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← Question derivation index · Original question batch
R0006-03863 · q-20261007-advance-20261009-r06-definite-integral-0063
On all R, f(x)=(13)x+(0). On a=-2≤b=-1, use N=13 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)=(13)x+(0). On a=-2≤b=-1, use N=13 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(-1, -2)
After
interval width = 1
Conditions: none
Compute the nonnegative interval width,allowing zero width.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
partition step := div(1, 13)
After
partition step = 1/13
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, 1/13)
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(-2, 0)
After
left sample0 x = -2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left sample0 mx := mul(13, -2)
After
left sample0 mx = -26
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(-26, 0)
After
left sample0 value = -26
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
left sample0 rectangle := mul(-26, 1/13)
After
left sample0 rectangle = -2
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[-2, -26, -2]
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, 1/13)
After
left sample1 offset = 1/13
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(-2, 1/13)
After
left sample1 x = -25/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(13, -25/13)
After
left sample1 mx = -25
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(-25, 0)
After
left sample1 value = -25
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
left sample1 rectangle := mul(-25, 1/13)
After
left sample1 rectangle = -25/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[-25/13, -25, -25/13]
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, 1/13)
After
left sample2 offset = 2/13
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(-2, 2/13)
After
left sample2 x = -24/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(13, -24/13)
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, 0)
After
left sample2 value = -24
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
left sample2 rectangle := mul(-24, 1/13)
After
left sample2 rectangle = -24/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[-24/13, -24, -24/13]
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, 1/13)
After
left sample3 offset = 3/13
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(-2, 3/13)
After
left sample3 x = -23/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(13, -23/13)
After
left sample3 mx = -23
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(-23, 0)
After
left sample3 value = -23
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
left sample3 rectangle := mul(-23, 1/13)
After
left sample3 rectangle = -23/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[-23/13, -23, -23/13]
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, 1/13)
After
left sample4 offset = 4/13
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(-2, 4/13)
After
left sample4 x = -22/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(13, -22/13)
After
left sample4 mx = -22
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(-22, 0)
After
left sample4 value = -22
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
left sample4 rectangle := mul(-22, 1/13)
After
left sample4 rectangle = -22/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[-22/13, -22, -22/13]
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, 1/13)
After
left sample5 offset = 5/13
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(-2, 5/13)
After
left sample5 x = -21/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(13, -21/13)
After
left sample5 mx = -21
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(-21, 0)
After
left sample5 value = -21
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
left sample5 rectangle := mul(-21, 1/13)
After
left sample5 rectangle = -21/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[-21/13, -21, -21/13]
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, 1/13)
After
left sample6 offset = 6/13
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(-2, 6/13)
After
left sample6 x = -20/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
left sample6 mx := mul(13, -20/13)
After
left sample6 mx = -20
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left sample6 value := sum(-20, 0)
After
left sample6 value = -20
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
left sample6 rectangle := mul(-20, 1/13)
After
left sample6 rectangle = -20/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
[-20/13, -20, -20/13]
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, 1/13)
After
left sample7 offset = 7/13
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(-2, 7/13)
After
left sample7 x = -19/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
left sample7 mx := mul(13, -19/13)
After
left sample7 mx = -19
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
left sample7 value := sum(-19, 0)
After
left sample7 value = -19
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
left sample7 rectangle := mul(-19, 1/13)
After
left sample7 rectangle = -19/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
[-19/13, -19, -19/13]
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, 1/13)
After
left sample8 offset = 8/13
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(-2, 8/13)
After
left sample8 x = -18/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
left sample8 mx := mul(13, -18/13)
After
left sample8 mx = -18
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
left sample8 value := sum(-18, 0)
After
left sample8 value = -18
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
left sample8 rectangle := mul(-18, 1/13)
After
left sample8 rectangle = -18/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
[-18/13, -18, -18/13]
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, 1/13)
After
left sample9 offset = 9/13
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(-2, 9/13)
After
left sample9 x = -17/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
left sample9 mx := mul(13, -17/13)
After
left sample9 mx = -17
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
left sample9 value := sum(-17, 0)
After
left sample9 value = -17
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
left sample9 rectangle := mul(-17, 1/13)
After
left sample9 rectangle = -17/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
[-17/13, -17, -17/13]
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, 1/13)
After
left sample10 offset = 10/13
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(-2, 10/13)
After
left sample10 x = -16/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
left sample10 mx := mul(13, -16/13)
After
left sample10 mx = -16
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
left sample10 value := sum(-16, 0)
After
left sample10 value = -16
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
left sample10 rectangle := mul(-16, 1/13)
After
left sample10 rectangle = -16/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
[-16/13, -16, -16/13]
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, 1/13)
After
left sample11 offset = 11/13
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(-2, 11/13)
After
left sample11 x = -15/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
left sample11 mx := mul(13, -15/13)
After
left sample11 mx = -15
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
left sample11 value := sum(-15, 0)
After
left sample11 value = -15
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
left sample11 rectangle := mul(-15, 1/13)
After
left sample11 rectangle = -15/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
[-15/13, -15, -15/13]
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, 1/13)
After
left sample12 offset = 12/13
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(-2, 12/13)
After
left sample12 x = -14/13
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
left sample12 mx := mul(13, -14/13)
After
left sample12 mx = -14
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
left sample12 value := sum(-14, 0)
After
left sample12 value = -14
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
left sample12 rectangle := mul(-14, 1/13)
After
left sample12 rectangle = -14/13
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
[-14/13, -14, -14/13]
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
finite left sum := sum(-2, -25/13, -24/13, -23/13, -22/13, -21/13, -20/13, -19/13, -18/13, -17/13, -16/13, -15/13, -14/13)
After
finite left sum = -20
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
affine integral primitive coefficient1 := div(0, 1)
After
affine integral primitive coefficient1 = 0
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
affine integral primitive coefficient2 := div(13, 2)
After
affine integral primitive coefficient2 = 13/2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
[0, 0, 13/2]
After
H′=the supplied polynomial;integrate from -2 to -1
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S084
Before
affine integral H(b) ascending coefficients=[0, 0, 13/2]
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, S085
Before
affine integral H(b) Horner multiply before degree1 := mul(13/2, -1)
After
affine integral H(b) Horner multiply before degree1 = -13/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S086
Before
affine integral H(b) Horner add degree1 := sum(-13/2, 0)
After
affine integral H(b) Horner add degree1 = -13/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
affine integral H(b) Horner multiply before degree0 := mul(-13/2, -1)
After
affine integral H(b) Horner multiply before degree0 = 13/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
affine integral H(b) Horner add degree0 := sum(13/2, 0)
After
affine integral H(b) Horner add degree0 = 13/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
affine integral H(a) ascending coefficients=[0, 0, 13/2]
After
evaluate at x=-2
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, S090
Before
affine integral H(a) Horner multiply before degree1 := mul(13/2, -2)
After
affine integral H(a) Horner multiply before degree1 = -13
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
affine integral H(a) Horner add degree1 := sum(-13, 0)
After
affine integral H(a) Horner add degree1 = -13
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S092
Before
affine integral H(a) Horner multiply before degree0 := mul(-13, -2)
After
affine integral H(a) Horner multiply before degree0 = 26
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
affine integral H(a) Horner add degree0 := sum(26, 0)
After
affine integral H(a) Horner add degree0 = 26
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S094
Before
affine integral oriented moment := sub(13/2, 26)
After
affine integral oriented moment = -39/2
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
integral minus left sum := sub(-39/2, -20)
After
integral minus left sum = 1/2
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, S096
Before
I=-39/2;S_N=-20;N=13
After
signed finite-sum difference=1/2
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, S097
S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S097 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S095 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S094 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S092 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S090 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S089 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S088 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S087 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S084 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Recompute finite left sum 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.1/2
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: 9cf4030df9616b1539b5fc66b2f2a8db18c2c991601e8bbf79032ec471f24a45candidate record SHA-256: 54199de6155ee8cdabe1480a5c22ddb558cd4e9bd7166d52b17433d5bfe06b15semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED