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← Question derivation index · Original question batch
R0006-03866 · q-20261007-advance-20261009-r06-definite-integral-0066
On all R, f(x)=(-16)x+(3). On a=1≤b=5, use N=16 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, f(x)=(-16)x+(3). On a=1≤b=5, use N=16 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.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.
Before
explicit real polynomial moment or constant integrand model
After
oriented I(a,b)=H(b)−H(a);do not sort endpoints
Conditions: none
Real polynomials are continuous and integrable;use termwise primitive endpoint differences for exact oriented integrals,not absolute area or a finite sum.
Rule: polynomial_moment_scope · Uses: G001
Before
interval width := sub(5, 1)
After
interval width = 4
Conditions: none
Compute the nonnegative interval width,allowing zero width.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
partition step := div(4, 16)
After
partition step = 1/4
Conditions: none
Divide by positive integer N to get the exact step.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
left sample0 offset := mul(0, 1/4)
After
left sample0 offset = 0
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
left sample0 x := sum(1, 0)
After
left sample0 x = 1
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left sample0 mx := mul(-16, 1)
After
left sample0 mx = -16
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(-16, 3)
After
left sample0 value = -13
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
left sample0 rectangle := mul(-13, 1/4)
After
left sample0 rectangle = -13/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[1, -13, -13/4]
After
finite left-sum term index=0
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S008
Before
left sample1 offset := mul(1, 1/4)
After
left sample1 offset = 1/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
left sample1 x := sum(1, 1/4)
After
left sample1 x = 5/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(-16, 5/4)
After
left sample1 mx = -20
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(-20, 3)
After
left sample1 value = -17
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
left sample1 rectangle := mul(-17, 1/4)
After
left sample1 rectangle = -17/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[5/4, -17, -17/4]
After
finite left-sum term index=1
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S014
Before
left sample2 offset := mul(2, 1/4)
After
left sample2 offset = 1/2
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
left sample2 x := sum(1, 1/2)
After
left sample2 x = 3/2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(-16, 3/2)
After
left sample2 mx = -24
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
left sample2 value := sum(-24, 3)
After
left sample2 value = -21
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
left sample2 rectangle := mul(-21, 1/4)
After
left sample2 rectangle = -21/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[3/2, -21, -21/4]
After
finite left-sum term index=2
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S020
Before
left sample3 offset := mul(3, 1/4)
After
left sample3 offset = 3/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
left sample3 x := sum(1, 3/4)
After
left sample3 x = 7/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(-16, 7/4)
After
left sample3 mx = -28
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(-28, 3)
After
left sample3 value = -25
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
left sample3 rectangle := mul(-25, 1/4)
After
left sample3 rectangle = -25/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[7/4, -25, -25/4]
After
finite left-sum term index=3
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S026
Before
left sample4 offset := mul(4, 1/4)
After
left sample4 offset = 1
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
left sample4 x := sum(1, 1)
After
left sample4 x = 2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(-16, 2)
After
left sample4 mx = -32
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(-32, 3)
After
left sample4 value = -29
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
left sample4 rectangle := mul(-29, 1/4)
After
left sample4 rectangle = -29/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[2, -29, -29/4]
After
finite left-sum term index=4
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S032
Before
left sample5 offset := mul(5, 1/4)
After
left sample5 offset = 5/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
left sample5 x := sum(1, 5/4)
After
left sample5 x = 9/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(-16, 9/4)
After
left sample5 mx = -36
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(-36, 3)
After
left sample5 value = -33
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
left sample5 rectangle := mul(-33, 1/4)
After
left sample5 rectangle = -33/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[9/4, -33, -33/4]
After
finite left-sum term index=5
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S038
Before
left sample6 offset := mul(6, 1/4)
After
left sample6 offset = 3/2
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
left sample6 x := sum(1, 3/2)
After
left sample6 x = 5/2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
left sample6 mx := mul(-16, 5/2)
After
left sample6 mx = -40
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left sample6 value := sum(-40, 3)
After
left sample6 value = -37
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
left sample6 rectangle := mul(-37, 1/4)
After
left sample6 rectangle = -37/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
[5/2, -37, -37/4]
After
finite left-sum term index=6
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S044
Before
left sample7 offset := mul(7, 1/4)
After
left sample7 offset = 7/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
left sample7 x := sum(1, 7/4)
After
left sample7 x = 11/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
left sample7 mx := mul(-16, 11/4)
After
left sample7 mx = -44
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
left sample7 value := sum(-44, 3)
After
left sample7 value = -41
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
left sample7 rectangle := mul(-41, 1/4)
After
left sample7 rectangle = -41/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
[11/4, -41, -41/4]
After
finite left-sum term index=7
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S050
Before
left sample8 offset := mul(8, 1/4)
After
left sample8 offset = 2
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
left sample8 x := sum(1, 2)
After
left sample8 x = 3
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
left sample8 mx := mul(-16, 3)
After
left sample8 mx = -48
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
left sample8 value := sum(-48, 3)
After
left sample8 value = -45
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
left sample8 rectangle := mul(-45, 1/4)
After
left sample8 rectangle = -45/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
[3, -45, -45/4]
After
finite left-sum term index=8
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S056
Before
left sample9 offset := mul(9, 1/4)
After
left sample9 offset = 9/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
left sample9 x := sum(1, 9/4)
After
left sample9 x = 13/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
left sample9 mx := mul(-16, 13/4)
After
left sample9 mx = -52
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
left sample9 value := sum(-52, 3)
After
left sample9 value = -49
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
left sample9 rectangle := mul(-49, 1/4)
After
left sample9 rectangle = -49/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
[13/4, -49, -49/4]
After
finite left-sum term index=9
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S062
Before
left sample10 offset := mul(10, 1/4)
After
left sample10 offset = 5/2
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
left sample10 x := sum(1, 5/2)
After
left sample10 x = 7/2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
left sample10 mx := mul(-16, 7/2)
After
left sample10 mx = -56
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
left sample10 value := sum(-56, 3)
After
left sample10 value = -53
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
left sample10 rectangle := mul(-53, 1/4)
After
left sample10 rectangle = -53/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
[7/2, -53, -53/4]
After
finite left-sum term index=10
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S068
Before
left sample11 offset := mul(11, 1/4)
After
left sample11 offset = 11/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
left sample11 x := sum(1, 11/4)
After
left sample11 x = 15/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
left sample11 mx := mul(-16, 15/4)
After
left sample11 mx = -60
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
left sample11 value := sum(-60, 3)
After
left sample11 value = -57
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
left sample11 rectangle := mul(-57, 1/4)
After
left sample11 rectangle = -57/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
[15/4, -57, -57/4]
After
finite left-sum term index=11
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S074
Before
left sample12 offset := mul(12, 1/4)
After
left sample12 offset = 3
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
left sample12 x := sum(1, 3)
After
left sample12 x = 4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
left sample12 mx := mul(-16, 4)
After
left sample12 mx = -64
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
left sample12 value := sum(-64, 3)
After
left sample12 value = -61
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
left sample12 rectangle := mul(-61, 1/4)
After
left sample12 rectangle = -61/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
[4, -61, -61/4]
After
finite left-sum term index=12
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S080
Before
left sample13 offset := mul(13, 1/4)
After
left sample13 offset = 13/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
left sample13 x := sum(1, 13/4)
After
left sample13 x = 17/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
left sample13 mx := mul(-16, 17/4)
After
left sample13 mx = -68
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
left sample13 value := sum(-68, 3)
After
left sample13 value = -65
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S084
Before
left sample13 rectangle := mul(-65, 1/4)
After
left sample13 rectangle = -65/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
[17/4, -65, -65/4]
After
finite left-sum term index=13
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S086
Before
left sample14 offset := mul(14, 1/4)
After
left sample14 offset = 7/2
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
left sample14 x := sum(1, 7/2)
After
left sample14 x = 9/2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
left sample14 mx := mul(-16, 9/2)
After
left sample14 mx = -72
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
left sample14 value := sum(-72, 3)
After
left sample14 value = -69
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S090
Before
left sample14 rectangle := mul(-69, 1/4)
After
left sample14 rectangle = -69/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
[9/2, -69, -69/4]
After
finite left-sum term index=14
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S092
Before
left sample15 offset := mul(15, 1/4)
After
left sample15 offset = 15/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
left sample15 x := sum(1, 15/4)
After
left sample15 x = 19/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S094
Before
left sample15 mx := mul(-16, 19/4)
After
left sample15 mx = -76
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
left sample15 value := sum(-76, 3)
After
left sample15 value = -73
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
left sample15 rectangle := mul(-73, 1/4)
After
left sample15 rectangle = -73/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
[19/4, -73, -73/4]
After
finite left-sum term index=15
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S098
Before
finite left sum := sum(-13/4, -17/4, -21/4, -25/4, -29/4, -33/4, -37/4, -41/4, -45/4, -49/4, -53/4, -57/4, -61/4, -65/4, -69/4, -73/4)
After
finite left sum = -172
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S099
Before
affine integral primitive coefficient1 := div(3, 1)
After
affine integral primitive coefficient1 = 3
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S100
Before
affine integral primitive coefficient2 := div(-16, 2)
After
affine integral primitive coefficient2 = -8
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
[0, 3, -8]
After
H′=the supplied polynomial;integrate from 1 to 5
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S102
Before
affine integral H(b) ascending coefficients=[0, 3, -8]
After
evaluate at x=5
Conditions: none
Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.
Rule: polynomial_evaluation_order · Uses: G001, S103
Before
affine integral H(b) Horner multiply before degree1 := mul(-8, 5)
After
affine integral H(b) Horner multiply before degree1 = -40
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S104
Before
affine integral H(b) Horner add degree1 := sum(-40, 3)
After
affine integral H(b) Horner add degree1 = -37
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S105
Before
affine integral H(b) Horner multiply before degree0 := mul(-37, 5)
After
affine integral H(b) Horner multiply before degree0 = -185
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S106
Before
affine integral H(b) Horner add degree0 := sum(-185, 0)
After
affine integral H(b) Horner add degree0 = -185
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
affine integral H(a) ascending coefficients=[0, 3, -8]
After
evaluate at x=1
Conditions: none
Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.
Rule: polynomial_evaluation_order · Uses: G001, S108
Before
affine integral H(a) Horner multiply before degree1 := mul(-8, 1)
After
affine integral H(a) Horner multiply before degree1 = -8
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
affine integral H(a) Horner add degree1 := sum(-8, 3)
After
affine integral H(a) Horner add degree1 = -5
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S110
Before
affine integral H(a) Horner multiply before degree0 := mul(-5, 1)
After
affine integral H(a) Horner multiply before degree0 = -5
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S111
Before
affine integral H(a) Horner add degree0 := sum(-5, 0)
After
affine integral H(a) Horner add degree0 = -5
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
affine integral oriented moment := sub(-185, -5)
After
affine integral oriented moment = -180
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
integral minus left sum := sub(-180, -172)
After
integral minus left sum = -8
Conditions: none
The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error.
Rule: exact_rational_arithmetic · Uses: G001, S114
Before
I=-180;S_N=-172;N=16
After
signed finite-sum difference=-8
Conditions: none
This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions.
Rule: finite_sum_vs_integral · Uses: G001, S115
S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S115 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S114 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S112 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S111 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S110 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S108 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S106 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S105 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S102 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S101 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S100 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S098 · premise_and_result_audit — Recompute left sample15 rectangle with exact rational arithmetic, not rounded decimals.S097 · premise_and_result_audit — Recompute left sample15 value with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Recompute left sample15 mx with exact rational arithmetic, not rounded decimals.S095 · premise_and_result_audit — Recompute left sample15 x with exact rational arithmetic, not rounded decimals.S094 · premise_and_result_audit — Recompute left sample15 offset with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S092 · premise_and_result_audit — Recompute left sample14 rectangle with exact rational arithmetic, not rounded decimals.S091 · premise_and_result_audit — Recompute left sample14 value with exact rational arithmetic, not rounded decimals.S090 · premise_and_result_audit — Recompute left sample14 mx with exact rational arithmetic, not rounded decimals.S089 · premise_and_result_audit — Recompute left sample14 x with exact rational arithmetic, not rounded decimals.S088 · premise_and_result_audit — Recompute left sample14 offset with exact rational arithmetic, not rounded decimals.S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S086 · premise_and_result_audit — Recompute left sample13 rectangle with exact rational arithmetic, not rounded decimals.S085 · premise_and_result_audit — Recompute left sample13 value with exact rational arithmetic, not rounded decimals.S084 · premise_and_result_audit — Recompute left sample13 mx with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Recompute left sample13 x with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Recompute left sample13 offset with exact rational arithmetic, not rounded decimals.S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S080 · premise_and_result_audit — Recompute left sample12 rectangle with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Recompute left sample12 value with exact rational arithmetic, not rounded decimals.S078 · premise_and_result_audit — Recompute left sample12 mx with exact rational arithmetic, not rounded decimals.S077 · premise_and_result_audit — Recompute left sample12 x with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Recompute left sample12 offset with exact rational arithmetic, not rounded decimals.S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S074 · premise_and_result_audit — Recompute left sample11 rectangle with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Recompute left sample11 value with exact rational arithmetic, not rounded decimals.S072 · premise_and_result_audit — Recompute left sample11 mx with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Recompute left sample11 x with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Recompute left sample11 offset with exact rational arithmetic, not rounded decimals.S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S068 · premise_and_result_audit — Recompute left sample10 rectangle with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Recompute left sample10 value with exact rational arithmetic, not rounded decimals.S066 · premise_and_result_audit — Recompute left sample10 mx with exact rational arithmetic, not rounded decimals.S065 · premise_and_result_audit — Recompute left sample10 x with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Recompute left sample10 offset with exact rational arithmetic, not rounded decimals.S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S062 · premise_and_result_audit — Recompute left sample9 rectangle with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Recompute left sample9 value with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Recompute left sample9 mx with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute left sample9 x with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute left sample9 offset with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S056 · premise_and_result_audit — Recompute left sample8 rectangle with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute left sample8 value with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Recompute left sample8 mx with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute left sample8 x with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute left sample8 offset with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S050 · premise_and_result_audit — Recompute left sample7 rectangle with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute left sample7 value with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute left sample7 mx with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute left sample7 x with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute left sample7 offset with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Recompute left sample6 rectangle with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute left sample6 value with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute left sample6 mx with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute left sample6 x with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute left sample6 offset with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S038 · premise_and_result_audit — Recompute left sample5 rectangle with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute left sample5 value with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute left sample5 mx with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute left sample5 x with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute left sample5 offset with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S032 · premise_and_result_audit — Recompute left sample4 rectangle with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute left sample4 value with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute left sample4 mx with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute left sample4 x with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute left sample4 offset with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Recompute left sample3 rectangle with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute left sample3 value with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute left sample3 mx with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute left sample3 x with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute left sample3 offset with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Recompute left sample2 rectangle with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute left sample2 value with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute left sample2 mx with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute left sample2 x with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute left sample2 offset with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Recompute left sample1 rectangle with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute left sample1 value with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute left sample1 mx with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute left sample1 x with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute left sample1 offset with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute left sample0 rectangle with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute left sample0 value with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute left sample0 mx with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute left sample0 x with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute left sample0 offset with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute partition step with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute interval width with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.-8/1
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f02b1874afc95a1c7013873a6fb308e12ea2460d9ceff596b02c6676c2ab9245candidate record SHA-256: fe0dc948147bb7f1ec32e1ace6fb4c5c183a14d4dabd56dcb8fbc0f42d0bb425semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED