# Question 32625: Finite Left-Endpoint Sum of an Affine Function [Return to catalogue page 164](https://ocme.evemisslab.com/en/questions/page/164/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r07-definite-integral-foundations/#q-20261007-advance-20261008-r07-definite-integral-0100) 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. R0064-32625 · q-20261007-advance-20261008-r07-definite-integral-0100 ## Original question and givens On all R, f(x)=(25)x+(12). For a=-2≤b=3 and N=50 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)=(25)x+(12). For a=-2≤b=3 and N=50 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(3, -2) After Left sum ORIGINAL upper minus lower endpoint = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Left sum ORIGINAL width over positive cell count := div(5, 50) After Left sum ORIGINAL width over positive cell count = 1/10 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, S002 S004 · calculationBefore Left sum ORIGINAL sample0 index times width := mul(0, 1/10) 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(-2, 0) After Left sum ORIGINAL sample0 lower plus offset = -2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Left sum ORIGINAL sample0 slope times x := mul(25, -2) After Left sum ORIGINAL sample0 slope times x = -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, S005 S007 · calculationBefore Left sum ORIGINAL sample0 affine value plus c := sum(-50, 12) After Left sum ORIGINAL sample0 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, S006 S008 · calculationBefore Left sum ORIGINAL sample0 value times cell width := mul(-38, 1/10) After Left sum ORIGINAL sample0 value times cell width = -19/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Left sum ORIGINAL accumulated term0 := sum(0, -19/5) After Left sum ORIGINAL accumulated term0 = -19/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [0, 50, 25, 12, -2, 3, 1/10] After derived = [-2/1, -38/1, -19/5, -19/5] 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, 1/10) After Left sum ORIGINAL sample1 index times width = 1/10 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, S010 S012 · calculationBefore Left sum ORIGINAL sample1 lower plus offset := sum(-2, 1/10) After Left sum ORIGINAL sample1 lower plus offset = -19/10 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, S011 S013 · calculationBefore Left sum ORIGINAL sample1 slope times x := mul(25, -19/10) After Left sum ORIGINAL sample1 slope times x = -95/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Left sum ORIGINAL sample1 affine value plus c := sum(-95/2, 12) After Left sum ORIGINAL sample1 affine value plus c = -71/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Left sum ORIGINAL sample1 value times cell width := mul(-71/2, 1/10) After Left sum ORIGINAL sample1 value times cell width = -71/20 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, S014 S016 · calculationBefore Left sum ORIGINAL accumulated term1 := sum(-19/5, -71/20) After Left sum ORIGINAL accumulated term1 = -147/20 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, S015 S017 · impliesBefore source roles = [1, 50, 25, 12, -2, 3, 1/10] After derived = [-19/10, -71/2, -71/20, -147/20] 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, 1/10) After Left sum ORIGINAL sample2 index times width = 1/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Left sum ORIGINAL sample2 lower plus offset := sum(-2, 1/5) After Left sum ORIGINAL sample2 lower plus offset = -9/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Left sum ORIGINAL sample2 slope times x := mul(25, -9/5) After Left sum ORIGINAL sample2 slope times x = -45 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Left sum ORIGINAL sample2 affine value plus c := sum(-45, 12) After Left sum ORIGINAL sample2 affine value plus c = -33 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Left sum ORIGINAL sample2 value times cell width := mul(-33, 1/10) After Left sum ORIGINAL sample2 value times cell width = -33/10 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, S021 S023 · calculationBefore Left sum ORIGINAL accumulated term2 := sum(-147/20, -33/10) After Left sum ORIGINAL accumulated term2 = -213/20 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, S022 S024 · impliesBefore source roles = [2, 50, 25, 12, -2, 3, 1/10] After derived = [-9/5, -33/1, -33/10, -213/20] 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, 1/10) After Left sum ORIGINAL sample3 index times width = 3/10 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, S024 S026 · calculationBefore Left sum ORIGINAL sample3 lower plus offset := sum(-2, 3/10) After Left sum ORIGINAL sample3 lower plus offset = -17/10 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, S025 S027 · calculationBefore Left sum ORIGINAL sample3 slope times x := mul(25, -17/10) After Left sum ORIGINAL sample3 slope times x = -85/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Left sum ORIGINAL sample3 affine value plus c := sum(-85/2, 12) After Left sum ORIGINAL sample3 affine value plus c = -61/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Left sum ORIGINAL sample3 value times cell width := mul(-61/2, 1/10) After Left sum ORIGINAL sample3 value times cell width = -61/20 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, S028 S030 · calculationBefore Left sum ORIGINAL accumulated term3 := sum(-213/20, -61/20) After Left sum ORIGINAL accumulated term3 = -137/10 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, S029 S031 · impliesBefore source roles = [3, 50, 25, 12, -2, 3, 1/10] After derived = [-17/10, -61/2, -61/20, -137/10] 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, 1/10) After Left sum ORIGINAL sample4 index times width = 2/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Left sum ORIGINAL sample4 lower plus offset := sum(-2, 2/5) After Left sum ORIGINAL sample4 lower plus offset = -8/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Left sum ORIGINAL sample4 slope times x := mul(25, -8/5) After Left sum ORIGINAL sample4 slope times x = -40 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Left sum ORIGINAL sample4 affine value plus c := sum(-40, 12) After Left sum ORIGINAL sample4 affine value plus c = -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, S034 S036 · calculationBefore Left sum ORIGINAL sample4 value times cell width := mul(-28, 1/10) After Left sum ORIGINAL sample4 value times cell width = -14/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Left sum ORIGINAL accumulated term4 := sum(-137/10, -14/5) After Left sum ORIGINAL accumulated term4 = -33/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore source roles = [4, 50, 25, 12, -2, 3, 1/10] After derived = [-8/5, -28/1, -14/5, -33/2] Conditions: none For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field. Step identifiers and 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, 1/10) After Left sum ORIGINAL sample5 index times width = 1/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Left sum ORIGINAL sample5 lower plus offset := sum(-2, 1/2) After Left sum ORIGINAL sample5 lower plus offset = -3/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Left sum ORIGINAL sample5 slope times x := mul(25, -3/2) After Left sum ORIGINAL sample5 slope times x = -75/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Left sum ORIGINAL sample5 affine value plus c := sum(-75/2, 12) After Left sum ORIGINAL sample5 affine value plus c = -51/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Left sum ORIGINAL sample5 value times cell width := mul(-51/2, 1/10) After Left sum ORIGINAL sample5 value times cell width = -51/20 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, S042 S044 · calculationBefore Left sum ORIGINAL accumulated term5 := sum(-33/2, -51/20) After Left sum ORIGINAL accumulated term5 = -381/20 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, S043 S045 · impliesBefore source roles = [5, 50, 25, 12, -2, 3, 1/10] After derived = [-3/2, -51/2, -51/20, -381/20] 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, 1/10) After Left sum ORIGINAL sample6 index times width = 3/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Left sum ORIGINAL sample6 lower plus offset := sum(-2, 3/5) After Left sum ORIGINAL sample6 lower plus offset = -7/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Left sum ORIGINAL sample6 slope times x := mul(25, -7/5) After Left sum ORIGINAL sample6 slope times x = -35 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.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(-35, 12) After Left sum ORIGINAL sample6 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Left sum ORIGINAL sample6 value times cell width := mul(-23, 1/10) After Left sum ORIGINAL sample6 value times cell width = -23/10 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, S049 S051 · calculationBefore Left sum ORIGINAL accumulated term6 := sum(-381/20, -23/10) After Left sum ORIGINAL accumulated term6 = -427/20 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, S050 S052 · impliesBefore source roles = [6, 50, 25, 12, -2, 3, 1/10] After derived = [-7/5, -23/1, -23/10, -427/20] 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, 1/10) After Left sum ORIGINAL sample7 index times width = 7/10 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, S052 S054 · calculationBefore Left sum ORIGINAL sample7 lower plus offset := sum(-2, 7/10) After Left sum ORIGINAL sample7 lower plus offset = -13/10 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, S053 S055 · calculationBefore Left sum ORIGINAL sample7 slope times x := mul(25, -13/10) After Left sum ORIGINAL sample7 slope times x = -65/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore Left sum ORIGINAL sample7 affine value plus c := sum(-65/2, 12) After Left sum ORIGINAL sample7 affine value plus c = -41/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Left sum ORIGINAL sample7 value times cell width := mul(-41/2, 1/10) After Left sum ORIGINAL sample7 value times cell width = -41/20 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, S056 S058 · calculationBefore Left sum ORIGINAL accumulated term7 := sum(-427/20, -41/20) After Left sum ORIGINAL accumulated term7 = -117/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · impliesBefore source roles = [7, 50, 25, 12, -2, 3, 1/10] After derived = [-13/10, -41/2, -41/20, -117/5] 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, 1/10) After Left sum ORIGINAL sample8 index times width = 4/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore Left sum ORIGINAL sample8 lower plus offset := sum(-2, 4/5) After Left sum ORIGINAL sample8 lower plus offset = -6/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore Left sum ORIGINAL sample8 slope times x := mul(25, -6/5) After Left sum ORIGINAL sample8 slope times x = -30 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore Left sum ORIGINAL sample8 affine value plus c := sum(-30, 12) After Left sum ORIGINAL sample8 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, S062 S064 · calculationBefore Left sum ORIGINAL sample8 value times cell width := mul(-18, 1/10) After Left sum ORIGINAL sample8 value times cell width = -9/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore Left sum ORIGINAL accumulated term8 := sum(-117/5, -9/5) After Left sum ORIGINAL accumulated term8 = -126/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · impliesBefore source roles = [8, 50, 25, 12, -2, 3, 1/10] After derived = [-6/5, -18/1, -9/5, -126/5] 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, 1/10) After Left sum ORIGINAL sample9 index times width = 9/10 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, S066 S068 · calculationBefore Left sum ORIGINAL sample9 lower plus offset := sum(-2, 9/10) After Left sum ORIGINAL sample9 lower plus offset = -11/10 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, S067 S069 · calculationBefore Left sum ORIGINAL sample9 slope times x := mul(25, -11/10) After Left sum ORIGINAL sample9 slope times x = -55/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore Left sum ORIGINAL sample9 affine value plus c := sum(-55/2, 12) After Left sum ORIGINAL sample9 affine value plus c = -31/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore Left sum ORIGINAL sample9 value times cell width := mul(-31/2, 1/10) After Left sum ORIGINAL sample9 value times cell width = -31/20 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, S070 S072 · calculationBefore Left sum ORIGINAL accumulated term9 := sum(-126/5, -31/20) After Left sum ORIGINAL accumulated term9 = -107/4 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, S071 S073 · impliesBefore source roles = [9, 50, 25, 12, -2, 3, 1/10] After derived = [-11/10, -31/2, -31/20, -107/4] 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, 1/10) After Left sum ORIGINAL sample10 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · calculationBefore Left sum ORIGINAL sample10 lower plus offset := sum(-2, 1) After Left sum ORIGINAL sample10 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore Left sum ORIGINAL sample10 slope times x := mul(25, -1) After Left sum ORIGINAL sample10 slope times x = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore Left sum ORIGINAL sample10 affine value plus c := sum(-25, 12) After Left sum ORIGINAL sample10 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore Left sum ORIGINAL sample10 value times cell width := mul(-13, 1/10) After Left sum ORIGINAL sample10 value times cell width = -13/10 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, S077 S079 · calculationBefore Left sum ORIGINAL accumulated term10 := sum(-107/4, -13/10) After Left sum ORIGINAL accumulated term10 = -561/20 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, S078 S080 · impliesBefore source roles = [10, 50, 25, 12, -2, 3, 1/10] After derived = [-1/1, -13/1, -13/10, -561/20] 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, 1/10) After Left sum ORIGINAL sample11 index times width = 11/10 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, S080 S082 · calculationBefore Left sum ORIGINAL sample11 lower plus offset := sum(-2, 11/10) After Left sum ORIGINAL sample11 lower plus offset = -9/10 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, S081 S083 · calculationBefore Left sum ORIGINAL sample11 slope times x := mul(25, -9/10) After Left sum ORIGINAL sample11 slope times x = -45/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore Left sum ORIGINAL sample11 affine value plus c := sum(-45/2, 12) After Left sum ORIGINAL sample11 affine value plus c = -21/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore Left sum ORIGINAL sample11 value times cell width := mul(-21/2, 1/10) After Left sum ORIGINAL sample11 value times cell width = -21/20 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, S084 S086 · calculationBefore Left sum ORIGINAL accumulated term11 := sum(-561/20, -21/20) After Left sum ORIGINAL accumulated term11 = -291/10 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, S085 S087 · impliesBefore source roles = [11, 50, 25, 12, -2, 3, 1/10] After derived = [-9/10, -21/2, -21/20, -291/10] 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, 1/10) After Left sum ORIGINAL sample12 index times width = 6/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · calculationBefore Left sum ORIGINAL sample12 lower plus offset := sum(-2, 6/5) After Left sum ORIGINAL sample12 lower plus offset = -4/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · calculationBefore Left sum ORIGINAL sample12 slope times x := mul(25, -4/5) After Left sum ORIGINAL sample12 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, S089 S091 · calculationBefore Left sum ORIGINAL sample12 affine value plus c := sum(-20, 12) After Left sum ORIGINAL sample12 affine value plus c = -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, S090 S092 · calculationBefore Left sum ORIGINAL sample12 value times cell width := mul(-8, 1/10) After Left sum ORIGINAL sample12 value times cell width = -4/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · calculationBefore Left sum ORIGINAL accumulated term12 := sum(-291/10, -4/5) After Left sum ORIGINAL accumulated term12 = -299/10 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, S092 S094 · impliesBefore source roles = [12, 50, 25, 12, -2, 3, 1/10] After derived = [-4/5, -8/1, -4/5, -299/10] 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, 1/10) After Left sum ORIGINAL sample13 index times width = 13/10 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, S094 S096 · calculationBefore Left sum ORIGINAL sample13 lower plus offset := sum(-2, 13/10) After Left sum ORIGINAL sample13 lower plus offset = -7/10 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, S095 S097 · calculationBefore Left sum ORIGINAL sample13 slope times x := mul(25, -7/10) After Left sum ORIGINAL sample13 slope times x = -35/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · calculationBefore Left sum ORIGINAL sample13 affine value plus c := sum(-35/2, 12) After Left sum ORIGINAL sample13 affine value plus c = -11/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097 S099 · calculationBefore Left sum ORIGINAL sample13 value times cell width := mul(-11/2, 1/10) After Left sum ORIGINAL sample13 value times cell width = -11/20 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, S098 S100 · calculationBefore Left sum ORIGINAL accumulated term13 := sum(-299/10, -11/20) After Left sum ORIGINAL accumulated term13 = -609/20 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, S099 S101 · impliesBefore source roles = [13, 50, 25, 12, -2, 3, 1/10] After derived = [-7/10, -11/2, -11/20, -609/20] 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, 1/10) After Left sum ORIGINAL sample14 index times width = 7/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · calculationBefore Left sum ORIGINAL sample14 lower plus offset := sum(-2, 7/5) After Left sum ORIGINAL sample14 lower plus offset = -3/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · calculationBefore Left sum ORIGINAL sample14 slope times x := mul(25, -3/5) After Left sum ORIGINAL sample14 slope times x = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · calculationBefore Left sum ORIGINAL sample14 affine value plus c := sum(-15, 12) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S104 S106 · calculationBefore Left sum ORIGINAL sample14 value times cell width := mul(-3, 1/10) After Left sum ORIGINAL sample14 value times cell width = -3/10 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, S105 S107 · calculationBefore Left sum ORIGINAL accumulated term14 := sum(-609/20, -3/10) After Left sum ORIGINAL accumulated term14 = -123/4 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, S106 S108 · impliesBefore source roles = [14, 50, 25, 12, -2, 3, 1/10] After derived = [-3/5, -3/1, -3/10, -123/4] 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, 1/10) After Left sum ORIGINAL sample15 index times width = 3/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S108 S110 · calculationBefore Left sum ORIGINAL sample15 lower plus offset := sum(-2, 3/2) After Left sum ORIGINAL sample15 lower plus offset = -1/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S109 S111 · calculationBefore Left sum ORIGINAL sample15 slope times x := mul(25, -1/2) After Left sum ORIGINAL sample15 slope times x = -25/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S110 S112 · calculationBefore Left sum ORIGINAL sample15 affine value plus c := sum(-25/2, 12) After Left sum ORIGINAL sample15 affine value plus c = -1/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111 S113 · calculationBefore Left sum ORIGINAL sample15 value times cell width := mul(-1/2, 1/10) After Left sum ORIGINAL sample15 value times cell width = -1/20 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, S112 S114 · calculationBefore Left sum ORIGINAL accumulated term15 := sum(-123/4, -1/20) After Left sum ORIGINAL accumulated term15 = -154/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S113 S115 · impliesBefore source roles = [15, 50, 25, 12, -2, 3, 1/10] After derived = [-1/2, -1/2, -1/20, -154/5] 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, 1/10) After Left sum ORIGINAL sample16 index times width = 8/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S115 S117 · calculationBefore Left sum ORIGINAL sample16 lower plus offset := sum(-2, 8/5) After Left sum ORIGINAL sample16 lower plus offset = -2/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S116 S118 · calculationBefore Left sum ORIGINAL sample16 slope times x := mul(25, -2/5) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S117 S119 · calculationBefore Left sum ORIGINAL sample16 affine value plus c := sum(-10, 12) After Left sum ORIGINAL sample16 affine value plus c = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S118 S120 · calculationBefore Left sum ORIGINAL sample16 value times cell width := mul(2, 1/10) After Left sum ORIGINAL sample16 value times cell width = 1/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S119 S121 · calculationBefore Left sum ORIGINAL accumulated term16 := sum(-154/5, 1/5) After Left sum ORIGINAL accumulated term16 = -153/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S120 S122 · impliesBefore source roles = [16, 50, 25, 12, -2, 3, 1/10] After derived = [-2/5, 2/1, 1/5, -153/5] 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, 1/10) After Left sum ORIGINAL sample17 index times width = 17/10 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, S122 S124 · calculationBefore Left sum ORIGINAL sample17 lower plus offset := sum(-2, 17/10) After Left sum ORIGINAL sample17 lower plus offset = -3/10 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, S123 S125 · calculationBefore Left sum ORIGINAL sample17 slope times x := mul(25, -3/10) After Left sum ORIGINAL sample17 slope times x = -15/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124 S126 · calculationBefore Left sum ORIGINAL sample17 affine value plus c := sum(-15/2, 12) After Left sum ORIGINAL sample17 affine value plus c = 9/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125 S127 · calculationBefore Left sum ORIGINAL sample17 value times cell width := mul(9/2, 1/10) After Left sum ORIGINAL sample17 value times cell width = 9/20 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, S126 S128 · calculationBefore Left sum ORIGINAL accumulated term17 := sum(-153/5, 9/20) After Left sum ORIGINAL accumulated term17 = -603/20 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, S127 S129 · impliesBefore source roles = [17, 50, 25, 12, -2, 3, 1/10] After derived = [-3/10, 9/2, 9/20, -603/20] 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, 1/10) After Left sum ORIGINAL sample18 index times width = 9/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S129 S131 · calculationBefore Left sum ORIGINAL sample18 lower plus offset := sum(-2, 9/5) After Left sum ORIGINAL sample18 lower plus offset = -1/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S130 S132 · calculationBefore Left sum ORIGINAL sample18 slope times x := mul(25, -1/5) After Left sum ORIGINAL sample18 slope times x = -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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S131 S133 · calculationBefore Left sum ORIGINAL sample18 affine value plus c := sum(-5, 12) After Left sum ORIGINAL sample18 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S132 S134 · calculationBefore Left sum ORIGINAL sample18 value times cell width := mul(7, 1/10) After Left sum ORIGINAL sample18 value times cell width = 7/10 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, S133 S135 · calculationBefore Left sum ORIGINAL accumulated term18 := sum(-603/20, 7/10) After Left sum ORIGINAL accumulated term18 = -589/20 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, S134 S136 · impliesBefore source roles = [18, 50, 25, 12, -2, 3, 1/10] After derived = [-1/5, 7/1, 7/10, -589/20] 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 · calculationBefore Left sum ORIGINAL sample19 index times width := mul(19, 1/10) After Left sum ORIGINAL sample19 index times width = 19/10 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, S136 S138 · calculationBefore Left sum ORIGINAL sample19 lower plus offset := sum(-2, 19/10) After Left sum ORIGINAL sample19 lower plus offset = -1/10 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, S137 S139 · calculationBefore Left sum ORIGINAL sample19 slope times x := mul(25, -1/10) After Left sum ORIGINAL sample19 slope times x = -5/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S138 S140 · calculationBefore Left sum ORIGINAL sample19 affine value plus c := sum(-5/2, 12) After Left sum ORIGINAL sample19 affine value plus c = 19/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S139 S141 · calculationBefore Left sum ORIGINAL sample19 value times cell width := mul(19/2, 1/10) After Left sum ORIGINAL sample19 value times cell width = 19/20 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, S140 S142 · calculationBefore Left sum ORIGINAL accumulated term19 := sum(-589/20, 19/20) After Left sum ORIGINAL accumulated term19 = -57/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S141 S143 · impliesBefore source roles = [19, 50, 25, 12, -2, 3, 1/10] After derived = [-1/10, 19/2, 19/20, -57/2] Conditions: none For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field. Step identifiers and dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S142 S144 · calculationBefore Left sum ORIGINAL sample20 index times width := mul(20, 1/10) After Left sum ORIGINAL sample20 index times width = 2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S143 S145 · calculationBefore Left sum ORIGINAL sample20 lower plus offset := sum(-2, 2) After Left sum ORIGINAL sample20 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, S144 S146 · calculationBefore Left sum ORIGINAL sample20 slope times x := mul(25, 0) After Left sum ORIGINAL sample20 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, S145 S147 · calculationBefore Left sum ORIGINAL sample20 affine value plus c := sum(0, 12) After Left sum ORIGINAL sample20 affine value plus c = 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, S146 S148 · calculationBefore Left sum ORIGINAL sample20 value times cell width := mul(12, 1/10) After Left sum ORIGINAL sample20 value times cell width = 6/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S147 S149 · calculationBefore Left sum ORIGINAL accumulated term20 := sum(-57/2, 6/5) After Left sum ORIGINAL accumulated term20 = -273/10 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, S148 S150 · impliesBefore source roles = [20, 50, 25, 12, -2, 3, 1/10] After derived = [0/1, 12/1, 6/5, -273/10] 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, S149 S151 · calculationBefore Left sum ORIGINAL sample21 index times width := mul(21, 1/10) After Left sum ORIGINAL sample21 index times width = 21/10 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, S150 S152 · calculationBefore Left sum ORIGINAL sample21 lower plus offset := sum(-2, 21/10) After Left sum ORIGINAL sample21 lower plus offset = 1/10 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, S151 S153 · calculationBefore Left sum ORIGINAL sample21 slope times x := mul(25, 1/10) After Left sum ORIGINAL sample21 slope times x = 5/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S152 S154 · calculationBefore Left sum ORIGINAL sample21 affine value plus c := sum(5/2, 12) After Left sum ORIGINAL sample21 affine value plus c = 29/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S153 S155 · calculationBefore Left sum ORIGINAL sample21 value times cell width := mul(29/2, 1/10) After Left sum ORIGINAL sample21 value times cell width = 29/20 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, S154 S156 · calculationBefore Left sum ORIGINAL accumulated term21 := sum(-273/10, 29/20) After Left sum ORIGINAL accumulated term21 = -517/20 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, S155 S157 · impliesBefore source roles = [21, 50, 25, 12, -2, 3, 1/10] After derived = [1/10, 29/2, 29/20, -517/20] 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, S156 S158 · calculationBefore Left sum ORIGINAL sample22 index times width := mul(22, 1/10) After Left sum ORIGINAL sample22 index times width = 11/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S157 S159 · calculationBefore Left sum ORIGINAL sample22 lower plus offset := sum(-2, 11/5) After Left sum ORIGINAL sample22 lower plus offset = 1/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S158 S160 · calculationBefore Left sum ORIGINAL sample22 slope times x := mul(25, 1/5) After Left sum ORIGINAL sample22 slope times x = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S159 S161 · calculationBefore Left sum ORIGINAL sample22 affine value plus c := sum(5, 12) After Left sum ORIGINAL sample22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S160 S162 · calculationBefore Left sum ORIGINAL sample22 value times cell width := mul(17, 1/10) After Left sum ORIGINAL sample22 value times cell width = 17/10 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, S161 S163 · calculationBefore Left sum ORIGINAL accumulated term22 := sum(-517/20, 17/10) After Left sum ORIGINAL accumulated term22 = -483/20 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, S162 S164 · impliesBefore source roles = [22, 50, 25, 12, -2, 3, 1/10] After derived = [1/5, 17/1, 17/10, -483/20] 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, S163 S165 · calculationBefore Left sum ORIGINAL sample23 index times width := mul(23, 1/10) After Left sum ORIGINAL sample23 index times width = 23/10 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, S164 S166 · calculationBefore Left sum ORIGINAL sample23 lower plus offset := sum(-2, 23/10) After Left sum ORIGINAL sample23 lower plus offset = 3/10 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, S165 S167 · calculationBefore Left sum ORIGINAL sample23 slope times x := mul(25, 3/10) After Left sum ORIGINAL sample23 slope times x = 15/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S166 S168 · calculationBefore Left sum ORIGINAL sample23 affine value plus c := sum(15/2, 12) After Left sum ORIGINAL sample23 affine value plus c = 39/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S167 S169 · calculationBefore Left sum ORIGINAL sample23 value times cell width := mul(39/2, 1/10) After Left sum ORIGINAL sample23 value times cell width = 39/20 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, S168 S170 · calculationBefore Left sum ORIGINAL accumulated term23 := sum(-483/20, 39/20) After Left sum ORIGINAL accumulated term23 = -111/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S169 S171 · impliesBefore source roles = [23, 50, 25, 12, -2, 3, 1/10] After derived = [3/10, 39/2, 39/20, -111/5] 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, S170 S172 · calculationBefore Left sum ORIGINAL sample24 index times width := mul(24, 1/10) After Left sum ORIGINAL sample24 index times width = 12/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S171 S173 · calculationBefore Left sum ORIGINAL sample24 lower plus offset := sum(-2, 12/5) After Left sum ORIGINAL sample24 lower plus offset = 2/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S172 S174 · calculationBefore Left sum ORIGINAL sample24 slope times x := mul(25, 2/5) After Left sum ORIGINAL sample24 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S173 S175 · calculationBefore Left sum ORIGINAL sample24 affine value plus c := sum(10, 12) After Left sum ORIGINAL sample24 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, S174 S176 · calculationBefore Left sum ORIGINAL sample24 value times cell width := mul(22, 1/10) After Left sum ORIGINAL sample24 value times cell width = 11/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S175 S177 · calculationBefore Left sum ORIGINAL accumulated term24 := sum(-111/5, 11/5) After Left sum ORIGINAL accumulated term24 = -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, S176 S178 · impliesBefore source roles = [24, 50, 25, 12, -2, 3, 1/10] After derived = [2/5, 22/1, 11/5, -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 dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S177 S179 · calculationBefore Left sum ORIGINAL sample25 index times width := mul(25, 1/10) After Left sum ORIGINAL sample25 index times width = 5/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S178 S180 · calculationBefore Left sum ORIGINAL sample25 lower plus offset := sum(-2, 5/2) After Left sum ORIGINAL sample25 lower plus offset = 1/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S179 S181 · calculationBefore Left sum ORIGINAL sample25 slope times x := mul(25, 1/2) After Left sum ORIGINAL sample25 slope times x = 25/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S180 S182 · calculationBefore Left sum ORIGINAL sample25 affine value plus c := sum(25/2, 12) After Left sum ORIGINAL sample25 affine value plus c = 49/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S181 S183 · calculationBefore Left sum ORIGINAL sample25 value times cell width := mul(49/2, 1/10) After Left sum ORIGINAL sample25 value times cell width = 49/20 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, S182 S184 · calculationBefore Left sum ORIGINAL accumulated term25 := sum(-20, 49/20) After Left sum ORIGINAL accumulated term25 = -351/20 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, S183 S185 · impliesBefore source roles = [25, 50, 25, 12, -2, 3, 1/10] After derived = [1/2, 49/2, 49/20, -351/20] 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, S184 S186 · calculationBefore Left sum ORIGINAL sample26 index times width := mul(26, 1/10) After Left sum ORIGINAL sample26 index times width = 13/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S185 S187 · calculationBefore Left sum ORIGINAL sample26 lower plus offset := sum(-2, 13/5) After Left sum ORIGINAL sample26 lower plus offset = 3/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S186 S188 · calculationBefore Left sum ORIGINAL sample26 slope times x := mul(25, 3/5) After Left sum ORIGINAL sample26 slope times x = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S187 S189 · calculationBefore Left sum ORIGINAL sample26 affine value plus c := sum(15, 12) After Left sum ORIGINAL sample26 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S188 S190 · calculationBefore Left sum ORIGINAL sample26 value times cell width := mul(27, 1/10) After Left sum ORIGINAL sample26 value times cell width = 27/10 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, S189 S191 · calculationBefore Left sum ORIGINAL accumulated term26 := sum(-351/20, 27/10) After Left sum ORIGINAL accumulated term26 = -297/20 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, S190 S192 · impliesBefore source roles = [26, 50, 25, 12, -2, 3, 1/10] After derived = [3/5, 27/1, 27/10, -297/20] 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, S191 S193 · calculationBefore Left sum ORIGINAL sample27 index times width := mul(27, 1/10) After Left sum ORIGINAL sample27 index times width = 27/10 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, S192 S194 · calculationBefore Left sum ORIGINAL sample27 lower plus offset := sum(-2, 27/10) After Left sum ORIGINAL sample27 lower plus offset = 7/10 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, S193 S195 · calculationBefore Left sum ORIGINAL sample27 slope times x := mul(25, 7/10) After Left sum ORIGINAL sample27 slope times x = 35/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S194 S196 · calculationBefore Left sum ORIGINAL sample27 affine value plus c := sum(35/2, 12) After Left sum ORIGINAL sample27 affine value plus c = 59/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S195 S197 · calculationBefore Left sum ORIGINAL sample27 value times cell width := mul(59/2, 1/10) After Left sum ORIGINAL sample27 value times cell width = 59/20 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, S196 S198 · calculationBefore Left sum ORIGINAL accumulated term27 := sum(-297/20, 59/20) After Left sum ORIGINAL accumulated term27 = -119/10 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, S197 S199 · impliesBefore source roles = [27, 50, 25, 12, -2, 3, 1/10] After derived = [7/10, 59/2, 59/20, -119/10] 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, S198 S200 · calculationBefore Left sum ORIGINAL sample28 index times width := mul(28, 1/10) After Left sum ORIGINAL sample28 index times width = 14/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S199 S201 · calculationBefore Left sum ORIGINAL sample28 lower plus offset := sum(-2, 14/5) After Left sum ORIGINAL sample28 lower plus offset = 4/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S200 S202 · calculationBefore Left sum ORIGINAL sample28 slope times x := mul(25, 4/5) After Left sum ORIGINAL sample28 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, S201 S203 · calculationBefore Left sum ORIGINAL sample28 affine value plus c := sum(20, 12) After Left sum ORIGINAL sample28 affine value plus c = 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, S202 S204 · calculationBefore Left sum ORIGINAL sample28 value times cell width := mul(32, 1/10) After Left sum ORIGINAL sample28 value times cell width = 16/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S203 S205 · calculationBefore Left sum ORIGINAL accumulated term28 := sum(-119/10, 16/5) After Left sum ORIGINAL accumulated term28 = -87/10 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, S204 S206 · impliesBefore source roles = [28, 50, 25, 12, -2, 3, 1/10] After derived = [4/5, 32/1, 16/5, -87/10] 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, S205 S207 · calculationBefore Left sum ORIGINAL sample29 index times width := mul(29, 1/10) After Left sum ORIGINAL sample29 index times width = 29/10 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, S206 S208 · calculationBefore Left sum ORIGINAL sample29 lower plus offset := sum(-2, 29/10) After Left sum ORIGINAL sample29 lower plus offset = 9/10 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, S207 S209 · calculationBefore Left sum ORIGINAL sample29 slope times x := mul(25, 9/10) After Left sum ORIGINAL sample29 slope times x = 45/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S208 S210 · calculationBefore Left sum ORIGINAL sample29 affine value plus c := sum(45/2, 12) After Left sum ORIGINAL sample29 affine value plus c = 69/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S209 S211 · calculationBefore Left sum ORIGINAL sample29 value times cell width := mul(69/2, 1/10) After Left sum ORIGINAL sample29 value times cell width = 69/20 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, S210 S212 · calculationBefore Left sum ORIGINAL accumulated term29 := sum(-87/10, 69/20) After Left sum ORIGINAL accumulated term29 = -21/4 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, S211 S213 · impliesBefore source roles = [29, 50, 25, 12, -2, 3, 1/10] After derived = [9/10, 69/2, 69/20, -21/4] 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, S212 S214 · calculationBefore Left sum ORIGINAL sample30 index times width := mul(30, 1/10) After Left sum ORIGINAL sample30 index times width = 3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S213 S215 · calculationBefore Left sum ORIGINAL sample30 lower plus offset := sum(-2, 3) After Left sum ORIGINAL sample30 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S214 S216 · calculationBefore Left sum ORIGINAL sample30 slope times x := mul(25, 1) After Left sum ORIGINAL sample30 slope times x = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S215 S217 · calculationBefore Left sum ORIGINAL sample30 affine value plus c := sum(25, 12) After Left sum ORIGINAL sample30 affine value plus c = 37 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S216 S218 · calculationBefore Left sum ORIGINAL sample30 value times cell width := mul(37, 1/10) After Left sum ORIGINAL sample30 value times cell width = 37/10 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, S217 S219 · calculationBefore Left sum ORIGINAL accumulated term30 := sum(-21/4, 37/10) After Left sum ORIGINAL accumulated term30 = -31/20 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, S218 S220 · impliesBefore source roles = [30, 50, 25, 12, -2, 3, 1/10] After derived = [1/1, 37/1, 37/10, -31/20] 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, S219 S221 · calculationBefore Left sum ORIGINAL sample31 index times width := mul(31, 1/10) After Left sum ORIGINAL sample31 index times width = 31/10 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, S220 S222 · calculationBefore Left sum ORIGINAL sample31 lower plus offset := sum(-2, 31/10) After Left sum ORIGINAL sample31 lower plus offset = 11/10 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, S221 S223 · calculationBefore Left sum ORIGINAL sample31 slope times x := mul(25, 11/10) After Left sum ORIGINAL sample31 slope times x = 55/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S222 S224 · calculationBefore Left sum ORIGINAL sample31 affine value plus c := sum(55/2, 12) After Left sum ORIGINAL sample31 affine value plus c = 79/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S223 S225 · calculationBefore Left sum ORIGINAL sample31 value times cell width := mul(79/2, 1/10) After Left sum ORIGINAL sample31 value times cell width = 79/20 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, S224 S226 · calculationBefore Left sum ORIGINAL accumulated term31 := sum(-31/20, 79/20) After Left sum ORIGINAL accumulated term31 = 12/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S225 S227 · impliesBefore source roles = [31, 50, 25, 12, -2, 3, 1/10] After derived = [11/10, 79/2, 79/20, 12/5] 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, S226 S228 · calculationBefore Left sum ORIGINAL sample32 index times width := mul(32, 1/10) After Left sum ORIGINAL sample32 index times width = 16/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S227 S229 · calculationBefore Left sum ORIGINAL sample32 lower plus offset := sum(-2, 16/5) After Left sum ORIGINAL sample32 lower plus offset = 6/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S228 S230 · calculationBefore Left sum ORIGINAL sample32 slope times x := mul(25, 6/5) After Left sum ORIGINAL sample32 slope times x = 30 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S229 S231 · calculationBefore Left sum ORIGINAL sample32 affine value plus c := sum(30, 12) After Left sum ORIGINAL sample32 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, S230 S232 · calculationBefore Left sum ORIGINAL sample32 value times cell width := mul(42, 1/10) After Left sum ORIGINAL sample32 value times cell width = 21/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S231 S233 · calculationBefore Left sum ORIGINAL accumulated term32 := sum(12/5, 21/5) After Left sum ORIGINAL accumulated term32 = 33/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S232 S234 · impliesBefore source roles = [32, 50, 25, 12, -2, 3, 1/10] After derived = [6/5, 42/1, 21/5, 33/5] 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, S233 S235 · calculationBefore Left sum ORIGINAL sample33 index times width := mul(33, 1/10) After Left sum ORIGINAL sample33 index times width = 33/10 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, S234 S236 · calculationBefore Left sum ORIGINAL sample33 lower plus offset := sum(-2, 33/10) After Left sum ORIGINAL sample33 lower plus offset = 13/10 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, S235 S237 · calculationBefore Left sum ORIGINAL sample33 slope times x := mul(25, 13/10) After Left sum ORIGINAL sample33 slope times x = 65/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S236 S238 · calculationBefore Left sum ORIGINAL sample33 affine value plus c := sum(65/2, 12) After Left sum ORIGINAL sample33 affine value plus c = 89/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S237 S239 · calculationBefore Left sum ORIGINAL sample33 value times cell width := mul(89/2, 1/10) After Left sum ORIGINAL sample33 value times cell width = 89/20 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, S238 S240 · calculationBefore Left sum ORIGINAL accumulated term33 := sum(33/5, 89/20) After Left sum ORIGINAL accumulated term33 = 221/20 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, S239 S241 · impliesBefore source roles = [33, 50, 25, 12, -2, 3, 1/10] After derived = [13/10, 89/2, 89/20, 221/20] 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, S240 S242 · calculationBefore Left sum ORIGINAL sample34 index times width := mul(34, 1/10) After Left sum ORIGINAL sample34 index times width = 17/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S241 S243 · calculationBefore Left sum ORIGINAL sample34 lower plus offset := sum(-2, 17/5) After Left sum ORIGINAL sample34 lower plus offset = 7/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S242 S244 · calculationBefore Left sum ORIGINAL sample34 slope times x := mul(25, 7/5) After Left sum ORIGINAL sample34 slope times x = 35 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.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, S243 S245 · calculationBefore Left sum ORIGINAL sample34 affine value plus c := sum(35, 12) After Left sum ORIGINAL sample34 affine value plus c = 47 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.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, S244 S246 · calculationBefore Left sum ORIGINAL sample34 value times cell width := mul(47, 1/10) After Left sum ORIGINAL sample34 value times cell width = 47/10 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, S245 S247 · calculationBefore Left sum ORIGINAL accumulated term34 := sum(221/20, 47/10) After Left sum ORIGINAL accumulated term34 = 63/4 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, S246 S248 · impliesBefore source roles = [34, 50, 25, 12, -2, 3, 1/10] After derived = [7/5, 47/1, 47/10, 63/4] 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, S247 S249 · calculationBefore Left sum ORIGINAL sample35 index times width := mul(35, 1/10) After Left sum ORIGINAL sample35 index times width = 7/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S248 S250 · calculationBefore Left sum ORIGINAL sample35 lower plus offset := sum(-2, 7/2) After Left sum ORIGINAL sample35 lower plus offset = 3/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S249 S251 · calculationBefore Left sum ORIGINAL sample35 slope times x := mul(25, 3/2) After Left sum ORIGINAL sample35 slope times x = 75/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S250 S252 · calculationBefore Left sum ORIGINAL sample35 affine value plus c := sum(75/2, 12) After Left sum ORIGINAL sample35 affine value plus c = 99/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S251 S253 · calculationBefore Left sum ORIGINAL sample35 value times cell width := mul(99/2, 1/10) After Left sum ORIGINAL sample35 value times cell width = 99/20 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, S252 S254 · calculationBefore Left sum ORIGINAL accumulated term35 := sum(63/4, 99/20) After Left sum ORIGINAL accumulated term35 = 207/10 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, S253 S255 · impliesBefore source roles = [35, 50, 25, 12, -2, 3, 1/10] After derived = [3/2, 99/2, 99/20, 207/10] 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, S254 S256 · calculationBefore Left sum ORIGINAL sample36 index times width := mul(36, 1/10) After Left sum ORIGINAL sample36 index times width = 18/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S255 S257 · calculationBefore Left sum ORIGINAL sample36 lower plus offset := sum(-2, 18/5) After Left sum ORIGINAL sample36 lower plus offset = 8/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S256 S258 · calculationBefore Left sum ORIGINAL sample36 slope times x := mul(25, 8/5) After Left sum ORIGINAL sample36 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, S257 S259 · calculationBefore Left sum ORIGINAL sample36 affine value plus c := sum(40, 12) After Left sum ORIGINAL sample36 affine value plus c = 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, S258 S260 · calculationBefore Left sum ORIGINAL sample36 value times cell width := mul(52, 1/10) After Left sum ORIGINAL sample36 value times cell width = 26/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S259 S261 · calculationBefore Left sum ORIGINAL accumulated term36 := sum(207/10, 26/5) After Left sum ORIGINAL accumulated term36 = 259/10 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, S260 S262 · impliesBefore source roles = [36, 50, 25, 12, -2, 3, 1/10] After derived = [8/5, 52/1, 26/5, 259/10] 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, S261 S263 · calculationBefore Left sum ORIGINAL sample37 index times width := mul(37, 1/10) After Left sum ORIGINAL sample37 index times width = 37/10 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, S262 S264 · calculationBefore Left sum ORIGINAL sample37 lower plus offset := sum(-2, 37/10) After Left sum ORIGINAL sample37 lower plus offset = 17/10 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, S263 S265 · calculationBefore Left sum ORIGINAL sample37 slope times x := mul(25, 17/10) After Left sum ORIGINAL sample37 slope times x = 85/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S264 S266 · calculationBefore Left sum ORIGINAL sample37 affine value plus c := sum(85/2, 12) After Left sum ORIGINAL sample37 affine value plus c = 109/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S265 S267 · calculationBefore Left sum ORIGINAL sample37 value times cell width := mul(109/2, 1/10) After Left sum ORIGINAL sample37 value times cell width = 109/20 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, S266 S268 · calculationBefore Left sum ORIGINAL accumulated term37 := sum(259/10, 109/20) After Left sum ORIGINAL accumulated term37 = 627/20 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, S267 S269 · impliesBefore source roles = [37, 50, 25, 12, -2, 3, 1/10] After derived = [17/10, 109/2, 109/20, 627/20] 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, S268 S270 · calculationBefore Left sum ORIGINAL sample38 index times width := mul(38, 1/10) After Left sum ORIGINAL sample38 index times width = 19/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S269 S271 · calculationBefore Left sum ORIGINAL sample38 lower plus offset := sum(-2, 19/5) After Left sum ORIGINAL sample38 lower plus offset = 9/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S270 S272 · calculationBefore Left sum ORIGINAL sample38 slope times x := mul(25, 9/5) After Left sum ORIGINAL sample38 slope times x = 45 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S271 S273 · calculationBefore Left sum ORIGINAL sample38 affine value plus c := sum(45, 12) After Left sum ORIGINAL sample38 affine value plus c = 57 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S272 S274 · calculationBefore Left sum ORIGINAL sample38 value times cell width := mul(57, 1/10) After Left sum ORIGINAL sample38 value times cell width = 57/10 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, S273 S275 · calculationBefore Left sum ORIGINAL accumulated term38 := sum(627/20, 57/10) After Left sum ORIGINAL accumulated term38 = 741/20 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, S274 S276 · impliesBefore source roles = [38, 50, 25, 12, -2, 3, 1/10] After derived = [9/5, 57/1, 57/10, 741/20] 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, S275 S277 · calculationBefore Left sum ORIGINAL sample39 index times width := mul(39, 1/10) After Left sum ORIGINAL sample39 index times width = 39/10 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, S276 S278 · calculationBefore Left sum ORIGINAL sample39 lower plus offset := sum(-2, 39/10) After Left sum ORIGINAL sample39 lower plus offset = 19/10 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, S277 S279 · calculationBefore Left sum ORIGINAL sample39 slope times x := mul(25, 19/10) After Left sum ORIGINAL sample39 slope times x = 95/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S278 S280 · calculationBefore Left sum ORIGINAL sample39 affine value plus c := sum(95/2, 12) After Left sum ORIGINAL sample39 affine value plus c = 119/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S279 S281 · calculationBefore Left sum ORIGINAL sample39 value times cell width := mul(119/2, 1/10) After Left sum ORIGINAL sample39 value times cell width = 119/20 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, S280 S282 · calculationBefore Left sum ORIGINAL accumulated term39 := sum(741/20, 119/20) After Left sum ORIGINAL accumulated term39 = 43 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.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, S281 S283 · impliesBefore source roles = [39, 50, 25, 12, -2, 3, 1/10] After derived = [19/10, 119/2, 119/20, 43/1] 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, S282 S284 · calculationBefore Left sum ORIGINAL sample40 index times width := mul(40, 1/10) After Left sum ORIGINAL sample40 index times width = 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, S283 S285 · calculationBefore Left sum ORIGINAL sample40 lower plus offset := sum(-2, 4) After Left sum ORIGINAL sample40 lower plus offset = 2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S284 S286 · calculationBefore Left sum ORIGINAL sample40 slope times x := mul(25, 2) After Left sum ORIGINAL sample40 slope times x = 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, S285 S287 · calculationBefore Left sum ORIGINAL sample40 affine value plus c := sum(50, 12) After Left sum ORIGINAL sample40 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, S286 S288 · calculationBefore Left sum ORIGINAL sample40 value times cell width := mul(62, 1/10) After Left sum ORIGINAL sample40 value times cell width = 31/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S287 S289 · calculationBefore Left sum ORIGINAL accumulated term40 := sum(43, 31/5) After Left sum ORIGINAL accumulated term40 = 246/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S288 S290 · impliesBefore source roles = [40, 50, 25, 12, -2, 3, 1/10] After derived = [2/1, 62/1, 31/5, 246/5] 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, S289 S291 · calculationBefore Left sum ORIGINAL sample41 index times width := mul(41, 1/10) After Left sum ORIGINAL sample41 index times width = 41/10 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, S290 S292 · calculationBefore Left sum ORIGINAL sample41 lower plus offset := sum(-2, 41/10) After Left sum ORIGINAL sample41 lower plus offset = 21/10 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, S291 S293 · calculationBefore Left sum ORIGINAL sample41 slope times x := mul(25, 21/10) After Left sum ORIGINAL sample41 slope times x = 105/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S292 S294 · calculationBefore Left sum ORIGINAL sample41 affine value plus c := sum(105/2, 12) After Left sum ORIGINAL sample41 affine value plus c = 129/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S293 S295 · calculationBefore Left sum ORIGINAL sample41 value times cell width := mul(129/2, 1/10) After Left sum ORIGINAL sample41 value times cell width = 129/20 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, S294 S296 · calculationBefore Left sum ORIGINAL accumulated term41 := sum(246/5, 129/20) After Left sum ORIGINAL accumulated term41 = 1113/20 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, S295 S297 · impliesBefore source roles = [41, 50, 25, 12, -2, 3, 1/10] After derived = [21/10, 129/2, 129/20, 1113/20] 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, S296 S298 · calculationBefore Left sum ORIGINAL sample42 index times width := mul(42, 1/10) After Left sum ORIGINAL sample42 index times width = 21/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S297 S299 · calculationBefore Left sum ORIGINAL sample42 lower plus offset := sum(-2, 21/5) After Left sum ORIGINAL sample42 lower plus offset = 11/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S298 S300 · calculationBefore Left sum ORIGINAL sample42 slope times x := mul(25, 11/5) After Left sum ORIGINAL sample42 slope times x = 55 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.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, S299 S301 · calculationBefore Left sum ORIGINAL sample42 affine value plus c := sum(55, 12) After Left sum ORIGINAL sample42 affine value plus c = 67 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.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, S300 S302 · calculationBefore Left sum ORIGINAL sample42 value times cell width := mul(67, 1/10) After Left sum ORIGINAL sample42 value times cell width = 67/10 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, S301 S303 · calculationBefore Left sum ORIGINAL accumulated term42 := sum(1113/20, 67/10) After Left sum ORIGINAL accumulated term42 = 1247/20 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, S302 S304 · impliesBefore source roles = [42, 50, 25, 12, -2, 3, 1/10] After derived = [11/5, 67/1, 67/10, 1247/20] 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, S303 S305 · calculationBefore Left sum ORIGINAL sample43 index times width := mul(43, 1/10) After Left sum ORIGINAL sample43 index times width = 43/10 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, S304 S306 · calculationBefore Left sum ORIGINAL sample43 lower plus offset := sum(-2, 43/10) After Left sum ORIGINAL sample43 lower plus offset = 23/10 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, S305 S307 · calculationBefore Left sum ORIGINAL sample43 slope times x := mul(25, 23/10) After Left sum ORIGINAL sample43 slope times x = 115/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S306 S308 · calculationBefore Left sum ORIGINAL sample43 affine value plus c := sum(115/2, 12) After Left sum ORIGINAL sample43 affine value plus c = 139/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S307 S309 · calculationBefore Left sum ORIGINAL sample43 value times cell width := mul(139/2, 1/10) After Left sum ORIGINAL sample43 value times cell width = 139/20 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, S308 S310 · calculationBefore Left sum ORIGINAL accumulated term43 := sum(1247/20, 139/20) After Left sum ORIGINAL accumulated term43 = 693/10 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, S309 S311 · impliesBefore source roles = [43, 50, 25, 12, -2, 3, 1/10] After derived = [23/10, 139/2, 139/20, 693/10] 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, S310 S312 · calculationBefore Left sum ORIGINAL sample44 index times width := mul(44, 1/10) After Left sum ORIGINAL sample44 index times width = 22/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S311 S313 · calculationBefore Left sum ORIGINAL sample44 lower plus offset := sum(-2, 22/5) After Left sum ORIGINAL sample44 lower plus offset = 12/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S312 S314 · calculationBefore Left sum ORIGINAL sample44 slope times x := mul(25, 12/5) After Left sum ORIGINAL sample44 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, S313 S315 · calculationBefore Left sum ORIGINAL sample44 affine value plus c := sum(60, 12) After Left sum ORIGINAL sample44 affine value plus c = 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, S314 S316 · calculationBefore Left sum ORIGINAL sample44 value times cell width := mul(72, 1/10) After Left sum ORIGINAL sample44 value times cell width = 36/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S315 S317 · calculationBefore Left sum ORIGINAL accumulated term44 := sum(693/10, 36/5) After Left sum ORIGINAL accumulated term44 = 153/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S316 S318 · impliesBefore source roles = [44, 50, 25, 12, -2, 3, 1/10] After derived = [12/5, 72/1, 36/5, 153/2] Conditions: none For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field. Step identifiers and dependenciesRule: LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum · Uses: G001, S317 S319 · calculationBefore Left sum ORIGINAL sample45 index times width := mul(45, 1/10) After Left sum ORIGINAL sample45 index times width = 9/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S318 S320 · calculationBefore Left sum ORIGINAL sample45 lower plus offset := sum(-2, 9/2) After Left sum ORIGINAL sample45 lower plus offset = 5/2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S319 S321 · calculationBefore Left sum ORIGINAL sample45 slope times x := mul(25, 5/2) After Left sum ORIGINAL sample45 slope times x = 125/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S320 S322 · calculationBefore Left sum ORIGINAL sample45 affine value plus c := sum(125/2, 12) After Left sum ORIGINAL sample45 affine value plus c = 149/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S321 S323 · calculationBefore Left sum ORIGINAL sample45 value times cell width := mul(149/2, 1/10) After Left sum ORIGINAL sample45 value times cell width = 149/20 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, S322 S324 · calculationBefore Left sum ORIGINAL accumulated term45 := sum(153/2, 149/20) After Left sum ORIGINAL accumulated term45 = 1679/20 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, S323 S325 · impliesBefore source roles = [45, 50, 25, 12, -2, 3, 1/10] After derived = [5/2, 149/2, 149/20, 1679/20] 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, S324 S326 · calculationBefore Left sum ORIGINAL sample46 index times width := mul(46, 1/10) After Left sum ORIGINAL sample46 index times width = 23/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S325 S327 · calculationBefore Left sum ORIGINAL sample46 lower plus offset := sum(-2, 23/5) After Left sum ORIGINAL sample46 lower plus offset = 13/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S326 S328 · calculationBefore Left sum ORIGINAL sample46 slope times x := mul(25, 13/5) After Left sum ORIGINAL sample46 slope times x = 65 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S327 S329 · calculationBefore Left sum ORIGINAL sample46 affine value plus c := sum(65, 12) After Left sum ORIGINAL sample46 affine value plus c = 77 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S328 S330 · calculationBefore Left sum ORIGINAL sample46 value times cell width := mul(77, 1/10) After Left sum ORIGINAL sample46 value times cell width = 77/10 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, S329 S331 · calculationBefore Left sum ORIGINAL accumulated term46 := sum(1679/20, 77/10) After Left sum ORIGINAL accumulated term46 = 1833/20 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, S330 S332 · impliesBefore source roles = [46, 50, 25, 12, -2, 3, 1/10] After derived = [13/5, 77/1, 77/10, 1833/20] 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, S331 S333 · calculationBefore Left sum ORIGINAL sample47 index times width := mul(47, 1/10) After Left sum ORIGINAL sample47 index times width = 47/10 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, S332 S334 · calculationBefore Left sum ORIGINAL sample47 lower plus offset := sum(-2, 47/10) After Left sum ORIGINAL sample47 lower plus offset = 27/10 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, S333 S335 · calculationBefore Left sum ORIGINAL sample47 slope times x := mul(25, 27/10) After Left sum ORIGINAL sample47 slope times x = 135/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S334 S336 · calculationBefore Left sum ORIGINAL sample47 affine value plus c := sum(135/2, 12) After Left sum ORIGINAL sample47 affine value plus c = 159/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S335 S337 · calculationBefore Left sum ORIGINAL sample47 value times cell width := mul(159/2, 1/10) After Left sum ORIGINAL sample47 value times cell width = 159/20 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, S336 S338 · calculationBefore Left sum ORIGINAL accumulated term47 := sum(1833/20, 159/20) After Left sum ORIGINAL accumulated term47 = 498/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S337 S339 · impliesBefore source roles = [47, 50, 25, 12, -2, 3, 1/10] After derived = [27/10, 159/2, 159/20, 498/5] 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, S338 S340 · calculationBefore Left sum ORIGINAL sample48 index times width := mul(48, 1/10) After Left sum ORIGINAL sample48 index times width = 24/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S339 S341 · calculationBefore Left sum ORIGINAL sample48 lower plus offset := sum(-2, 24/5) After Left sum ORIGINAL sample48 lower plus offset = 14/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S340 S342 · calculationBefore Left sum ORIGINAL sample48 slope times x := mul(25, 14/5) After Left sum ORIGINAL sample48 slope times x = 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, S341 S343 · calculationBefore Left sum ORIGINAL sample48 affine value plus c := sum(70, 12) After Left sum ORIGINAL sample48 affine value plus c = 82 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.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, S342 S344 · calculationBefore Left sum ORIGINAL sample48 value times cell width := mul(82, 1/10) After Left sum ORIGINAL sample48 value times cell width = 41/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S343 S345 · calculationBefore Left sum ORIGINAL accumulated term48 := sum(498/5, 41/5) After Left sum ORIGINAL accumulated term48 = 539/5 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S344 S346 · impliesBefore source roles = [48, 50, 25, 12, -2, 3, 1/10] After derived = [14/5, 82/1, 41/5, 539/5] 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, S345 S347 · calculationBefore Left sum ORIGINAL sample49 index times width := mul(49, 1/10) After Left sum ORIGINAL sample49 index times width = 49/10 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, S346 S348 · calculationBefore Left sum ORIGINAL sample49 lower plus offset := sum(-2, 49/10) After Left sum ORIGINAL sample49 lower plus offset = 29/10 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, S347 S349 · calculationBefore Left sum ORIGINAL sample49 slope times x := mul(25, 29/10) After Left sum ORIGINAL sample49 slope times x = 145/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S348 S350 · calculationBefore Left sum ORIGINAL sample49 affine value plus c := sum(145/2, 12) After Left sum ORIGINAL sample49 affine value plus c = 169/2 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S349 S351 · calculationBefore Left sum ORIGINAL sample49 value times cell width := mul(169/2, 1/10) After Left sum ORIGINAL sample49 value times cell width = 169/20 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, S350 S352 · calculationBefore Left sum ORIGINAL accumulated term49 := sum(539/5, 169/20) After Left sum ORIGINAL accumulated term49 = 465/4 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, S351 S353 · impliesBefore source roles = [49, 50, 25, 12, -2, 3, 1/10] After derived = [29/10, 169/2, 169/20, 465/4] 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, S352 S354 · impliesBefore source roles = [50, 25, 12, -2, 3] After derived = [1/10, 465/4] 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, S353 ## Reverse checks - S354 · 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. - S353 · 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. - S352 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term49 with exact rational arithmetic, not rounded decimals. - S351 · premise_and_result_audit — Recompute Left sum ORIGINAL sample49 value times cell width with exact rational arithmetic, not rounded decimals. - S350 · premise_and_result_audit — Recompute Left sum ORIGINAL sample49 affine value plus c with exact rational arithmetic, not rounded decimals. - S349 · premise_and_result_audit — Recompute Left sum ORIGINAL sample49 slope times x with exact rational arithmetic, not rounded decimals. - S348 · premise_and_result_audit — Recompute Left sum ORIGINAL sample49 lower plus offset with exact rational arithmetic, not rounded decimals. - S347 · premise_and_result_audit — Recompute Left sum ORIGINAL sample49 index times width with exact rational arithmetic, not rounded decimals. - S346 · 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. - S345 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term48 with exact rational arithmetic, not rounded decimals. - S344 · premise_and_result_audit — Recompute Left sum ORIGINAL sample48 value times cell width with exact rational arithmetic, not rounded decimals. - S343 · premise_and_result_audit — Recompute Left sum ORIGINAL sample48 affine value plus c with exact rational arithmetic, not rounded decimals. - S342 · premise_and_result_audit — Recompute Left sum ORIGINAL sample48 slope times x with exact rational arithmetic, not rounded decimals. - S341 · premise_and_result_audit — Recompute Left sum ORIGINAL sample48 lower plus offset with exact rational arithmetic, not rounded decimals. - S340 · premise_and_result_audit — Recompute Left sum ORIGINAL sample48 index times width with exact rational arithmetic, not rounded decimals. - S339 · 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. - S338 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term47 with exact rational arithmetic, not rounded decimals. - S337 · premise_and_result_audit — Recompute Left sum ORIGINAL sample47 value times cell width with exact rational arithmetic, not rounded decimals. - S336 · premise_and_result_audit — Recompute Left sum ORIGINAL sample47 affine value plus c with exact rational arithmetic, not rounded decimals. - S335 · premise_and_result_audit — Recompute Left sum ORIGINAL sample47 slope times x with exact rational arithmetic, not rounded decimals. - S334 · premise_and_result_audit — Recompute Left sum ORIGINAL sample47 lower plus offset with exact rational arithmetic, not rounded decimals. - S333 · premise_and_result_audit — Recompute Left sum ORIGINAL sample47 index times width with exact rational arithmetic, not rounded decimals. - S332 · 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. - S331 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term46 with exact rational arithmetic, not rounded decimals. - S330 · premise_and_result_audit — Recompute Left sum ORIGINAL sample46 value times cell width with exact rational arithmetic, not rounded decimals. - S329 · premise_and_result_audit — Recompute Left sum ORIGINAL sample46 affine value plus c with exact rational arithmetic, not rounded decimals. - S328 · premise_and_result_audit — Recompute Left sum ORIGINAL sample46 slope times x with exact rational arithmetic, not rounded decimals. - S327 · premise_and_result_audit — Recompute Left sum ORIGINAL sample46 lower plus offset with exact rational arithmetic, not rounded decimals. - S326 · premise_and_result_audit — Recompute Left sum ORIGINAL sample46 index times width with exact rational arithmetic, not rounded decimals. - S325 · 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. - S324 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term45 with exact rational arithmetic, not rounded decimals. - S323 · premise_and_result_audit — Recompute Left sum ORIGINAL sample45 value times cell width with exact rational arithmetic, not rounded decimals. - S322 · premise_and_result_audit — Recompute Left sum ORIGINAL sample45 affine value plus c with exact rational arithmetic, not rounded decimals. - S321 · premise_and_result_audit — Recompute Left sum ORIGINAL sample45 slope times x with exact rational arithmetic, not rounded decimals. - S320 · premise_and_result_audit — Recompute Left sum ORIGINAL sample45 lower plus offset with exact rational arithmetic, not rounded decimals. - S319 · premise_and_result_audit — Recompute Left sum ORIGINAL sample45 index times width with exact rational arithmetic, not rounded decimals. - S318 · 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. - S317 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term44 with exact rational arithmetic, not rounded decimals. - S316 · premise_and_result_audit — Recompute Left sum ORIGINAL sample44 value times cell width with exact rational arithmetic, not rounded decimals. - S315 · premise_and_result_audit — Recompute Left sum ORIGINAL sample44 affine value plus c with exact rational arithmetic, not rounded decimals. - S314 · premise_and_result_audit — Recompute Left sum ORIGINAL sample44 slope times x with exact rational arithmetic, not rounded decimals. - S313 · premise_and_result_audit — Recompute Left sum ORIGINAL sample44 lower plus offset with exact rational arithmetic, not rounded decimals. - S312 · premise_and_result_audit — Recompute Left sum ORIGINAL sample44 index times width with exact rational arithmetic, not rounded decimals. - S311 · 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. - S310 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term43 with exact rational arithmetic, not rounded decimals. - S309 · premise_and_result_audit — Recompute Left sum ORIGINAL sample43 value times cell width with exact rational arithmetic, not rounded decimals. - S308 · premise_and_result_audit — Recompute Left sum ORIGINAL sample43 affine value plus c with exact rational arithmetic, not rounded decimals. - S307 · premise_and_result_audit — Recompute Left sum ORIGINAL sample43 slope times x with exact rational arithmetic, not rounded decimals. - S306 · premise_and_result_audit — Recompute Left sum ORIGINAL sample43 lower plus offset with exact rational arithmetic, not rounded decimals. - S305 · premise_and_result_audit — Recompute Left sum ORIGINAL sample43 index times width with exact rational arithmetic, not rounded decimals. - S304 · 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. - S303 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term42 with exact rational arithmetic, not rounded decimals. - S302 · premise_and_result_audit — Recompute Left sum ORIGINAL sample42 value times cell width with exact rational arithmetic, not rounded decimals. - S301 · premise_and_result_audit — Recompute Left sum ORIGINAL sample42 affine value plus c with exact rational arithmetic, not rounded decimals. - S300 · premise_and_result_audit — Recompute Left sum ORIGINAL sample42 slope times x with exact rational arithmetic, not rounded decimals. - S299 · premise_and_result_audit — Recompute Left sum ORIGINAL sample42 lower plus offset with exact rational arithmetic, not rounded decimals. - S298 · premise_and_result_audit — Recompute Left sum ORIGINAL sample42 index times width with exact rational arithmetic, not rounded decimals. - S297 · 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. - S296 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term41 with exact rational arithmetic, not rounded decimals. - S295 · premise_and_result_audit — Recompute Left sum ORIGINAL sample41 value times cell width with exact rational arithmetic, not rounded decimals. - S294 · premise_and_result_audit — Recompute Left sum ORIGINAL sample41 affine value plus c with exact rational arithmetic, not rounded decimals. - S293 · premise_and_result_audit — Recompute Left sum ORIGINAL sample41 slope times x with exact rational arithmetic, not rounded decimals. - S292 · premise_and_result_audit — Recompute Left sum ORIGINAL sample41 lower plus offset with exact rational arithmetic, not rounded decimals. - S291 · premise_and_result_audit — Recompute Left sum ORIGINAL sample41 index times width with exact rational arithmetic, not rounded decimals. - S290 · 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. - S289 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term40 with exact rational arithmetic, not rounded decimals. - S288 · premise_and_result_audit — Recompute Left sum ORIGINAL sample40 value times cell width with exact rational arithmetic, not rounded decimals. - S287 · premise_and_result_audit — Recompute Left sum ORIGINAL sample40 affine value plus c with exact rational arithmetic, not rounded decimals. - S286 · premise_and_result_audit — Recompute Left sum ORIGINAL sample40 slope times x with exact rational arithmetic, not rounded decimals. - S285 · premise_and_result_audit — Recompute Left sum ORIGINAL sample40 lower plus offset with exact rational arithmetic, not rounded decimals. - S284 · premise_and_result_audit — Recompute Left sum ORIGINAL sample40 index times width with exact rational arithmetic, not rounded decimals. - S283 · 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. - S282 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term39 with exact rational arithmetic, not rounded decimals. - S281 · premise_and_result_audit — Recompute Left sum ORIGINAL sample39 value times cell width with exact rational arithmetic, not rounded decimals. - S280 · premise_and_result_audit — Recompute Left sum ORIGINAL sample39 affine value plus c with exact rational arithmetic, not rounded decimals. - S279 · premise_and_result_audit — Recompute Left sum ORIGINAL sample39 slope times x with exact rational arithmetic, not rounded decimals. - S278 · premise_and_result_audit — Recompute Left sum ORIGINAL sample39 lower plus offset with exact rational arithmetic, not rounded decimals. - S277 · premise_and_result_audit — Recompute Left sum ORIGINAL sample39 index times width with exact rational arithmetic, not rounded decimals. - S276 · 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. - S275 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term38 with exact rational arithmetic, not rounded decimals. - S274 · premise_and_result_audit — Recompute Left sum ORIGINAL sample38 value times cell width with exact rational arithmetic, not rounded decimals. - S273 · premise_and_result_audit — Recompute Left sum ORIGINAL sample38 affine value plus c with exact rational arithmetic, not rounded decimals. - S272 · premise_and_result_audit — Recompute Left sum ORIGINAL sample38 slope times x with exact rational arithmetic, not rounded decimals. - S271 · premise_and_result_audit — Recompute Left sum ORIGINAL sample38 lower plus offset with exact rational arithmetic, not rounded decimals. - S270 · premise_and_result_audit — Recompute Left sum ORIGINAL sample38 index times width with exact rational arithmetic, not rounded decimals. - S269 · 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. - S268 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term37 with exact rational arithmetic, not rounded decimals. - S267 · premise_and_result_audit — Recompute Left sum ORIGINAL sample37 value times cell width with exact rational arithmetic, not rounded decimals. - S266 · premise_and_result_audit — Recompute Left sum ORIGINAL sample37 affine value plus c with exact rational arithmetic, not rounded decimals. - S265 · premise_and_result_audit — Recompute Left sum ORIGINAL sample37 slope times x with exact rational arithmetic, not rounded decimals. - S264 · premise_and_result_audit — Recompute Left sum ORIGINAL sample37 lower plus offset with exact rational arithmetic, not rounded decimals. - S263 · premise_and_result_audit — Recompute Left sum ORIGINAL sample37 index times width with exact rational arithmetic, not rounded decimals. - S262 · 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. - S261 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term36 with exact rational arithmetic, not rounded decimals. - S260 · premise_and_result_audit — Recompute Left sum ORIGINAL sample36 value times cell width with exact rational arithmetic, not rounded decimals. - S259 · premise_and_result_audit — Recompute Left sum ORIGINAL sample36 affine value plus c with exact rational arithmetic, not rounded decimals. - S258 · premise_and_result_audit — Recompute Left sum ORIGINAL sample36 slope times x with exact rational arithmetic, not rounded decimals. - S257 · premise_and_result_audit — Recompute Left sum ORIGINAL sample36 lower plus offset with exact rational arithmetic, not rounded decimals. - S256 · premise_and_result_audit — Recompute Left sum ORIGINAL sample36 index times width with exact rational arithmetic, not rounded decimals. - S255 · 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. - S254 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term35 with exact rational arithmetic, not rounded decimals. - S253 · premise_and_result_audit — Recompute Left sum ORIGINAL sample35 value times cell width with exact rational arithmetic, not rounded decimals. - S252 · premise_and_result_audit — Recompute Left sum ORIGINAL sample35 affine value plus c with exact rational arithmetic, not rounded decimals. - S251 · premise_and_result_audit — Recompute Left sum ORIGINAL sample35 slope times x with exact rational arithmetic, not rounded decimals. - S250 · premise_and_result_audit — Recompute Left sum ORIGINAL sample35 lower plus offset with exact rational arithmetic, not rounded decimals. - S249 · premise_and_result_audit — Recompute Left sum ORIGINAL sample35 index times width with exact rational arithmetic, not rounded decimals. - S248 · 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. - S247 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term34 with exact rational arithmetic, not rounded decimals. - S246 · premise_and_result_audit — Recompute Left sum ORIGINAL sample34 value times cell width with exact rational arithmetic, not rounded decimals. - S245 · premise_and_result_audit — Recompute Left sum ORIGINAL sample34 affine value plus c with exact rational arithmetic, not rounded decimals. - S244 · premise_and_result_audit — Recompute Left sum ORIGINAL sample34 slope times x with exact rational arithmetic, not rounded decimals. - S243 · premise_and_result_audit — Recompute Left sum ORIGINAL sample34 lower plus offset with exact rational arithmetic, not rounded decimals. - S242 · premise_and_result_audit — Recompute Left sum ORIGINAL sample34 index times width with exact rational arithmetic, not rounded decimals. - S241 · 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. - S240 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term33 with exact rational arithmetic, not rounded decimals. - S239 · premise_and_result_audit — Recompute Left sum ORIGINAL sample33 value times cell width with exact rational arithmetic, not rounded decimals. - S238 · premise_and_result_audit — Recompute Left sum ORIGINAL sample33 affine value plus c with exact rational arithmetic, not rounded decimals. - S237 · premise_and_result_audit — Recompute Left sum ORIGINAL sample33 slope times x with exact rational arithmetic, not rounded decimals. - S236 · premise_and_result_audit — Recompute Left sum ORIGINAL sample33 lower plus offset with exact rational arithmetic, not rounded decimals. - S235 · premise_and_result_audit — Recompute Left sum ORIGINAL sample33 index times width with exact rational arithmetic, not rounded decimals. - S234 · 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. - S233 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term32 with exact rational arithmetic, not rounded decimals. - S232 · premise_and_result_audit — Recompute Left sum ORIGINAL sample32 value times cell width with exact rational arithmetic, not rounded decimals. - S231 · premise_and_result_audit — Recompute Left sum ORIGINAL sample32 affine value plus c with exact rational arithmetic, not rounded decimals. - S230 · premise_and_result_audit — Recompute Left sum ORIGINAL sample32 slope times x with exact rational arithmetic, not rounded decimals. - S229 · premise_and_result_audit — Recompute Left sum ORIGINAL sample32 lower plus offset with exact rational arithmetic, not rounded decimals. - S228 · premise_and_result_audit — Recompute Left sum ORIGINAL sample32 index times width with exact rational arithmetic, not rounded decimals. - S227 · 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. - S226 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term31 with exact rational arithmetic, not rounded decimals. - S225 · premise_and_result_audit — Recompute Left sum ORIGINAL sample31 value times cell width with exact rational arithmetic, not rounded decimals. - S224 · premise_and_result_audit — Recompute Left sum ORIGINAL sample31 affine value plus c with exact rational arithmetic, not rounded decimals. - S223 · premise_and_result_audit — Recompute Left sum ORIGINAL sample31 slope times x with exact rational arithmetic, not rounded decimals. - S222 · premise_and_result_audit — Recompute Left sum ORIGINAL sample31 lower plus offset with exact rational arithmetic, not rounded decimals. - S221 · premise_and_result_audit — Recompute Left sum ORIGINAL sample31 index times width with exact rational arithmetic, not rounded decimals. - S220 · 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. - S219 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term30 with exact rational arithmetic, not rounded decimals. - S218 · premise_and_result_audit — Recompute Left sum ORIGINAL sample30 value times cell width with exact rational arithmetic, not rounded decimals. - S217 · premise_and_result_audit — Recompute Left sum ORIGINAL sample30 affine value plus c with exact rational arithmetic, not rounded decimals. - S216 · premise_and_result_audit — Recompute Left sum ORIGINAL sample30 slope times x with exact rational arithmetic, not rounded decimals. - S215 · premise_and_result_audit — Recompute Left sum ORIGINAL sample30 lower plus offset with exact rational arithmetic, not rounded decimals. - S214 · premise_and_result_audit — Recompute Left sum ORIGINAL sample30 index times width with exact rational arithmetic, not rounded decimals. - S213 · 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. - S212 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term29 with exact rational arithmetic, not rounded decimals. - S211 · premise_and_result_audit — Recompute Left sum ORIGINAL sample29 value times cell width with exact rational arithmetic, not rounded decimals. - S210 · premise_and_result_audit — Recompute Left sum ORIGINAL sample29 affine value plus c with exact rational arithmetic, not rounded decimals. - S209 · premise_and_result_audit — Recompute Left sum ORIGINAL sample29 slope times x with exact rational arithmetic, not rounded decimals. - S208 · premise_and_result_audit — Recompute Left sum ORIGINAL sample29 lower plus offset with exact rational arithmetic, not rounded decimals. - S207 · premise_and_result_audit — Recompute Left sum ORIGINAL sample29 index times width with exact rational arithmetic, not rounded decimals. - S206 · 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. - S205 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term28 with exact rational arithmetic, not rounded decimals. - S204 · premise_and_result_audit — Recompute Left sum ORIGINAL sample28 value times cell width with exact rational arithmetic, not rounded decimals. - S203 · premise_and_result_audit — Recompute Left sum ORIGINAL sample28 affine value plus c with exact rational arithmetic, not rounded decimals. - S202 · premise_and_result_audit — Recompute Left sum ORIGINAL sample28 slope times x with exact rational arithmetic, not rounded decimals. - S201 · premise_and_result_audit — Recompute Left sum ORIGINAL sample28 lower plus offset with exact rational arithmetic, not rounded decimals. - S200 · premise_and_result_audit — Recompute Left sum ORIGINAL sample28 index times width with exact rational arithmetic, not rounded decimals. - S199 · 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. - S198 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term27 with exact rational arithmetic, not rounded decimals. - S197 · premise_and_result_audit — Recompute Left sum ORIGINAL sample27 value times cell width with exact rational arithmetic, not rounded decimals. - S196 · premise_and_result_audit — Recompute Left sum ORIGINAL sample27 affine value plus c with exact rational arithmetic, not rounded decimals. - S195 · premise_and_result_audit — Recompute Left sum ORIGINAL sample27 slope times x with exact rational arithmetic, not rounded decimals. - S194 · premise_and_result_audit — Recompute Left sum ORIGINAL sample27 lower plus offset with exact rational arithmetic, not rounded decimals. - S193 · premise_and_result_audit — Recompute Left sum ORIGINAL sample27 index times width with exact rational arithmetic, not rounded decimals. - S192 · 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. - S191 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term26 with exact rational arithmetic, not rounded decimals. - S190 · premise_and_result_audit — Recompute Left sum ORIGINAL sample26 value times cell width with exact rational arithmetic, not rounded decimals. - S189 · premise_and_result_audit — Recompute Left sum ORIGINAL sample26 affine value plus c with exact rational arithmetic, not rounded decimals. - S188 · premise_and_result_audit — Recompute Left sum ORIGINAL sample26 slope times x with exact rational arithmetic, not rounded decimals. - S187 · premise_and_result_audit — Recompute Left sum ORIGINAL sample26 lower plus offset with exact rational arithmetic, not rounded decimals. - S186 · premise_and_result_audit — Recompute Left sum ORIGINAL sample26 index times width with exact rational arithmetic, not rounded decimals. - S185 · 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. - S184 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term25 with exact rational arithmetic, not rounded decimals. - S183 · premise_and_result_audit — Recompute Left sum ORIGINAL sample25 value times cell width with exact rational arithmetic, not rounded decimals. - S182 · premise_and_result_audit — Recompute Left sum ORIGINAL sample25 affine value plus c with exact rational arithmetic, not rounded decimals. - S181 · premise_and_result_audit — Recompute Left sum ORIGINAL sample25 slope times x with exact rational arithmetic, not rounded decimals. - S180 · premise_and_result_audit — Recompute Left sum ORIGINAL sample25 lower plus offset with exact rational arithmetic, not rounded decimals. - S179 · premise_and_result_audit — Recompute Left sum ORIGINAL sample25 index times width with exact rational arithmetic, not rounded decimals. - S178 · 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. - S177 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term24 with exact rational arithmetic, not rounded decimals. - S176 · premise_and_result_audit — Recompute Left sum ORIGINAL sample24 value times cell width with exact rational arithmetic, not rounded decimals. - S175 · premise_and_result_audit — Recompute Left sum ORIGINAL sample24 affine value plus c with exact rational arithmetic, not rounded decimals. - S174 · premise_and_result_audit — Recompute Left sum ORIGINAL sample24 slope times x with exact rational arithmetic, not rounded decimals. - S173 · premise_and_result_audit — Recompute Left sum ORIGINAL sample24 lower plus offset with exact rational arithmetic, not rounded decimals. - S172 · premise_and_result_audit — Recompute Left sum ORIGINAL sample24 index times width with exact rational arithmetic, not rounded decimals. - S171 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S170 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term23 with exact rational arithmetic, not rounded decimals. - S169 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 value times cell width with exact rational arithmetic, not rounded decimals. - S168 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 affine value plus c with exact rational arithmetic, not rounded decimals. - S167 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 slope times x with exact rational arithmetic, not rounded decimals. - S166 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 lower plus offset with exact rational arithmetic, not rounded decimals. - S165 · premise_and_result_audit — Recompute Left sum ORIGINAL sample23 index times width with exact rational arithmetic, not rounded decimals. - S164 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S163 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term22 with exact rational arithmetic, not rounded decimals. - S162 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 value times cell width with exact rational arithmetic, not rounded decimals. - S161 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 affine value plus c with exact rational arithmetic, not rounded decimals. - S160 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 slope times x with exact rational arithmetic, not rounded decimals. - S159 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 lower plus offset with exact rational arithmetic, not rounded decimals. - S158 · premise_and_result_audit — Recompute Left sum ORIGINAL sample22 index times width with exact rational arithmetic, not rounded decimals. - S157 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S156 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term21 with exact rational arithmetic, not rounded decimals. - S155 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 value times cell width with exact rational arithmetic, not rounded decimals. - S154 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 affine value plus c with exact rational arithmetic, not rounded decimals. - S153 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 slope times x with exact rational arithmetic, not rounded decimals. - S152 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 lower plus offset with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Recompute Left sum ORIGINAL sample21 index times width with exact rational arithmetic, not rounded decimals. - S150 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S149 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term20 with exact rational arithmetic, not rounded decimals. - S148 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 value times cell width with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 affine value plus c with exact rational arithmetic, not rounded decimals. - S146 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 slope times x with exact rational arithmetic, not rounded decimals. - S145 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 lower plus offset with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute Left sum ORIGINAL sample20 index times width with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S142 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term19 with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 value times cell width with exact rational arithmetic, not rounded decimals. - S140 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 affine value plus c with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 slope times x with exact rational arithmetic, not rounded decimals. - S138 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 lower plus offset with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Recompute Left sum ORIGINAL sample19 index times width with exact rational arithmetic, not rounded decimals. - S136 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S135 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term18 with exact rational arithmetic, not rounded decimals. - S134 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 value times cell width with exact rational arithmetic, not rounded decimals. - S133 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 affine value plus c with exact rational arithmetic, not rounded decimals. - S132 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 slope times x with exact rational arithmetic, not rounded decimals. - S131 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 lower plus offset with exact rational arithmetic, not rounded decimals. - S130 · premise_and_result_audit — Recompute Left sum ORIGINAL sample18 index times width with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S128 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term17 with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 value times cell width with exact rational arithmetic, not rounded decimals. - S126 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 affine value plus c with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 slope times x with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 lower plus offset with exact rational arithmetic, not rounded decimals. - S123 · premise_and_result_audit — Recompute Left sum ORIGINAL sample17 index times width with exact rational arithmetic, not rounded decimals. - S122 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S121 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term16 with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 value times cell width with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 affine value plus c with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 slope times x with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 lower plus offset with exact rational arithmetic, not rounded decimals. - S116 · premise_and_result_audit — Recompute Left sum ORIGINAL sample16 index times width with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S114 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term15 with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 value times cell width with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 affine value plus c with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 slope times x with exact rational arithmetic, not rounded decimals. - S110 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 lower plus offset with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute Left sum ORIGINAL sample15 index times width with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S107 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term14 with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 value times cell width with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 affine value plus c with exact rational arithmetic, not rounded decimals. - S104 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 slope times x with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 lower plus offset with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Recompute Left sum ORIGINAL sample14 index times width with exact rational arithmetic, not rounded decimals. - S101 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S100 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term13 with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 value times cell width with exact rational arithmetic, not rounded decimals. - S098 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 affine value plus c with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 slope times x with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 lower plus offset with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Recompute Left sum ORIGINAL sample13 index times width with exact rational arithmetic, not rounded decimals. - S094 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S093 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term12 with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 value times cell width with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 affine value plus c with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 slope times x with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 lower plus offset with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute Left sum ORIGINAL sample12 index times width with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S086 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term11 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 value times cell width with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 affine value plus c with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 slope times x with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 lower plus offset with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute Left sum ORIGINAL sample11 index times width with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S079 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term10 with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 value times cell width with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 affine value plus c with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 slope times x with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 lower plus offset with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute Left sum ORIGINAL sample10 index times width with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S072 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term9 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 value times cell width with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 affine value plus c with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 slope times x with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 lower plus offset with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute Left sum ORIGINAL sample9 index times width with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S065 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term8 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 value times cell width with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 affine value plus c with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 slope times x with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 lower plus offset with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute Left sum ORIGINAL sample8 index times width with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S058 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term7 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 value times cell width with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 affine value plus c with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 slope times x with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 lower plus offset with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Left sum ORIGINAL sample7 index times width with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S051 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term6 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 value times cell width with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 affine value plus c with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 slope times x with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 lower plus offset with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Left sum ORIGINAL sample6 index times width with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S044 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term5 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 value times cell width with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 affine value plus c with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 slope times x with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 lower plus offset with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Left sum ORIGINAL sample5 index times width with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S037 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term4 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 value times cell width with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 affine value plus c with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 slope times x with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 lower plus offset with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Left sum ORIGINAL sample4 index times width with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S030 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term3 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 value times cell width with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 affine value plus c with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 slope times x with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 lower plus offset with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Left sum ORIGINAL sample3 index times width with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S023 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term2 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 value times cell width with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 affine value plus c with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 slope times x with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 lower plus offset with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Left sum ORIGINAL sample2 index times width with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S016 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term1 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 value times cell width with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 affine value plus c with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 slope times x with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 lower plus offset with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Left sum ORIGINAL sample1 index times width with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S009 · premise_and_result_audit — Recompute Left sum ORIGINAL accumulated term0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 value times cell width with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 affine value plus c with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 slope times x with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 lower plus offset with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Left sum ORIGINAL sample0 index times width with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Left sum ORIGINAL width over positive cell count with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Left sum ORIGINAL upper minus lower endpoint with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. ## Result 465/4 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: 1664eca412eac2416d1eb84b1244342c91944c03ac1be08b95e49b7a3305df39 - candidate record SHA-256: 7f47d196eb7bb914d89f194581bbf7ecdf720bb1c06ea65c8174c35704960d59 - 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-0100.json)