plus offset = 52/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S095





S097 · calculationBefore

Left sum ORIGINAL sample13 slope times x := mul(19, 52/19)

After

Left sum ORIGINAL sample13 slope times x = 52

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096





S098 · calculationBefore

Left sum ORIGINAL sample13 affine value plus c := sum(52, 6)

After

Left sum ORIGINAL sample13 affine value plus c = 58

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097





S099 · calculationBefore

Left sum ORIGINAL sample13 value times cell width := mul(58, 4/19)

After

Left sum ORIGINAL sample13 value times cell width = 232/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S098





S100 · calculationBefore

Left sum ORIGINAL accumulated term13 := sum(1560/19, 232/19)

After

Left sum ORIGINAL accumulated term13 = 1792/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S099





S101 · impliesBefore

source roles = [13, 19, 19, 6, 0, 4, 4/19]

After

derived = [52/19, 58/1, 232/19, 1792/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S100





S102 · calculationBefore

Left sum ORIGINAL sample14 index times width := mul(14, 4/19)

After

Left sum ORIGINAL sample14 index times width = 56/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S101





S103 · calculationBefore

Left sum ORIGINAL sample14 lower plus offset := sum(0, 56/19)

After

Left sum ORIGINAL sample14 lower plus offset = 56/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S102





S104 · calculationBefore

Left sum ORIGINAL sample14 slope times x := mul(19, 56/19)

After

Left sum ORIGINAL sample14 slope times x = 56

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103





S105 · calculationBefore

Left sum ORIGINAL sample14 affine value plus c := sum(56, 6)

After

Left sum ORIGINAL sample14 affine value plus c = 62

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S104





S106 · calculationBefore

Left sum ORIGINAL sample14 value times cell width := mul(62, 4/19)

After

Left sum ORIGINAL sample14 value times cell width = 248/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S105





S107 · calculationBefore

Left sum ORIGINAL accumulated term14 := sum(1792/19, 248/19)

After

Left sum ORIGINAL accumulated term14 = 2040/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S106





S108 · impliesBefore

source roles = [14, 19, 19, 6, 0, 4, 4/19]

After

derived = [56/19, 62/1, 248/19, 2040/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S107





S109 · calculationBefore

Left sum ORIGINAL sample15 index times width := mul(15, 4/19)

After

Left sum ORIGINAL sample15 index times width = 60/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S108





S110 · calculationBefore

Left sum ORIGINAL sample15 lower plus offset := sum(0, 60/19)

After

Left sum ORIGINAL sample15 lower plus offset = 60/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S109





S111 · calculationBefore

Left sum ORIGINAL sample15 slope times x := mul(19, 60/19)

After

Left sum ORIGINAL sample15 slope times x = 60

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S110





S112 · calculationBefore

Left sum ORIGINAL sample15 affine value plus c := sum(60, 6)

After

Left sum ORIGINAL sample15 affine value plus c = 66

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111





S113 · calculationBefore

Left sum ORIGINAL sample15 value times cell width := mul(66, 4/19)

After

Left sum ORIGINAL sample15 value times cell width = 264/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S112





S114 · calculationBefore

Left sum ORIGINAL accumulated term15 := sum(2040/19, 264/19)

After

Left sum ORIGINAL accumulated term15 = 2304/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S113





S115 · impliesBefore

source roles = [15, 19, 19, 6, 0, 4, 4/19]

After

derived = [60/19, 66/1, 264/19, 2304/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S114





S116 · calculationBefore

Left sum ORIGINAL sample16 index times width := mul(16, 4/19)

After

Left sum ORIGINAL sample16 index times width = 64/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S115





S117 · calculationBefore

Left sum ORIGINAL sample16 lower plus offset := sum(0, 64/19)

After

Left sum ORIGINAL sample16 lower plus offset = 64/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S116





S118 · calculationBefore

Left sum ORIGINAL sample16 slope times x := mul(19, 64/19)

After

Left sum ORIGINAL sample16 slope times x = 64

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117





S119 · calculationBefore

Left sum ORIGINAL sample16 affine value plus c := sum(64, 6)

After

Left sum ORIGINAL sample16 affine value plus c = 70

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S118





S120 · calculationBefore

Left sum ORIGINAL sample16 value times cell width := mul(70, 4/19)

After

Left sum ORIGINAL sample16 value times cell width = 280/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S119





S121 · calculationBefore

Left sum ORIGINAL accumulated term16 := sum(2304/19, 280/19)

After

Left sum ORIGINAL accumulated term16 = 136

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S120





S122 · impliesBefore

source roles = [16, 19, 19, 6, 0, 4, 4/19]

After

derived = [64/19, 70/1, 280/19, 136/1]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S121





S123 · calculationBefore

Left sum ORIGINAL sample17 index times width := mul(17, 4/19)

After

Left sum ORIGINAL sample17 index times width = 68/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S122





S124 · calculationBefore

Left sum ORIGINAL sample17 lower plus offset := sum(0, 68/19)

After

Left sum ORIGINAL sample17 lower plus offset = 68/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S123





S125 · calculationBefore

Left sum ORIGINAL sample17 slope times x := mul(19, 68/19)

After

Left sum ORIGINAL sample17 slope times x = 68

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124





S126 · calculationBefore

Left sum ORIGINAL sample17 affine value plus c := sum(68, 6)

After

Left sum ORIGINAL sample17 affine value plus c = 74

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125





S127 · calculationBefore

Left sum ORIGINAL sample17 value times cell width := mul(74, 4/19)

After

Left sum ORIGINAL sample17 value times cell width = 296/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S126





S128 · calculationBefore

Left sum ORIGINAL accumulated term17 := sum(136, 296/19)

After

Left sum ORIGINAL accumulated term17 = 2880/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S127





S129 · impliesBefore

source roles = [17, 19, 19, 6, 0, 4, 4/19]

After

derived = [68/19, 74/1, 296/19, 2880/19]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S128





S130 · calculationBefore

Left sum ORIGINAL sample18 index times width := mul(18, 4/19)

After

Left sum ORIGINAL sample18 index times width = 72/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S129





S131 · calculationBefore

Left sum ORIGINAL sample18 lower plus offset := sum(0, 72/19)

After

Left sum ORIGINAL sample18 lower plus offset = 72/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S130





S132 · calculationBefore

Left sum ORIGINAL sample18 slope times x := mul(19, 72/19)

After

Left sum ORIGINAL sample18 slope times x = 72

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S131





S133 · calculationBefore

Left sum ORIGINAL sample18 affine value plus c := sum(72, 6)

After

Left sum ORIGINAL sample18 affine value plus c = 78

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S132





S134 · calculationBefore

Left sum ORIGINAL sample18 value times cell width := mul(78, 4/19)

After

Left sum ORIGINAL sample18 value times cell width = 312/19

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S133





S135 · calculationBefore

Left sum ORIGINAL accumulated term18 := sum(2880/19, 312/19)

After

Left sum ORIGINAL accumulated term18 = 168

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S134





S136 · impliesBefore

source roles = [18, 19, 19, 6, 0, 4, 4/19]

After

derived = [72/19, 78/1, 312/19, 168/1]

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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S135





S137 · impliesBefore

source roles = [19, 19, 6, 0, 4]

After

derived = [4/19, 168/1]

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 dependenciesRule: LeftSum_FIXED_finite_partition_NOT_integral_or_limit_claim · Uses: G001, S136









## Reverse checks


- S137 · 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.
- 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

168/1

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: c7151cc4dc90f381e155bf7ac4b88ae5b81f90b79d1191b60ce2b70ba1029670
- candidate record SHA-256: e21f7104f7ef706f5528e706dd725d978eb3433d69219d09374862388ddb9946
- semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
- question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r07-definite-integral-0094.json)