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R0004-02556 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0081
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=2, b=-3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=6. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
G001 — Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=2, b=-3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=6. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).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
n=10; a=2; b=-3
After
S={0,...,9}; T(z)=(a·z+b) mod nConditions: none
Remainders lie in0,...,n−1,so every state has one successor in the same finite set;negative dividends do not produce negative canonical remainders.
Rule: finite_self_map · Uses: G001
Before
start=0
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S001
Before
point0 step1 product := mul(2, 0)
After
point0 step1 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
point0 step1 affine value := sum(0, -3)
After
point0 step1 affine value = -3
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Euclidean division of -3 by 10
After
-3=-1·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S004
Before
point0 step2 product := mul(2, 7)
After
point0 step2 product = 14
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
point0 step2 affine value := sum(14, -3)
After
point0 step2 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Euclidean division of 11 by 10
After
11=1·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S007
Before
point0 step3 product := mul(2, 1)
After
point0 step3 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
point0 step3 affine value := sum(2, -3)
After
point0 step3 affine value = -1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Euclidean division of -1 by 10
After
-1=-1·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S010
Before
point0 step4 product := mul(2, 9)
After
point0 step4 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
point0 step4 affine value := sum(18, -3)
After
point0 step4 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
Euclidean division of 15 by 10
After
15=1·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S013
Before
point0 step5 product := mul(2, 5)
After
point0 step5 product = 10
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
point0 step5 affine value := sum(10, -3)
After
point0 step5 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S016
Before
distinct trace=[0, 7, 1, 9, 5]; first repeated state=7
After
cycle_start=1; starting-point least period=null
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S017
Before
start=1
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S018
Before
point1 step1 product := mul(2, 1)
After
point1 step1 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
point1 step1 affine value := sum(2, -3)
After
point1 step1 affine value = -1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
Euclidean division of -1 by 10
After
-1=-1·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S021
Before
point1 step2 product := mul(2, 9)
After
point1 step2 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
point1 step2 affine value := sum(18, -3)
After
point1 step2 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
Euclidean division of 15 by 10
After
15=1·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S024
Before
point1 step3 product := mul(2, 5)
After
point1 step3 product = 10
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
point1 step3 affine value := sum(10, -3)
After
point1 step3 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S027
Before
point1 step4 product := mul(2, 7)
After
point1 step4 product = 14
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
point1 step4 affine value := sum(14, -3)
After
point1 step4 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
Euclidean division of 11 by 10
After
11=1·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S030
Before
distinct trace=[1, 9, 5, 7]; first repeated state=1
After
cycle_start=0; starting-point least period=4
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S031
Before
start=2
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S032
Before
point2 step1 product := mul(2, 2)
After
point2 step1 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
point2 step1 affine value := sum(4, -3)
After
point2 step1 affine value = 1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
Euclidean division of 1 by 10
After
1=0·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S035
Before
point2 step2 product := mul(2, 1)
After
point2 step2 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
point2 step2 affine value := sum(2, -3)
After
point2 step2 affine value = -1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
Euclidean division of -1 by 10
After
-1=-1·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S038
Before
point2 step3 product := mul(2, 9)
After
point2 step3 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
point2 step3 affine value := sum(18, -3)
After
point2 step3 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
Euclidean division of 15 by 10
After
15=1·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S041
Before
point2 step4 product := mul(2, 5)
After
point2 step4 product = 10
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
point2 step4 affine value := sum(10, -3)
After
point2 step4 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S044
Before
point2 step5 product := mul(2, 7)
After
point2 step5 product = 14
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
point2 step5 affine value := sum(14, -3)
After
point2 step5 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
Euclidean division of 11 by 10
After
11=1·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S047
Before
distinct trace=[2, 1, 9, 5, 7]; first repeated state=1
After
cycle_start=1; starting-point least period=null
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S048
Before
start=3
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S049
Before
point3 step1 product := mul(2, 3)
After
point3 step1 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
point3 step1 affine value := sum(6, -3)
After
point3 step1 affine value = 3
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
Euclidean division of 3 by 10
After
3=0·10+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S052
Before
distinct trace=[3]; first repeated state=3
After
cycle_start=0; starting-point least period=1
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S053
Before
start=4
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S054
Before
point4 step1 product := mul(2, 4)
After
point4 step1 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
point4 step1 affine value := sum(8, -3)
After
point4 step1 affine value = 5
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
Euclidean division of 5 by 10
After
5=0·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S057
Before
point4 step2 product := mul(2, 5)
After
point4 step2 product = 10
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
point4 step2 affine value := sum(10, -3)
After
point4 step2 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S060
Before
point4 step3 product := mul(2, 7)
After
point4 step3 product = 14
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
point4 step3 affine value := sum(14, -3)
After
point4 step3 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
Euclidean division of 11 by 10
After
11=1·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S063
Before
point4 step4 product := mul(2, 1)
After
point4 step4 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
point4 step4 affine value := sum(2, -3)
After
point4 step4 affine value = -1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
Euclidean division of -1 by 10
After
-1=-1·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S066
Before
point4 step5 product := mul(2, 9)
After
point4 step5 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
point4 step5 affine value := sum(18, -3)
After
point4 step5 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S068
Before
Euclidean division of 15 by 10
After
15=1·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S069
Before
distinct trace=[4, 5, 7, 1, 9]; first repeated state=5
After
cycle_start=1; starting-point least period=null
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S070
Before
start=5
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S071
Before
point5 step1 product := mul(2, 5)
After
point5 step1 product = 10
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
point5 step1 affine value := sum(10, -3)
After
point5 step1 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S074
Before
point5 step2 product := mul(2, 7)
After
point5 step2 product = 14
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
point5 step2 affine value := sum(14, -3)
After
point5 step2 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
Euclidean division of 11 by 10
After
11=1·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S077
Before
point5 step3 product := mul(2, 1)
After
point5 step3 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
point5 step3 affine value := sum(2, -3)
After
point5 step3 affine value = -1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
Euclidean division of -1 by 10
After
-1=-1·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S080
Before
point5 step4 product := mul(2, 9)
After
point5 step4 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
point5 step4 affine value := sum(18, -3)
After
point5 step4 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
Euclidean division of 15 by 10
After
15=1·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S083
Before
distinct trace=[5, 7, 1, 9]; first repeated state=5
After
cycle_start=0; starting-point least period=4
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S084
Before
start=6
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S085
Before
point6 step1 product := mul(2, 6)
After
point6 step1 product = 12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S086
Before
point6 step1 affine value := sum(12, -3)
After
point6 step1 affine value = 9
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
Euclidean division of 9 by 10
After
9=0·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S088
Before
point6 step2 product := mul(2, 9)
After
point6 step2 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
point6 step2 affine value := sum(18, -3)
After
point6 step2 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S090
Before
Euclidean division of 15 by 10
After
15=1·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S091
Before
point6 step3 product := mul(2, 5)
After
point6 step3 product = 10
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S092
Before
point6 step3 affine value := sum(10, -3)
After
point6 step3 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S094
Before
point6 step4 product := mul(2, 7)
After
point6 step4 product = 14
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
point6 step4 affine value := sum(14, -3)
After
point6 step4 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
Euclidean division of 11 by 10
After
11=1·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S097
Before
point6 step5 product := mul(2, 1)
After
point6 step5 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S098
Before
point6 step5 affine value := sum(2, -3)
After
point6 step5 affine value = -1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S099
Before
Euclidean division of -1 by 10
After
-1=-1·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S100
Before
distinct trace=[6, 9, 5, 7, 1]; first repeated state=9
After
cycle_start=1; starting-point least period=null
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S101
Before
start=7
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S102
Before
point7 step1 product := mul(2, 7)
After
point7 step1 product = 14
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
point7 step1 affine value := sum(14, -3)
After
point7 step1 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S104
Before
Euclidean division of 11 by 10
After
11=1·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S105
Before
point7 step2 product := mul(2, 1)
After
point7 step2 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S106
Before
point7 step2 affine value := sum(2, -3)
After
point7 step2 affine value = -1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
Euclidean division of -1 by 10
After
-1=-1·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S108
Before
point7 step3 product := mul(2, 9)
After
point7 step3 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
point7 step3 affine value := sum(18, -3)
After
point7 step3 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S110
Before
Euclidean division of 15 by 10
After
15=1·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S111
Before
point7 step4 product := mul(2, 5)
After
point7 step4 product = 10
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
point7 step4 affine value := sum(10, -3)
After
point7 step4 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S114
Before
distinct trace=[7, 1, 9, 5]; first repeated state=7
After
cycle_start=0; starting-point least period=4
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S115
Before
start=8
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S116
Before
point8 step1 product := mul(2, 8)
After
point8 step1 product = 16
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S117
Before
point8 step1 affine value := sum(16, -3)
After
point8 step1 affine value = 13
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S118
Before
Euclidean division of 13 by 10
After
13=1·10+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S119
Before
point8 step2 product := mul(2, 3)
After
point8 step2 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S120
Before
point8 step2 affine value := sum(6, -3)
After
point8 step2 affine value = 3
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
Euclidean division of 3 by 10
After
3=0·10+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S122
Before
distinct trace=[8, 3]; first repeated state=3
After
cycle_start=1; starting-point least period=null
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S123
Before
start=9
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S124
Before
point9 step1 product := mul(2, 9)
After
point9 step1 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S125
Before
point9 step1 affine value := sum(18, -3)
After
point9 step1 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S126
Before
Euclidean division of 15 by 10
After
15=1·10+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S127
Before
point9 step2 product := mul(2, 5)
After
point9 step2 product = 10
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S128
Before
point9 step2 affine value := sum(10, -3)
After
point9 step2 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S129
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S130
Before
point9 step3 product := mul(2, 7)
After
point9 step3 product = 14
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S131
Before
point9 step3 affine value := sum(14, -3)
After
point9 step3 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S132
Before
Euclidean division of 11 by 10
After
11=1·10+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S133
Before
point9 step4 product := mul(2, 1)
After
point9 step4 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S134
Before
point9 step4 affine value := sum(2, -3)
After
point9 step4 affine value = -1
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S135
Before
Euclidean division of -1 by 10
After
-1=-1·10+9
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S136
Before
distinct trace=[9, 5, 7, 1]; first repeated state=9
After
cycle_start=0; starting-point least period=4
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S137
Before
[[0, nonperiodic point], [1, 4], [2, nonperiodic point], [3, 1], [4, nonperiodic point], [5, 4], [6, nonperiodic point], [7, 4], [8, nonperiodic point], [9, 4]]
After
least positive period≤6: [1, 3, 5, 7, 9]
Conditions: none
Cover all n states;select only periodic points whose least positive period is at most the bound,not transient points that merely enter a cycle.
Rule: least_period_bound_filter · Uses: G001, S138
S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S138 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S136 · premise_and_result_audit — Recompute point9 step4 affine value with exact rational arithmetic, not rounded decimals.S135 · premise_and_result_audit — Recompute point9 step4 product with exact rational arithmetic, not rounded decimals.S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S133 · premise_and_result_audit — Recompute point9 step3 affine value with exact rational arithmetic, not rounded decimals.S132 · premise_and_result_audit — Recompute point9 step3 product with exact rational arithmetic, not rounded decimals.S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S130 · premise_and_result_audit — Recompute point9 step2 affine value with exact rational arithmetic, not rounded decimals.S129 · premise_and_result_audit — Recompute point9 step2 product with exact rational arithmetic, not rounded decimals.S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S127 · premise_and_result_audit — Recompute point9 step1 affine value with exact rational arithmetic, not rounded decimals.S126 · premise_and_result_audit — Recompute point9 step1 product with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S124 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S122 · premise_and_result_audit — Recompute point8 step2 affine value with exact rational arithmetic, not rounded decimals.S121 · premise_and_result_audit — Recompute point8 step2 product with exact rational arithmetic, not rounded decimals.S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S119 · premise_and_result_audit — Recompute point8 step1 affine value with exact rational arithmetic, not rounded decimals.S118 · premise_and_result_audit — Recompute point8 step1 product with exact rational arithmetic, not rounded decimals.S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S116 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S114 · premise_and_result_audit — Recompute point7 step4 affine value with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Recompute point7 step4 product with exact rational arithmetic, not rounded decimals.S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S111 · premise_and_result_audit — Recompute point7 step3 affine value with exact rational arithmetic, not rounded decimals.S110 · premise_and_result_audit — Recompute point7 step3 product 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 point7 step2 affine value with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Recompute point7 step2 product with exact rational arithmetic, not rounded decimals.S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S105 · premise_and_result_audit — Recompute point7 step1 affine value with exact rational arithmetic, not rounded decimals.S104 · premise_and_result_audit — Recompute point7 step1 product with exact rational arithmetic, not rounded decimals.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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S100 · premise_and_result_audit — Recompute point6 step5 affine value with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Recompute point6 step5 product with exact rational arithmetic, not rounded decimals.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 point6 step4 affine value with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Recompute point6 step4 product with exact rational arithmetic, not rounded decimals.S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S094 · premise_and_result_audit — Recompute point6 step3 affine value with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Recompute point6 step3 product with exact rational arithmetic, not rounded decimals.S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S091 · premise_and_result_audit — Recompute point6 step2 affine value with exact rational arithmetic, not rounded decimals.S090 · premise_and_result_audit — Recompute point6 step2 product with exact rational arithmetic, not rounded decimals.S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S088 · premise_and_result_audit — Recompute point6 step1 affine value with exact rational arithmetic, not rounded decimals.S087 · premise_and_result_audit — Recompute point6 step1 product 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S083 · premise_and_result_audit — Recompute point5 step4 affine value with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Recompute point5 step4 product 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 point5 step3 affine value with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Recompute point5 step3 product with exact rational arithmetic, not rounded decimals.S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S077 · premise_and_result_audit — Recompute point5 step2 affine value with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Recompute point5 step2 product 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 point5 step1 affine value with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Recompute point5 step1 product with exact rational arithmetic, not rounded decimals.S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S071 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S069 · premise_and_result_audit — Recompute point4 step5 affine value with exact rational arithmetic, not rounded decimals.S068 · premise_and_result_audit — Recompute point4 step5 product with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S066 · premise_and_result_audit — Recompute point4 step4 affine value with exact rational arithmetic, not rounded decimals.S065 · premise_and_result_audit — Recompute point4 step4 product with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S063 · premise_and_result_audit — Recompute point4 step3 affine value with exact rational arithmetic, not rounded decimals.S062 · premise_and_result_audit — Recompute point4 step3 product with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S060 · premise_and_result_audit — Recompute point4 step2 affine value with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute point4 step2 product with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S057 · premise_and_result_audit — Recompute point4 step1 affine value with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Recompute point4 step1 product with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S054 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S052 · premise_and_result_audit — Recompute point3 step1 affine value with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Recompute point3 step1 product with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S049 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S047 · premise_and_result_audit — Recompute point2 step5 affine value with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute point2 step5 product 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 point2 step4 affine value with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute point2 step4 product with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S041 · premise_and_result_audit — Recompute point2 step3 affine value with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute point2 step3 product 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 point2 step2 affine value with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute point2 step2 product with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Recompute point2 step1 affine value with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute point2 step1 product 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Recompute point1 step4 affine value with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute point1 step4 product with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Recompute point1 step3 affine value with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute point1 step3 product with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Recompute point1 step2 affine value with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute point1 step2 product with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S021 · premise_and_result_audit — Recompute point1 step1 affine value with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute point1 step1 product with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Recompute point0 step5 affine value with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute point0 step5 product with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Recompute point0 step4 affine value with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute point0 step4 product with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute point0 step3 affine value with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute point0 step3 product with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute point0 step2 affine value with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute point0 step2 product with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Recompute point0 step1 affine value with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute point0 step1 product with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(10, 2, -3, 6, [[0, nonperiodic-point], [1, 4], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, 4], [6, nonperiodic-point], [7, 4], [8, nonperiodic-point], [9, 4]], [1, 3, 5, 7, 9])
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: c9ee1ff35e53b7bc88c529786c3e1ab6299dc5462903c5e4b9210ca51c8372cbcandidate record SHA-256: 1e7d61678d09c46bb3b55a9d3e1cd2dade3c29000a04aaf7e6def7962bd917c2semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED