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R0006-03861 · q-20261007-advance-20261009-r06-definite-integral-0061
On all R, f(x)=(11)x+(-2). On a=0≤b=5, use N=11 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)=(11)x+(-2). On a=0≤b=5, use N=11 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, 0)
After
interval width = 5
Conditions: none
Compute the nonnegative interval width,allowing zero width.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
partition step := div(5, 11)
After
partition step = 5/11
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, 5/11)
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(0, 0)
After
left sample0 x = 0
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left sample0 mx := mul(11, 0)
After
left sample0 mx = 0
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(0, -2)
After
left sample0 value = -2
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
left sample0 rectangle := mul(-2, 5/11)
After
left sample0 rectangle = -10/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[0, -2, -10/11]
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, 5/11)
After
left sample1 offset = 5/11
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(0, 5/11)
After
left sample1 x = 5/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(11, 5/11)
After
left sample1 mx = 5
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(5, -2)
After
left sample1 value = 3
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
left sample1 rectangle := mul(3, 5/11)
After
left sample1 rectangle = 15/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[5/11, 3, 15/11]
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, 5/11)
After
left sample2 offset = 10/11
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(0, 10/11)
After
left sample2 x = 10/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(11, 10/11)
After
left sample2 mx = 10
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
left sample2 value := sum(10, -2)
After
left sample2 value = 8
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
left sample2 rectangle := mul(8, 5/11)
After
left sample2 rectangle = 40/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[10/11, 8, 40/11]
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, 5/11)
After
left sample3 offset = 15/11
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(0, 15/11)
After
left sample3 x = 15/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(11, 15/11)
After
left sample3 mx = 15
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(15, -2)
After
left sample3 value = 13
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
left sample3 rectangle := mul(13, 5/11)
After
left sample3 rectangle = 65/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[15/11, 13, 65/11]
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, 5/11)
After
left sample4 offset = 20/11
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(0, 20/11)
After
left sample4 x = 20/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(11, 20/11)
After
left sample4 mx = 20
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(20, -2)
After
left sample4 value = 18
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
left sample4 rectangle := mul(18, 5/11)
After
left sample4 rectangle = 90/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[20/11, 18, 90/11]
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, 5/11)
After
left sample5 offset = 25/11
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(0, 25/11)
After
left sample5 x = 25/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(11, 25/11)
After
left sample5 mx = 25
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(25, -2)
After
left sample5 value = 23
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
left sample5 rectangle := mul(23, 5/11)
After
left sample5 rectangle = 115/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[25/11, 23, 115/11]
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, 5/11)
After
left sample6 offset = 30/11
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(0, 30/11)
After
left sample6 x = 30/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
left sample6 mx := mul(11, 30/11)
After
left sample6 mx = 30
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left sample6 value := sum(30, -2)
After
left sample6 value = 28
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
left sample6 rectangle := mul(28, 5/11)
After
left sample6 rectangle = 140/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
[30/11, 28, 140/11]
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, 5/11)
After
left sample7 offset = 35/11
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(0, 35/11)
After
left sample7 x = 35/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
left sample7 mx := mul(11, 35/11)
After
left sample7 mx = 35
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
left sample7 value := sum(35, -2)
After
left sample7 value = 33
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
left sample7 rectangle := mul(33, 5/11)
After
left sample7 rectangle = 15
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
[35/11, 33, 15]
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, 5/11)
After
left sample8 offset = 40/11
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(0, 40/11)
After
left sample8 x = 40/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
left sample8 mx := mul(11, 40/11)
After
left sample8 mx = 40
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
left sample8 value := sum(40, -2)
After
left sample8 value = 38
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
left sample8 rectangle := mul(38, 5/11)
After
left sample8 rectangle = 190/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
[40/11, 38, 190/11]
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, 5/11)
After
left sample9 offset = 45/11
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(0, 45/11)
After
left sample9 x = 45/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
left sample9 mx := mul(11, 45/11)
After
left sample9 mx = 45
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
left sample9 value := sum(45, -2)
After
left sample9 value = 43
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
left sample9 rectangle := mul(43, 5/11)
After
left sample9 rectangle = 215/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
[45/11, 43, 215/11]
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, 5/11)
After
left sample10 offset = 50/11
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(0, 50/11)
After
left sample10 x = 50/11
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
left sample10 mx := mul(11, 50/11)
After
left sample10 mx = 50
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
left sample10 value := sum(50, -2)
After
left sample10 value = 48
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
left sample10 rectangle := mul(48, 5/11)
After
left sample10 rectangle = 240/11
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
[50/11, 48, 240/11]
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
finite left sum := sum(-10/11, 15/11, 40/11, 65/11, 90/11, 115/11, 140/11, 15, 190/11, 215/11, 240/11)
After
finite left sum = 115
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
affine integral primitive coefficient1 := div(-2, 1)
After
affine integral primitive coefficient1 = -2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
affine integral primitive coefficient2 := div(11, 2)
After
affine integral primitive coefficient2 = 11/2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
[0, -2, 11/2]
After
H′=the supplied polynomial;integrate from 0 to 5
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S072
Before
affine integral H(b) ascending coefficients=[0, -2, 11/2]
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, S073
Before
affine integral H(b) Horner multiply before degree1 := mul(11/2, 5)
After
affine integral H(b) Horner multiply before degree1 = 55/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S074
Before
affine integral H(b) Horner add degree1 := sum(55/2, -2)
After
affine integral H(b) Horner add degree1 = 51/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
affine integral H(b) Horner multiply before degree0 := mul(51/2, 5)
After
affine integral H(b) Horner multiply before degree0 = 255/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
affine integral H(b) Horner add degree0 := sum(255/2, 0)
After
affine integral H(b) Horner add degree0 = 255/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
affine integral H(a) ascending coefficients=[0, -2, 11/2]
After
evaluate at x=0
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, S078
Before
affine integral H(a) Horner multiply before degree1 := mul(11/2, 0)
After
affine integral H(a) Horner multiply before degree1 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
affine integral H(a) Horner add degree1 := sum(0, -2)
After
affine integral H(a) Horner add degree1 = -2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S080
Before
affine integral H(a) Horner multiply before degree0 := mul(-2, 0)
After
affine integral H(a) Horner multiply before degree0 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
affine integral H(a) Horner add degree0 := sum(0, 0)
After
affine integral H(a) Horner add degree0 = 0
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
affine integral oriented moment := sub(255/2, 0)
After
affine integral oriented moment = 255/2
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
integral minus left sum := sub(255/2, 115)
After
integral minus left sum = 25/2
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, S084
Before
I=255/2;S_N=115;N=11
After
signed finite-sum difference=25/2
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, S085
S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S085 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S084 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S081 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S080 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S078 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S077 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S075 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S072 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Recompute finite left sum 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.25/2
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: 23ea6919486002bbeab3a414d81975452c51e558bb2ed2e37bb71eb29ac2bd11candidate record SHA-256: 77d74a712992e74e1f2ce86a013623f08cae2147c258d9e0e85d350566e358d0semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED