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R0004-02572 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0097
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-3, b=13. 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=4. 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(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-3, b=13. 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=4. 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=8; a=-3; b=13
After
S={0,...,7}; 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(-3, 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, 13)
After
point0 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, S003
Before
Euclidean division of 13 by 8
After
13=1·8+5
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(-3, 5)
After
point0 step2 product = -15
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(-15, 13)
After
point0 step2 affine value = -2
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 -2 by 8
After
-2=-1·8+6
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(-3, 6)
After
point0 step3 product = -18
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(-18, 13)
After
point0 step3 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, S009
Before
Euclidean division of -5 by 8
After
-5=-1·8+3
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(-3, 3)
After
point0 step4 product = -9
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(-9, 13)
After
point0 step4 affine value = 4
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 4 by 8
After
4=0·8+4
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(-3, 4)
After
point0 step5 product = -12
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(-12, 13)
After
point0 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, S015
Before
Euclidean division of 1 by 8
After
1=0·8+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S016
Before
point0 step6 product := mul(-3, 1)
After
point0 step6 product = -3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
point0 step6 affine value := sum(-3, 13)
After
point0 step6 affine value = 10
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
Euclidean division of 10 by 8
After
10=1·8+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S019
Before
point0 step7 product := mul(-3, 2)
After
point0 step7 product = -6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
point0 step7 affine value := sum(-6, 13)
After
point0 step7 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, S021
Before
Euclidean division of 7 by 8
After
7=0·8+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S022
Before
point0 step8 product := mul(-3, 7)
After
point0 step8 product = -21
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
point0 step8 affine value := sum(-21, 13)
After
point0 step8 affine value = -8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
Euclidean division of -8 by 8
After
-8=-1·8+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S025
Before
distinct trace=[0, 5, 6, 3, 4, 1, 2, 7]; first repeated state=0
After
cycle_start=0; starting-point least period=8
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, S026
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, S027
Before
point1 step1 product := mul(-3, 1)
After
point1 step1 product = -3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
point1 step1 affine value := sum(-3, 13)
After
point1 step1 affine value = 10
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 10 by 8
After
10=1·8+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S030
Before
point1 step2 product := mul(-3, 2)
After
point1 step2 product = -6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
point1 step2 affine value := sum(-6, 13)
After
point1 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, S032
Before
Euclidean division of 7 by 8
After
7=0·8+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S033
Before
point1 step3 product := mul(-3, 7)
After
point1 step3 product = -21
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
point1 step3 affine value := sum(-21, 13)
After
point1 step3 affine value = -8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
Euclidean division of -8 by 8
After
-8=-1·8+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S036
Before
point1 step4 product := mul(-3, 0)
After
point1 step4 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
point1 step4 affine value := sum(0, 13)
After
point1 step4 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, S038
Before
Euclidean division of 13 by 8
After
13=1·8+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S039
Before
point1 step5 product := mul(-3, 5)
After
point1 step5 product = -15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
point1 step5 affine value := sum(-15, 13)
After
point1 step5 affine value = -2
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
Euclidean division of -2 by 8
After
-2=-1·8+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S042
Before
point1 step6 product := mul(-3, 6)
After
point1 step6 product = -18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
point1 step6 affine value := sum(-18, 13)
After
point1 step6 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, S044
Before
Euclidean division of -5 by 8
After
-5=-1·8+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S045
Before
point1 step7 product := mul(-3, 3)
After
point1 step7 product = -9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
point1 step7 affine value := sum(-9, 13)
After
point1 step7 affine value = 4
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
Euclidean division of 4 by 8
After
4=0·8+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S048
Before
point1 step8 product := mul(-3, 4)
After
point1 step8 product = -12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
point1 step8 affine value := sum(-12, 13)
After
point1 step8 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, S050
Before
Euclidean division of 1 by 8
After
1=0·8+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S051
Before
distinct trace=[1, 2, 7, 0, 5, 6, 3, 4]; first repeated state=1
After
cycle_start=0; starting-point least period=8
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, S052
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, S053
Before
point2 step1 product := mul(-3, 2)
After
point2 step1 product = -6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
point2 step1 affine value := sum(-6, 13)
After
point2 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, S055
Before
Euclidean division of 7 by 8
After
7=0·8+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S056
Before
point2 step2 product := mul(-3, 7)
After
point2 step2 product = -21
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
point2 step2 affine value := sum(-21, 13)
After
point2 step2 affine value = -8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
Euclidean division of -8 by 8
After
-8=-1·8+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S059
Before
point2 step3 product := mul(-3, 0)
After
point2 step3 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
point2 step3 affine value := sum(0, 13)
After
point2 step3 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, S061
Before
Euclidean division of 13 by 8
After
13=1·8+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S062
Before
point2 step4 product := mul(-3, 5)
After
point2 step4 product = -15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
point2 step4 affine value := sum(-15, 13)
After
point2 step4 affine value = -2
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
Euclidean division of -2 by 8
After
-2=-1·8+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S065
Before
point2 step5 product := mul(-3, 6)
After
point2 step5 product = -18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
point2 step5 affine value := sum(-18, 13)
After
point2 step5 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, S067
Before
Euclidean division of -5 by 8
After
-5=-1·8+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S068
Before
point2 step6 product := mul(-3, 3)
After
point2 step6 product = -9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
point2 step6 affine value := sum(-9, 13)
After
point2 step6 affine value = 4
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
Euclidean division of 4 by 8
After
4=0·8+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S071
Before
point2 step7 product := mul(-3, 4)
After
point2 step7 product = -12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
point2 step7 affine value := sum(-12, 13)
After
point2 step7 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, S073
Before
Euclidean division of 1 by 8
After
1=0·8+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S074
Before
point2 step8 product := mul(-3, 1)
After
point2 step8 product = -3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
point2 step8 affine value := sum(-3, 13)
After
point2 step8 affine value = 10
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 10 by 8
After
10=1·8+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S077
Before
distinct trace=[2, 7, 0, 5, 6, 3, 4, 1]; first repeated state=2
After
cycle_start=0; starting-point least period=8
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, S078
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, S079
Before
point3 step1 product := mul(-3, 3)
After
point3 step1 product = -9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S080
Before
point3 step1 affine value := sum(-9, 13)
After
point3 step1 affine value = 4
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
Euclidean division of 4 by 8
After
4=0·8+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S082
Before
point3 step2 product := mul(-3, 4)
After
point3 step2 product = -12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
point3 step2 affine value := sum(-12, 13)
After
point3 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, S084
Before
Euclidean division of 1 by 8
After
1=0·8+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S085
Before
point3 step3 product := mul(-3, 1)
After
point3 step3 product = -3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S086
Before
point3 step3 affine value := sum(-3, 13)
After
point3 step3 affine value = 10
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 10 by 8
After
10=1·8+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S088
Before
point3 step4 product := mul(-3, 2)
After
point3 step4 product = -6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
point3 step4 affine value := sum(-6, 13)
After
point3 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, S090
Before
Euclidean division of 7 by 8
After
7=0·8+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S091
Before
point3 step5 product := mul(-3, 7)
After
point3 step5 product = -21
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S092
Before
point3 step5 affine value := sum(-21, 13)
After
point3 step5 affine value = -8
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 -8 by 8
After
-8=-1·8+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S094
Before
point3 step6 product := mul(-3, 0)
After
point3 step6 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
point3 step6 affine value := sum(0, 13)
After
point3 step6 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, S096
Before
Euclidean division of 13 by 8
After
13=1·8+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S097
Before
point3 step7 product := mul(-3, 5)
After
point3 step7 product = -15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S098
Before
point3 step7 affine value := sum(-15, 13)
After
point3 step7 affine value = -2
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 -2 by 8
After
-2=-1·8+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S100
Before
point3 step8 product := mul(-3, 6)
After
point3 step8 product = -18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
point3 step8 affine value := sum(-18, 13)
After
point3 step8 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, S102
Before
Euclidean division of -5 by 8
After
-5=-1·8+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S103
Before
distinct trace=[3, 4, 1, 2, 7, 0, 5, 6]; first repeated state=3
After
cycle_start=0; starting-point least period=8
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, S104
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, S105
Before
point4 step1 product := mul(-3, 4)
After
point4 step1 product = -12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S106
Before
point4 step1 affine value := sum(-12, 13)
After
point4 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, S107
Before
Euclidean division of 1 by 8
After
1=0·8+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S108
Before
point4 step2 product := mul(-3, 1)
After
point4 step2 product = -3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
point4 step2 affine value := sum(-3, 13)
After
point4 step2 affine value = 10
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 10 by 8
After
10=1·8+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S111
Before
point4 step3 product := mul(-3, 2)
After
point4 step3 product = -6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
point4 step3 affine value := sum(-6, 13)
After
point4 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, S113
Before
Euclidean division of 7 by 8
After
7=0·8+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
point4 step4 product := mul(-3, 7)
After
point4 step4 product = -21
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S115
Before
point4 step4 affine value := sum(-21, 13)
After
point4 step4 affine value = -8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S116
Before
Euclidean division of -8 by 8
After
-8=-1·8+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S117
Before
point4 step5 product := mul(-3, 0)
After
point4 step5 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S118
Before
point4 step5 affine value := sum(0, 13)
After
point4 step5 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, S119
Before
Euclidean division of 13 by 8
After
13=1·8+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S120
Before
point4 step6 product := mul(-3, 5)
After
point4 step6 product = -15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
point4 step6 affine value := sum(-15, 13)
After
point4 step6 affine value = -2
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S122
Before
Euclidean division of -2 by 8
After
-2=-1·8+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S123
Before
point4 step7 product := mul(-3, 6)
After
point4 step7 product = -18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S124
Before
point4 step7 affine value := sum(-18, 13)
After
point4 step7 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, S125
Before
Euclidean division of -5 by 8
After
-5=-1·8+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S126
Before
point4 step8 product := mul(-3, 3)
After
point4 step8 product = -9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S127
Before
point4 step8 affine value := sum(-9, 13)
After
point4 step8 affine value = 4
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S128
Before
Euclidean division of 4 by 8
After
4=0·8+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S129
Before
distinct trace=[4, 1, 2, 7, 0, 5, 6, 3]; first repeated state=4
After
cycle_start=0; starting-point least period=8
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, S130
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, S131
Before
point5 step1 product := mul(-3, 5)
After
point5 step1 product = -15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S132
Before
point5 step1 affine value := sum(-15, 13)
After
point5 step1 affine value = -2
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S133
Before
Euclidean division of -2 by 8
After
-2=-1·8+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S134
Before
point5 step2 product := mul(-3, 6)
After
point5 step2 product = -18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S135
Before
point5 step2 affine value := sum(-18, 13)
After
point5 step2 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, S136
Before
Euclidean division of -5 by 8
After
-5=-1·8+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S137
Before
point5 step3 product := mul(-3, 3)
After
point5 step3 product = -9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S138
Before
point5 step3 affine value := sum(-9, 13)
After
point5 step3 affine value = 4
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S139
Before
Euclidean division of 4 by 8
After
4=0·8+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S140
Before
point5 step4 product := mul(-3, 4)
After
point5 step4 product = -12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S141
Before
point5 step4 affine value := sum(-12, 13)
After
point5 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, S142
Before
Euclidean division of 1 by 8
After
1=0·8+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S143
Before
point5 step5 product := mul(-3, 1)
After
point5 step5 product = -3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S144
Before
point5 step5 affine value := sum(-3, 13)
After
point5 step5 affine value = 10
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S145
Before
Euclidean division of 10 by 8
After
10=1·8+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S146
Before
point5 step6 product := mul(-3, 2)
After
point5 step6 product = -6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S147
Before
point5 step6 affine value := sum(-6, 13)
After
point5 step6 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, S148
Before
Euclidean division of 7 by 8
After
7=0·8+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S149
Before
point5 step7 product := mul(-3, 7)
After
point5 step7 product = -21
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S150
Before
point5 step7 affine value := sum(-21, 13)
After
point5 step7 affine value = -8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S151
Before
Euclidean division of -8 by 8
After
-8=-1·8+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S152
Before
point5 step8 product := mul(-3, 0)
After
point5 step8 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S153
Before
point5 step8 affine value := sum(0, 13)
After
point5 step8 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, S154
Before
Euclidean division of 13 by 8
After
13=1·8+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S155
Before
distinct trace=[5, 6, 3, 4, 1, 2, 7, 0]; first repeated state=5
After
cycle_start=0; starting-point least period=8
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, S156
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, S157
Before
point6 step1 product := mul(-3, 6)
After
point6 step1 product = -18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S158
Before
point6 step1 affine value := sum(-18, 13)
After
point6 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, S159
Before
Euclidean division of -5 by 8
After
-5=-1·8+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S160
Before
point6 step2 product := mul(-3, 3)
After
point6 step2 product = -9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S161
Before
point6 step2 affine value := sum(-9, 13)
After
point6 step2 affine value = 4
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S162
Before
Euclidean division of 4 by 8
After
4=0·8+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S163
Before
point6 step3 product := mul(-3, 4)
After
point6 step3 product = -12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S164
Before
point6 step3 affine value := sum(-12, 13)
After
point6 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, S165
Before
Euclidean division of 1 by 8
After
1=0·8+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S166
Before
point6 step4 product := mul(-3, 1)
After
point6 step4 product = -3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S167
Before
point6 step4 affine value := sum(-3, 13)
After
point6 step4 affine value = 10
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S168
Before
Euclidean division of 10 by 8
After
10=1·8+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S169
Before
point6 step5 product := mul(-3, 2)
After
point6 step5 product = -6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S170
Before
point6 step5 affine value := sum(-6, 13)
After
point6 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, S171
Before
Euclidean division of 7 by 8
After
7=0·8+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S172
Before
point6 step6 product := mul(-3, 7)
After
point6 step6 product = -21
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S173
Before
point6 step6 affine value := sum(-21, 13)
After
point6 step6 affine value = -8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S174
Before
Euclidean division of -8 by 8
After
-8=-1·8+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S175
Before
point6 step7 product := mul(-3, 0)
After
point6 step7 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S176
Before
point6 step7 affine value := sum(0, 13)
After
point6 step7 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, S177
Before
Euclidean division of 13 by 8
After
13=1·8+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S178
Before
point6 step8 product := mul(-3, 5)
After
point6 step8 product = -15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S179
Before
point6 step8 affine value := sum(-15, 13)
After
point6 step8 affine value = -2
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S180
Before
Euclidean division of -2 by 8
After
-2=-1·8+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S181
Before
distinct trace=[6, 3, 4, 1, 2, 7, 0, 5]; first repeated state=6
After
cycle_start=0; starting-point least period=8
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, S182
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, S183
Before
point7 step1 product := mul(-3, 7)
After
point7 step1 product = -21
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S184
Before
point7 step1 affine value := sum(-21, 13)
After
point7 step1 affine value = -8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S185
Before
Euclidean division of -8 by 8
After
-8=-1·8+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S186
Before
point7 step2 product := mul(-3, 0)
After
point7 step2 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S187
Before
point7 step2 affine value := sum(0, 13)
After
point7 step2 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, S188
Before
Euclidean division of 13 by 8
After
13=1·8+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S189
Before
point7 step3 product := mul(-3, 5)
After
point7 step3 product = -15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S190
Before
point7 step3 affine value := sum(-15, 13)
After
point7 step3 affine value = -2
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S191
Before
Euclidean division of -2 by 8
After
-2=-1·8+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S192
Before
point7 step4 product := mul(-3, 6)
After
point7 step4 product = -18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S193
Before
point7 step4 affine value := sum(-18, 13)
After
point7 step4 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, S194
Before
Euclidean division of -5 by 8
After
-5=-1·8+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S195
Before
point7 step5 product := mul(-3, 3)
After
point7 step5 product = -9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S196
Before
point7 step5 affine value := sum(-9, 13)
After
point7 step5 affine value = 4
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S197
Before
Euclidean division of 4 by 8
After
4=0·8+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S198
Before
point7 step6 product := mul(-3, 4)
After
point7 step6 product = -12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S199
Before
point7 step6 affine value := sum(-12, 13)
After
point7 step6 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, S200
Before
Euclidean division of 1 by 8
After
1=0·8+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S201
Before
point7 step7 product := mul(-3, 1)
After
point7 step7 product = -3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S202
Before
point7 step7 affine value := sum(-3, 13)
After
point7 step7 affine value = 10
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S203
Before
Euclidean division of 10 by 8
After
10=1·8+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S204
Before
point7 step8 product := mul(-3, 2)
After
point7 step8 product = -6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S205
Before
point7 step8 affine value := sum(-6, 13)
After
point7 step8 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, S206
Before
Euclidean division of 7 by 8
After
7=0·8+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S207
Before
distinct trace=[7, 0, 5, 6, 3, 4, 1, 2]; first repeated state=7
After
cycle_start=0; starting-point least period=8
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, S208
Before
[[0, 8], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8]]
After
least positive period≤4: []
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, S209
S210 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S209 · 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.S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S207 · premise_and_result_audit — Recompute point7 step8 affine value with exact rational arithmetic, not rounded decimals.S206 · premise_and_result_audit — Recompute point7 step8 product with exact rational arithmetic, not rounded decimals.S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S204 · premise_and_result_audit — Recompute point7 step7 affine value with exact rational arithmetic, not rounded decimals.S203 · premise_and_result_audit — Recompute point7 step7 product with exact rational arithmetic, not rounded decimals.S202 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S201 · premise_and_result_audit — Recompute point7 step6 affine value with exact rational arithmetic, not rounded decimals.S200 · premise_and_result_audit — Recompute point7 step6 product with exact rational arithmetic, not rounded decimals.S199 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S198 · premise_and_result_audit — Recompute point7 step5 affine value with exact rational arithmetic, not rounded decimals.S197 · premise_and_result_audit — Recompute point7 step5 product with exact rational arithmetic, not rounded decimals.S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S195 · premise_and_result_audit — Recompute point7 step4 affine value with exact rational arithmetic, not rounded decimals.S194 · premise_and_result_audit — Recompute point7 step4 product with exact rational arithmetic, not rounded decimals.S193 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S192 · premise_and_result_audit — Recompute point7 step3 affine value with exact rational arithmetic, not rounded decimals.S191 · premise_and_result_audit — Recompute point7 step3 product with exact rational arithmetic, not rounded decimals.S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S189 · premise_and_result_audit — Recompute point7 step2 affine value with exact rational arithmetic, not rounded decimals.S188 · premise_and_result_audit — Recompute point7 step2 product with exact rational arithmetic, not rounded decimals.S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S186 · premise_and_result_audit — Recompute point7 step1 affine value with exact rational arithmetic, not rounded decimals.S185 · premise_and_result_audit — Recompute point7 step1 product with exact rational arithmetic, not rounded decimals.S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S183 · 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.S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S181 · premise_and_result_audit — Recompute point6 step8 affine value with exact rational arithmetic, not rounded decimals.S180 · premise_and_result_audit — Recompute point6 step8 product with exact rational arithmetic, not rounded decimals.S179 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S178 · premise_and_result_audit — Recompute point6 step7 affine value with exact rational arithmetic, not rounded decimals.S177 · premise_and_result_audit — Recompute point6 step7 product with exact rational arithmetic, not rounded decimals.S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S175 · premise_and_result_audit — Recompute point6 step6 affine value with exact rational arithmetic, not rounded decimals.S174 · premise_and_result_audit — Recompute point6 step6 product with exact rational arithmetic, not rounded decimals.S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S172 · premise_and_result_audit — Recompute point6 step5 affine value with exact rational arithmetic, not rounded decimals.S171 · premise_and_result_audit — Recompute point6 step5 product with exact rational arithmetic, not rounded decimals.S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S169 · premise_and_result_audit — Recompute point6 step4 affine value with exact rational arithmetic, not rounded decimals.S168 · premise_and_result_audit — Recompute point6 step4 product with exact rational arithmetic, not rounded decimals.S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S166 · premise_and_result_audit — Recompute point6 step3 affine value with exact rational arithmetic, not rounded decimals.S165 · premise_and_result_audit — Recompute point6 step3 product with exact rational arithmetic, not rounded decimals.S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S163 · premise_and_result_audit — Recompute point6 step2 affine value with exact rational arithmetic, not rounded decimals.S162 · premise_and_result_audit — Recompute point6 step2 product with exact rational arithmetic, not rounded decimals.S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S160 · premise_and_result_audit — Recompute point6 step1 affine value with exact rational arithmetic, not rounded decimals.S159 · premise_and_result_audit — Recompute point6 step1 product with exact rational arithmetic, not rounded decimals.S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S157 · 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.S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S155 · premise_and_result_audit — Recompute point5 step8 affine value with exact rational arithmetic, not rounded decimals.S154 · premise_and_result_audit — Recompute point5 step8 product with exact rational arithmetic, not rounded decimals.S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S152 · premise_and_result_audit — Recompute point5 step7 affine value with exact rational arithmetic, not rounded decimals.S151 · premise_and_result_audit — Recompute point5 step7 product with exact rational arithmetic, not rounded decimals.S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S149 · premise_and_result_audit — Recompute point5 step6 affine value with exact rational arithmetic, not rounded decimals.S148 · premise_and_result_audit — Recompute point5 step6 product with exact rational arithmetic, not rounded decimals.S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S146 · premise_and_result_audit — Recompute point5 step5 affine value with exact rational arithmetic, not rounded decimals.S145 · premise_and_result_audit — Recompute point5 step5 product with exact rational arithmetic, not rounded decimals.S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S143 · premise_and_result_audit — Recompute point5 step4 affine value with exact rational arithmetic, not rounded decimals.S142 · premise_and_result_audit — Recompute point5 step4 product with exact rational arithmetic, not rounded decimals.S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S140 · premise_and_result_audit — Recompute point5 step3 affine value with exact rational arithmetic, not rounded decimals.S139 · premise_and_result_audit — Recompute point5 step3 product with exact rational arithmetic, not rounded decimals.S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S137 · premise_and_result_audit — Recompute point5 step2 affine value with exact rational arithmetic, not rounded decimals.S136 · premise_and_result_audit — Recompute point5 step2 product with exact rational arithmetic, not rounded decimals.S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S134 · premise_and_result_audit — Recompute point5 step1 affine value with exact rational arithmetic, not rounded decimals.S133 · premise_and_result_audit — Recompute point5 step1 product with exact rational arithmetic, not rounded decimals.S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S131 · 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.S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S129 · premise_and_result_audit — Recompute point4 step8 affine value with exact rational arithmetic, not rounded decimals.S128 · premise_and_result_audit — Recompute point4 step8 product with exact rational arithmetic, not rounded decimals.S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S126 · premise_and_result_audit — Recompute point4 step7 affine value with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Recompute point4 step7 product with exact rational arithmetic, not rounded decimals.S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S123 · premise_and_result_audit — Recompute point4 step6 affine value with exact rational arithmetic, not rounded decimals.S122 · premise_and_result_audit — Recompute point4 step6 product with exact rational arithmetic, not rounded decimals.S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S120 · premise_and_result_audit — Recompute point4 step5 affine value with exact rational arithmetic, not rounded decimals.S119 · premise_and_result_audit — Recompute point4 step5 product with exact rational arithmetic, not rounded decimals.S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S117 · premise_and_result_audit — Recompute point4 step4 affine value with exact rational arithmetic, not rounded decimals.S116 · premise_and_result_audit — Recompute point4 step4 product with exact rational arithmetic, not rounded decimals.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 point4 step3 affine value with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Recompute point4 step3 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 point4 step2 affine value with exact rational arithmetic, not rounded decimals.S110 · premise_and_result_audit — Recompute point4 step2 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 point4 step1 affine value with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Recompute point4 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S103 · premise_and_result_audit — Recompute point3 step8 affine value with exact rational arithmetic, not rounded decimals.S102 · premise_and_result_audit — Recompute point3 step8 product with exact rational arithmetic, not rounded decimals.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 point3 step7 affine value with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Recompute point3 step7 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 point3 step6 affine value with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Recompute point3 step6 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 point3 step5 affine value with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Recompute point3 step5 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 point3 step4 affine value with exact rational arithmetic, not rounded decimals.S090 · premise_and_result_audit — Recompute point3 step4 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 point3 step3 affine value with exact rational arithmetic, not rounded decimals.S087 · premise_and_result_audit — Recompute point3 step3 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 — Recompute point3 step2 affine value with exact rational arithmetic, not rounded decimals.S084 · premise_and_result_audit — Recompute point3 step2 product with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S082 · premise_and_result_audit — Recompute point3 step1 affine value with exact rational arithmetic, not rounded decimals.S081 · premise_and_result_audit — Recompute point3 step1 product with exact rational arithmetic, not rounded decimals.S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S079 · 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.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 point2 step8 affine value with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Recompute point2 step8 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 point2 step7 affine value with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Recompute point2 step7 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 — Recompute point2 step6 affine value with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Recompute point2 step6 product 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 point2 step5 affine value with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Recompute point2 step5 product with exact rational arithmetic, not rounded decimals.S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S065 · premise_and_result_audit — Recompute point2 step4 affine value with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Recompute point2 step4 product 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 point2 step3 affine value with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Recompute point2 step3 product with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S059 · premise_and_result_audit — Recompute point2 step2 affine value with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute point2 step2 product 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 point2 step1 affine value with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute point2 step1 product with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S053 · 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.S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S051 · premise_and_result_audit — Recompute point1 step8 affine value with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute point1 step8 product with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S048 · premise_and_result_audit — Recompute point1 step7 affine value with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute point1 step7 product with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S045 · premise_and_result_audit — Recompute point1 step6 affine value with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Recompute point1 step6 product with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S042 · premise_and_result_audit — Recompute point1 step5 affine value with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute point1 step5 product with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S039 · premise_and_result_audit — Recompute point1 step4 affine value with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute point1 step4 product with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S036 · premise_and_result_audit — Recompute point1 step3 affine value with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute point1 step3 product with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S033 · premise_and_result_audit — Recompute point1 step2 affine value with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute point1 step2 product with exact rational arithmetic, not rounded decimals.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 step1 affine value with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute point1 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Recompute point0 step8 affine value with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute point0 step8 product with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Recompute point0 step7 affine value with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute point0 step7 product with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Recompute point0 step6 affine value with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute point0 step6 product with exact rational arithmetic, not rounded decimals.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.(8, -3, 13, 4, [[0, 8], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8]], [])
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: 695246bfd35a007ec0cf502fa38f3c6253fccdc99eb1c03b12c3a099fc5c4471candidate record SHA-256: 79e86bcb6bdbad14ee5a4bb2845cf61344d657dbfb6241a18ac953f57a1bbc94semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED