← Question derivation index · Original question batch
R0004-02555 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0080
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, b=-4. 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=5. 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(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, b=-4. 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=5. 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=9; a=1; b=-4
After
S={0,...,8}; 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(1, 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, -4)
After
point0 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, S003
Before
Euclidean division of -4 by 9
After
-4=-1·9+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(1, 5)
After
point0 step2 product = 5
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(5, -4)
After
point0 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, S006
Before
Euclidean division of 1 by 9
After
1=0·9+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(1, 1)
After
point0 step3 product = 1
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(1, -4)
After
point0 step3 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, S009
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
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(1, 6)
After
point0 step4 product = 6
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(6, -4)
After
point0 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, S012
Before
Euclidean division of 2 by 9
After
2=0·9+2
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(1, 2)
After
point0 step5 product = 2
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(2, -4)
After
point0 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, S015
Before
Euclidean division of -2 by 9
After
-2=-1·9+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
point0 step6 product := mul(1, 7)
After
point0 step6 product = 7
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(7, -4)
After
point0 step6 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, S018
Before
Euclidean division of 3 by 9
After
3=0·9+3
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(1, 3)
After
point0 step7 product = 3
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(3, -4)
After
point0 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, S021
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
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(1, 8)
After
point0 step8 product = 8
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(8, -4)
After
point0 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, S024
Before
Euclidean division of 4 by 9
After
4=0·9+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S025
Before
point0 step9 product := mul(1, 4)
After
point0 step9 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
point0 step9 affine value := sum(4, -4)
After
point0 step9 affine value = 0
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
Euclidean division of 0 by 9
After
0=0·9+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S028
Before
distinct trace=[0, 5, 1, 6, 2, 7, 3, 8, 4]; first repeated state=0
After
cycle_start=0; starting-point least period=9
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, S029
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, S030
Before
point1 step1 product := mul(1, 1)
After
point1 step1 product = 1
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
point1 step1 affine value := sum(1, -4)
After
point1 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, S032
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
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 step2 product := mul(1, 6)
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, S034
Before
point1 step2 affine value := sum(6, -4)
After
point1 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, S035
Before
Euclidean division of 2 by 9
After
2=0·9+2
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 step3 product := mul(1, 2)
After
point1 step3 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
point1 step3 affine value := sum(2, -4)
After
point1 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, S038
Before
Euclidean division of -2 by 9
After
-2=-1·9+7
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 step4 product := mul(1, 7)
After
point1 step4 product = 7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
point1 step4 affine value := sum(7, -4)
After
point1 step4 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, S041
Before
Euclidean division of 3 by 9
After
3=0·9+3
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 step5 product := mul(1, 3)
After
point1 step5 product = 3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
point1 step5 affine value := sum(3, -4)
After
point1 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, S044
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
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 step6 product := mul(1, 8)
After
point1 step6 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
point1 step6 affine value := sum(8, -4)
After
point1 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, S047
Before
Euclidean division of 4 by 9
After
4=0·9+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 step7 product := mul(1, 4)
After
point1 step7 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
point1 step7 affine value := sum(4, -4)
After
point1 step7 affine value = 0
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 0 by 9
After
0=0·9+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S051
Before
point1 step8 product := mul(1, 0)
After
point1 step8 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
point1 step8 affine value := sum(0, -4)
After
point1 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, S053
Before
Euclidean division of -4 by 9
After
-4=-1·9+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S054
Before
point1 step9 product := mul(1, 5)
After
point1 step9 product = 5
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
point1 step9 affine value := sum(5, -4)
After
point1 step9 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, S056
Before
Euclidean division of 1 by 9
After
1=0·9+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S057
Before
distinct trace=[1, 6, 2, 7, 3, 8, 4, 0, 5]; first repeated state=1
After
cycle_start=0; starting-point least period=9
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, S058
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, S059
Before
point2 step1 product := mul(1, 2)
After
point2 step1 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
point2 step1 affine value := sum(2, -4)
After
point2 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, S061
Before
Euclidean division of -2 by 9
After
-2=-1·9+7
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 step2 product := mul(1, 7)
After
point2 step2 product = 7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
point2 step2 affine value := sum(7, -4)
After
point2 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, S064
Before
Euclidean division of 3 by 9
After
3=0·9+3
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 step3 product := mul(1, 3)
After
point2 step3 product = 3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
point2 step3 affine value := sum(3, -4)
After
point2 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, S067
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
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 step4 product := mul(1, 8)
After
point2 step4 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
point2 step4 affine value := sum(8, -4)
After
point2 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, S070
Before
Euclidean division of 4 by 9
After
4=0·9+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 step5 product := mul(1, 4)
After
point2 step5 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
point2 step5 affine value := sum(4, -4)
After
point2 step5 affine value = 0
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 0 by 9
After
0=0·9+0
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 step6 product := mul(1, 0)
After
point2 step6 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
point2 step6 affine value := sum(0, -4)
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, S076
Before
Euclidean division of -4 by 9
After
-4=-1·9+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S077
Before
point2 step7 product := mul(1, 5)
After
point2 step7 product = 5
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
point2 step7 affine value := sum(5, -4)
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, S079
Before
Euclidean division of 1 by 9
After
1=0·9+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S080
Before
point2 step8 product := mul(1, 1)
After
point2 step8 product = 1
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
point2 step8 affine value := sum(1, -4)
After
point2 step8 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, S082
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S083
Before
point2 step9 product := mul(1, 6)
After
point2 step9 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S084
Before
point2 step9 affine value := sum(6, -4)
After
point2 step9 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, S085
Before
Euclidean division of 2 by 9
After
2=0·9+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S086
Before
distinct trace=[2, 7, 3, 8, 4, 0, 5, 1, 6]; first repeated state=2
After
cycle_start=0; starting-point least period=9
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, S087
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, S088
Before
point3 step1 product := mul(1, 3)
After
point3 step1 product = 3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
point3 step1 affine value := sum(3, -4)
After
point3 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, S090
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
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 step2 product := mul(1, 8)
After
point3 step2 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S092
Before
point3 step2 affine value := sum(8, -4)
After
point3 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, S093
Before
Euclidean division of 4 by 9
After
4=0·9+4
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 step3 product := mul(1, 4)
After
point3 step3 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
point3 step3 affine value := sum(4, -4)
After
point3 step3 affine value = 0
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 0 by 9
After
0=0·9+0
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 step4 product := mul(1, 0)
After
point3 step4 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S098
Before
point3 step4 affine value := sum(0, -4)
After
point3 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, S099
Before
Euclidean division of -4 by 9
After
-4=-1·9+5
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 step5 product := mul(1, 5)
After
point3 step5 product = 5
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
point3 step5 affine value := sum(5, -4)
After
point3 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, S102
Before
Euclidean division of 1 by 9
After
1=0·9+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S103
Before
point3 step6 product := mul(1, 1)
After
point3 step6 product = 1
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S104
Before
point3 step6 affine value := sum(1, -4)
After
point3 step6 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, S105
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S106
Before
point3 step7 product := mul(1, 6)
After
point3 step7 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
point3 step7 affine value := sum(6, -4)
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, S108
Before
Euclidean division of 2 by 9
After
2=0·9+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S109
Before
point3 step8 product := mul(1, 2)
After
point3 step8 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S110
Before
point3 step8 affine value := sum(2, -4)
After
point3 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, S111
Before
Euclidean division of -2 by 9
After
-2=-1·9+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S112
Before
point3 step9 product := mul(1, 7)
After
point3 step9 product = 7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
point3 step9 affine value := sum(7, -4)
After
point3 step9 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, S114
Before
Euclidean division of 3 by 9
After
3=0·9+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S115
Before
distinct trace=[3, 8, 4, 0, 5, 1, 6, 2, 7]; first repeated state=3
After
cycle_start=0; starting-point least period=9
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, S116
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, S117
Before
point4 step1 product := mul(1, 4)
After
point4 step1 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S118
Before
point4 step1 affine value := sum(4, -4)
After
point4 step1 affine value = 0
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 0 by 9
After
0=0·9+0
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 step2 product := mul(1, 0)
After
point4 step2 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
point4 step2 affine value := sum(0, -4)
After
point4 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, S122
Before
Euclidean division of -4 by 9
After
-4=-1·9+5
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 step3 product := mul(1, 5)
After
point4 step3 product = 5
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S124
Before
point4 step3 affine value := sum(5, -4)
After
point4 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, S125
Before
Euclidean division of 1 by 9
After
1=0·9+1
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 step4 product := mul(1, 1)
After
point4 step4 product = 1
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S127
Before
point4 step4 affine value := sum(1, -4)
After
point4 step4 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, S128
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S129
Before
point4 step5 product := mul(1, 6)
After
point4 step5 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S130
Before
point4 step5 affine value := sum(6, -4)
After
point4 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, S131
Before
Euclidean division of 2 by 9
After
2=0·9+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S132
Before
point4 step6 product := mul(1, 2)
After
point4 step6 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S133
Before
point4 step6 affine value := sum(2, -4)
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, S134
Before
Euclidean division of -2 by 9
After
-2=-1·9+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S135
Before
point4 step7 product := mul(1, 7)
After
point4 step7 product = 7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S136
Before
point4 step7 affine value := sum(7, -4)
After
point4 step7 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, S137
Before
Euclidean division of 3 by 9
After
3=0·9+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S138
Before
point4 step8 product := mul(1, 3)
After
point4 step8 product = 3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S139
Before
point4 step8 affine value := sum(3, -4)
After
point4 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, S140
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S141
Before
point4 step9 product := mul(1, 8)
After
point4 step9 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S142
Before
point4 step9 affine value := sum(8, -4)
After
point4 step9 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, S143
Before
Euclidean division of 4 by 9
After
4=0·9+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S144
Before
distinct trace=[4, 0, 5, 1, 6, 2, 7, 3, 8]; first repeated state=4
After
cycle_start=0; starting-point least period=9
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, S145
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, S146
Before
point5 step1 product := mul(1, 5)
After
point5 step1 product = 5
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S147
Before
point5 step1 affine value := sum(5, -4)
After
point5 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, S148
Before
Euclidean division of 1 by 9
After
1=0·9+1
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 step2 product := mul(1, 1)
After
point5 step2 product = 1
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S150
Before
point5 step2 affine value := sum(1, -4)
After
point5 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, S151
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
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 step3 product := mul(1, 6)
After
point5 step3 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S153
Before
point5 step3 affine value := sum(6, -4)
After
point5 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, S154
Before
Euclidean division of 2 by 9
After
2=0·9+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S155
Before
point5 step4 product := mul(1, 2)
After
point5 step4 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S156
Before
point5 step4 affine value := sum(2, -4)
After
point5 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, S157
Before
Euclidean division of -2 by 9
After
-2=-1·9+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S158
Before
point5 step5 product := mul(1, 7)
After
point5 step5 product = 7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S159
Before
point5 step5 affine value := sum(7, -4)
After
point5 step5 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, S160
Before
Euclidean division of 3 by 9
After
3=0·9+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S161
Before
point5 step6 product := mul(1, 3)
After
point5 step6 product = 3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S162
Before
point5 step6 affine value := sum(3, -4)
After
point5 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, S163
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S164
Before
point5 step7 product := mul(1, 8)
After
point5 step7 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S165
Before
point5 step7 affine value := sum(8, -4)
After
point5 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, S166
Before
Euclidean division of 4 by 9
After
4=0·9+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S167
Before
point5 step8 product := mul(1, 4)
After
point5 step8 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S168
Before
point5 step8 affine value := sum(4, -4)
After
point5 step8 affine value = 0
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S169
Before
Euclidean division of 0 by 9
After
0=0·9+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S170
Before
point5 step9 product := mul(1, 0)
After
point5 step9 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S171
Before
point5 step9 affine value := sum(0, -4)
After
point5 step9 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, S172
Before
Euclidean division of -4 by 9
After
-4=-1·9+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S173
Before
distinct trace=[5, 1, 6, 2, 7, 3, 8, 4, 0]; first repeated state=5
After
cycle_start=0; starting-point least period=9
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, S174
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, S175
Before
point6 step1 product := mul(1, 6)
After
point6 step1 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S176
Before
point6 step1 affine value := sum(6, -4)
After
point6 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, S177
Before
Euclidean division of 2 by 9
After
2=0·9+2
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 step2 product := mul(1, 2)
After
point6 step2 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S179
Before
point6 step2 affine value := sum(2, -4)
After
point6 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, S180
Before
Euclidean division of -2 by 9
After
-2=-1·9+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S181
Before
point6 step3 product := mul(1, 7)
After
point6 step3 product = 7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S182
Before
point6 step3 affine value := sum(7, -4)
After
point6 step3 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, S183
Before
Euclidean division of 3 by 9
After
3=0·9+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S184
Before
point6 step4 product := mul(1, 3)
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, S185
Before
point6 step4 affine value := sum(3, -4)
After
point6 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, S186
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S187
Before
point6 step5 product := mul(1, 8)
After
point6 step5 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S188
Before
point6 step5 affine value := sum(8, -4)
After
point6 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, S189
Before
Euclidean division of 4 by 9
After
4=0·9+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S190
Before
point6 step6 product := mul(1, 4)
After
point6 step6 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S191
Before
point6 step6 affine value := sum(4, -4)
After
point6 step6 affine value = 0
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S192
Before
Euclidean division of 0 by 9
After
0=0·9+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S193
Before
point6 step7 product := mul(1, 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, S194
Before
point6 step7 affine value := sum(0, -4)
After
point6 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, S195
Before
Euclidean division of -4 by 9
After
-4=-1·9+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S196
Before
point6 step8 product := mul(1, 5)
After
point6 step8 product = 5
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S197
Before
point6 step8 affine value := sum(5, -4)
After
point6 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, S198
Before
Euclidean division of 1 by 9
After
1=0·9+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S199
Before
point6 step9 product := mul(1, 1)
After
point6 step9 product = 1
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S200
Before
point6 step9 affine value := sum(1, -4)
After
point6 step9 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, S201
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S202
Before
distinct trace=[6, 2, 7, 3, 8, 4, 0, 5, 1]; first repeated state=6
After
cycle_start=0; starting-point least period=9
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, S203
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, S204
Before
point7 step1 product := mul(1, 7)
After
point7 step1 product = 7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S205
Before
point7 step1 affine value := sum(7, -4)
After
point7 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, S206
Before
Euclidean division of 3 by 9
After
3=0·9+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S207
Before
point7 step2 product := mul(1, 3)
After
point7 step2 product = 3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S208
Before
point7 step2 affine value := sum(3, -4)
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, S209
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S210
Before
point7 step3 product := mul(1, 8)
After
point7 step3 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S211
Before
point7 step3 affine value := sum(8, -4)
After
point7 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, S212
Before
Euclidean division of 4 by 9
After
4=0·9+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S213
Before
point7 step4 product := mul(1, 4)
After
point7 step4 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S214
Before
point7 step4 affine value := sum(4, -4)
After
point7 step4 affine value = 0
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S215
Before
Euclidean division of 0 by 9
After
0=0·9+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S216
Before
point7 step5 product := mul(1, 0)
After
point7 step5 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S217
Before
point7 step5 affine value := sum(0, -4)
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, S218
Before
Euclidean division of -4 by 9
After
-4=-1·9+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S219
Before
point7 step6 product := mul(1, 5)
After
point7 step6 product = 5
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S220
Before
point7 step6 affine value := sum(5, -4)
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, S221
Before
Euclidean division of 1 by 9
After
1=0·9+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S222
Before
point7 step7 product := mul(1, 1)
After
point7 step7 product = 1
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S223
Before
point7 step7 affine value := sum(1, -4)
After
point7 step7 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, S224
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S225
Before
point7 step8 product := mul(1, 6)
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, S226
Before
point7 step8 affine value := sum(6, -4)
After
point7 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, S227
Before
Euclidean division of 2 by 9
After
2=0·9+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S228
Before
point7 step9 product := mul(1, 2)
After
point7 step9 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S229
Before
point7 step9 affine value := sum(2, -4)
After
point7 step9 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, S230
Before
Euclidean division of -2 by 9
After
-2=-1·9+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S231
Before
distinct trace=[7, 3, 8, 4, 0, 5, 1, 6, 2]; first repeated state=7
After
cycle_start=0; starting-point least period=9
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, S232
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, S233
Before
point8 step1 product := mul(1, 8)
After
point8 step1 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S234
Before
point8 step1 affine value := sum(8, -4)
After
point8 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, S235
Before
Euclidean division of 4 by 9
After
4=0·9+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S236
Before
point8 step2 product := mul(1, 4)
After
point8 step2 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S237
Before
point8 step2 affine value := sum(4, -4)
After
point8 step2 affine value = 0
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S238
Before
Euclidean division of 0 by 9
After
0=0·9+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S239
Before
point8 step3 product := mul(1, 0)
After
point8 step3 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S240
Before
point8 step3 affine value := sum(0, -4)
After
point8 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, S241
Before
Euclidean division of -4 by 9
After
-4=-1·9+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S242
Before
point8 step4 product := mul(1, 5)
After
point8 step4 product = 5
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S243
Before
point8 step4 affine value := sum(5, -4)
After
point8 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, S244
Before
Euclidean division of 1 by 9
After
1=0·9+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S245
Before
point8 step5 product := mul(1, 1)
After
point8 step5 product = 1
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S246
Before
point8 step5 affine value := sum(1, -4)
After
point8 step5 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, S247
Before
Euclidean division of -3 by 9
After
-3=-1·9+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S248
Before
point8 step6 product := mul(1, 6)
After
point8 step6 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S249
Before
point8 step6 affine value := sum(6, -4)
After
point8 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, S250
Before
Euclidean division of 2 by 9
After
2=0·9+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S251
Before
point8 step7 product := mul(1, 2)
After
point8 step7 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S252
Before
point8 step7 affine value := sum(2, -4)
After
point8 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, S253
Before
Euclidean division of -2 by 9
After
-2=-1·9+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S254
Before
point8 step8 product := mul(1, 7)
After
point8 step8 product = 7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S255
Before
point8 step8 affine value := sum(7, -4)
After
point8 step8 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, S256
Before
Euclidean division of 3 by 9
After
3=0·9+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S257
Before
point8 step9 product := mul(1, 3)
After
point8 step9 product = 3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S258
Before
point8 step9 affine value := sum(3, -4)
After
point8 step9 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, S259
Before
Euclidean division of -1 by 9
After
-1=-1·9+8
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S260
Before
distinct trace=[8, 4, 0, 5, 1, 6, 2, 7, 3]; first repeated state=8
After
cycle_start=0; starting-point least period=9
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, S261
Before
[[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]]
After
least positive period≤5: []
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, S262
S263 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S262 · 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.S261 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S260 · premise_and_result_audit — Recompute point8 step9 affine value with exact rational arithmetic, not rounded decimals.S259 · premise_and_result_audit — Recompute point8 step9 product with exact rational arithmetic, not rounded decimals.S258 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S257 · premise_and_result_audit — Recompute point8 step8 affine value with exact rational arithmetic, not rounded decimals.S256 · premise_and_result_audit — Recompute point8 step8 product with exact rational arithmetic, not rounded decimals.S255 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S254 · premise_and_result_audit — Recompute point8 step7 affine value with exact rational arithmetic, not rounded decimals.S253 · premise_and_result_audit — Recompute point8 step7 product with exact rational arithmetic, not rounded decimals.S252 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S251 · premise_and_result_audit — Recompute point8 step6 affine value with exact rational arithmetic, not rounded decimals.S250 · premise_and_result_audit — Recompute point8 step6 product with exact rational arithmetic, not rounded decimals.S249 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S248 · premise_and_result_audit — Recompute point8 step5 affine value with exact rational arithmetic, not rounded decimals.S247 · premise_and_result_audit — Recompute point8 step5 product with exact rational arithmetic, not rounded decimals.S246 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S245 · premise_and_result_audit — Recompute point8 step4 affine value with exact rational arithmetic, not rounded decimals.S244 · premise_and_result_audit — Recompute point8 step4 product with exact rational arithmetic, not rounded decimals.S243 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S242 · premise_and_result_audit — Recompute point8 step3 affine value with exact rational arithmetic, not rounded decimals.S241 · premise_and_result_audit — Recompute point8 step3 product with exact rational arithmetic, not rounded decimals.S240 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S239 · premise_and_result_audit — Recompute point8 step2 affine value with exact rational arithmetic, not rounded decimals.S238 · premise_and_result_audit — Recompute point8 step2 product with exact rational arithmetic, not rounded decimals.S237 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S236 · premise_and_result_audit — Recompute point8 step1 affine value with exact rational arithmetic, not rounded decimals.S235 · premise_and_result_audit — Recompute point8 step1 product with exact rational arithmetic, not rounded decimals.S234 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S233 · 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.S232 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S231 · premise_and_result_audit — Recompute point7 step9 affine value with exact rational arithmetic, not rounded decimals.S230 · premise_and_result_audit — Recompute point7 step9 product with exact rational arithmetic, not rounded decimals.S229 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S228 · premise_and_result_audit — Recompute point7 step8 affine value with exact rational arithmetic, not rounded decimals.S227 · premise_and_result_audit — Recompute point7 step8 product with exact rational arithmetic, not rounded decimals.S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S225 · premise_and_result_audit — Recompute point7 step7 affine value with exact rational arithmetic, not rounded decimals.S224 · premise_and_result_audit — Recompute point7 step7 product with exact rational arithmetic, not rounded decimals.S223 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S222 · premise_and_result_audit — Recompute point7 step6 affine value with exact rational arithmetic, not rounded decimals.S221 · premise_and_result_audit — Recompute point7 step6 product with exact rational arithmetic, not rounded decimals.S220 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S219 · premise_and_result_audit — Recompute point7 step5 affine value with exact rational arithmetic, not rounded decimals.S218 · premise_and_result_audit — Recompute point7 step5 product with exact rational arithmetic, not rounded decimals.S217 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S216 · premise_and_result_audit — Recompute point7 step4 affine value with exact rational arithmetic, not rounded decimals.S215 · premise_and_result_audit — Recompute point7 step4 product with exact rational arithmetic, not rounded decimals.S214 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S213 · premise_and_result_audit — Recompute point7 step3 affine value with exact rational arithmetic, not rounded decimals.S212 · premise_and_result_audit — Recompute point7 step3 product with exact rational arithmetic, not rounded decimals.S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S210 · premise_and_result_audit — Recompute point7 step2 affine value with exact rational arithmetic, not rounded decimals.S209 · premise_and_result_audit — Recompute point7 step2 product with exact rational arithmetic, not rounded decimals.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 step1 affine value with exact rational arithmetic, not rounded decimals.S206 · premise_and_result_audit — Recompute point7 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S203 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S202 · premise_and_result_audit — Recompute point6 step9 affine value with exact rational arithmetic, not rounded decimals.S201 · premise_and_result_audit — Recompute point6 step9 product with exact rational arithmetic, not rounded decimals.S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S199 · premise_and_result_audit — Recompute point6 step8 affine value with exact rational arithmetic, not rounded decimals.S198 · premise_and_result_audit — Recompute point6 step8 product with exact rational arithmetic, not rounded decimals.S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S196 · premise_and_result_audit — Recompute point6 step7 affine value with exact rational arithmetic, not rounded decimals.S195 · premise_and_result_audit — Recompute point6 step7 product with exact rational arithmetic, not rounded decimals.S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S193 · premise_and_result_audit — Recompute point6 step6 affine value with exact rational arithmetic, not rounded decimals.S192 · premise_and_result_audit — Recompute point6 step6 product with exact rational arithmetic, not rounded decimals.S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S190 · premise_and_result_audit — Recompute point6 step5 affine value with exact rational arithmetic, not rounded decimals.S189 · premise_and_result_audit — Recompute point6 step5 product with exact rational arithmetic, not rounded decimals.S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S187 · premise_and_result_audit — Recompute point6 step4 affine value with exact rational arithmetic, not rounded decimals.S186 · premise_and_result_audit — Recompute point6 step4 product with exact rational arithmetic, not rounded decimals.S185 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S184 · premise_and_result_audit — Recompute point6 step3 affine value with exact rational arithmetic, not rounded decimals.S183 · premise_and_result_audit — Recompute point6 step3 product with exact rational arithmetic, not rounded decimals.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 step2 affine value with exact rational arithmetic, not rounded decimals.S180 · premise_and_result_audit — Recompute point6 step2 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 step1 affine value with exact rational arithmetic, not rounded decimals.S177 · premise_and_result_audit — Recompute point6 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S173 · premise_and_result_audit — Recompute point5 step9 affine value with exact rational arithmetic, not rounded decimals.S172 · premise_and_result_audit — Recompute point5 step9 product with exact rational arithmetic, not rounded decimals.S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S170 · premise_and_result_audit — Recompute point5 step8 affine value with exact rational arithmetic, not rounded decimals.S169 · premise_and_result_audit — Recompute point5 step8 product with exact rational arithmetic, not rounded decimals.S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S167 · premise_and_result_audit — Recompute point5 step7 affine value with exact rational arithmetic, not rounded decimals.S166 · premise_and_result_audit — Recompute point5 step7 product with exact rational arithmetic, not rounded decimals.S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S164 · premise_and_result_audit — Recompute point5 step6 affine value with exact rational arithmetic, not rounded decimals.S163 · premise_and_result_audit — Recompute point5 step6 product with exact rational arithmetic, not rounded decimals.S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S161 · premise_and_result_audit — Recompute point5 step5 affine value with exact rational arithmetic, not rounded decimals.S160 · premise_and_result_audit — Recompute point5 step5 product with exact rational arithmetic, not rounded decimals.S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S158 · premise_and_result_audit — Recompute point5 step4 affine value with exact rational arithmetic, not rounded decimals.S157 · premise_and_result_audit — Recompute point5 step4 product with exact rational arithmetic, not rounded decimals.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 step3 affine value with exact rational arithmetic, not rounded decimals.S154 · premise_and_result_audit — Recompute point5 step3 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 step2 affine value with exact rational arithmetic, not rounded decimals.S151 · premise_and_result_audit — Recompute point5 step2 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 step1 affine value with exact rational arithmetic, not rounded decimals.S148 · premise_and_result_audit — Recompute point5 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S144 · premise_and_result_audit — Recompute point4 step9 affine value with exact rational arithmetic, not rounded decimals.S143 · premise_and_result_audit — Recompute point4 step9 product with exact rational arithmetic, not rounded decimals.S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S141 · premise_and_result_audit — Recompute point4 step8 affine value with exact rational arithmetic, not rounded decimals.S140 · premise_and_result_audit — Recompute point4 step8 product with exact rational arithmetic, not rounded decimals.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 — Recompute point4 step7 affine value with exact rational arithmetic, not rounded decimals.S137 · premise_and_result_audit — Recompute point4 step7 product with exact rational arithmetic, not rounded decimals.S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S135 · premise_and_result_audit — Recompute point4 step6 affine value with exact rational arithmetic, not rounded decimals.S134 · premise_and_result_audit — Recompute point4 step6 product with exact rational arithmetic, not rounded decimals.S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S132 · premise_and_result_audit — Recompute point4 step5 affine value with exact rational arithmetic, not rounded decimals.S131 · premise_and_result_audit — Recompute point4 step5 product with exact rational arithmetic, not rounded decimals.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 step4 affine value with exact rational arithmetic, not rounded decimals.S128 · premise_and_result_audit — Recompute point4 step4 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 step3 affine value with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Recompute point4 step3 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 step2 affine value with exact rational arithmetic, not rounded decimals.S122 · premise_and_result_audit — Recompute point4 step2 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 step1 affine value with exact rational arithmetic, not rounded decimals.S119 · premise_and_result_audit — Recompute point4 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S115 · premise_and_result_audit — Recompute point3 step9 affine value with exact rational arithmetic, not rounded decimals.S114 · premise_and_result_audit — Recompute point3 step9 product with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S112 · premise_and_result_audit — Recompute point3 step8 affine value with exact rational arithmetic, not rounded decimals.S111 · premise_and_result_audit — Recompute point3 step8 product with exact rational arithmetic, not rounded decimals.S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S109 · premise_and_result_audit — Recompute point3 step7 affine value with exact rational arithmetic, not rounded decimals.S108 · premise_and_result_audit — Recompute point3 step7 product with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S106 · premise_and_result_audit — Recompute point3 step6 affine value with exact rational arithmetic, not rounded decimals.S105 · premise_and_result_audit — Recompute point3 step6 product with exact rational arithmetic, not rounded decimals.S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S103 · premise_and_result_audit — Recompute point3 step5 affine value with exact rational arithmetic, not rounded decimals.S102 · premise_and_result_audit — Recompute point3 step5 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 step4 affine value with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Recompute point3 step4 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 step3 affine value with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Recompute point3 step3 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 step2 affine value with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Recompute point3 step2 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 step1 affine value with exact rational arithmetic, not rounded decimals.S090 · premise_and_result_audit — Recompute point3 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S086 · premise_and_result_audit — Recompute point2 step9 affine value with exact rational arithmetic, not rounded decimals.S085 · premise_and_result_audit — Recompute point2 step9 product with exact rational arithmetic, not rounded decimals.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 point2 step8 affine value with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Recompute point2 step8 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 point2 step7 affine value with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Recompute point2 step7 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 point2 step6 affine value with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Recompute point2 step6 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 step5 affine value with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Recompute point2 step5 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 step4 affine value with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Recompute point2 step4 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 step3 affine value with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Recompute point2 step3 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 step2 affine value with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Recompute point2 step2 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 step1 affine value with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Recompute point2 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.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 point1 step9 affine value with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Recompute point1 step9 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 — Recompute point1 step8 affine value with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute point1 step8 product with exact rational arithmetic, not rounded decimals.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 step7 affine value with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute point1 step7 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 step6 affine value with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute point1 step6 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 step5 affine value with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Recompute point1 step5 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 step4 affine value with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute point1 step4 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 step3 affine value with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute point1 step3 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 step2 affine value with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute point1 step2 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 step1 affine value with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute point1 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Recompute point0 step9 affine value with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute point0 step9 product with exact rational arithmetic, not rounded decimals.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.(9, 1, -4, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 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: 66a82dbcee34b93742ef8236494393772782c1874b9d91661ef5da6a3701f944candidate record SHA-256: 78df8e025f95ad0b393146895bf6a7b5b15a3ded9e47d87850fe6d382b1d9fa1semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED