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R0004-02571 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0096
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. 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=3. 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(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. 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=3. 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=7; a=3; b=12
After
S={0,...,6}; T(z)=(a·z+b) mod nConditions: none
Remainders lie in0,...,n−1,so every state has one successor in the same finite set;negative dividends do not produce negative canonical remainders.
Rule: finite_self_map · Uses: G001
Before
start=0
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S001
Before
point0 step1 product := mul(3, 0)
After
point0 step1 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
point0 step1 affine value := sum(0, 12)
After
point0 step1 affine value = 12
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 12 by 7
After
12=1·7+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S004
Before
point0 step2 product := mul(3, 5)
After
point0 step2 product = 15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
point0 step2 affine value := sum(15, 12)
After
point0 step2 affine value = 27
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 27 by 7
After
27=3·7+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S007
Before
point0 step3 product := mul(3, 6)
After
point0 step3 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
point0 step3 affine value := sum(18, 12)
After
point0 step3 affine value = 30
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 30 by 7
After
30=4·7+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S010
Before
point0 step4 product := mul(3, 2)
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, 12)
After
point0 step4 affine value = 18
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 18 by 7
After
18=2·7+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S013
Before
point0 step5 product := mul(3, 4)
After
point0 step5 product = 12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
point0 step5 affine value := sum(12, 12)
After
point0 step5 affine value = 24
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 24 by 7
After
24=3·7+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S016
Before
point0 step6 product := mul(3, 3)
After
point0 step6 product = 9
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(9, 12)
After
point0 step6 affine value = 21
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 21 by 7
After
21=3·7+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S019
Before
distinct trace=[0, 5, 6, 2, 4, 3]; first repeated state=0
After
cycle_start=0; starting-point least period=6
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, S020
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, S021
Before
point1 step1 product := mul(3, 1)
After
point1 step1 product = 3
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
point1 step1 affine value := sum(3, 12)
After
point1 step1 affine value = 15
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
Euclidean division of 15 by 7
After
15=2·7+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S024
Before
distinct trace=[1]; first repeated state=1
After
cycle_start=0; starting-point least period=1
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S025
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, S026
Before
point2 step1 product := mul(3, 2)
After
point2 step1 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
point2 step1 affine value := sum(6, 12)
After
point2 step1 affine value = 18
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
Euclidean division of 18 by 7
After
18=2·7+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S029
Before
point2 step2 product := mul(3, 4)
After
point2 step2 product = 12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
point2 step2 affine value := sum(12, 12)
After
point2 step2 affine value = 24
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
Euclidean division of 24 by 7
After
24=3·7+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S032
Before
point2 step3 product := mul(3, 3)
After
point2 step3 product = 9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
point2 step3 affine value := sum(9, 12)
After
point2 step3 affine value = 21
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
Euclidean division of 21 by 7
After
21=3·7+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S035
Before
point2 step4 product := mul(3, 0)
After
point2 step4 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
point2 step4 affine value := sum(0, 12)
After
point2 step4 affine value = 12
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
Euclidean division of 12 by 7
After
12=1·7+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S038
Before
point2 step5 product := mul(3, 5)
After
point2 step5 product = 15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
point2 step5 affine value := sum(15, 12)
After
point2 step5 affine value = 27
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
Euclidean division of 27 by 7
After
27=3·7+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S041
Before
point2 step6 product := mul(3, 6)
After
point2 step6 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
point2 step6 affine value := sum(18, 12)
After
point2 step6 affine value = 30
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
Euclidean division of 30 by 7
After
30=4·7+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S044
Before
distinct trace=[2, 4, 3, 0, 5, 6]; first repeated state=2
After
cycle_start=0; starting-point least period=6
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, S045
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, S046
Before
point3 step1 product := mul(3, 3)
After
point3 step1 product = 9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
point3 step1 affine value := sum(9, 12)
After
point3 step1 affine value = 21
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
Euclidean division of 21 by 7
After
21=3·7+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S049
Before
point3 step2 product := mul(3, 0)
After
point3 step2 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
point3 step2 affine value := sum(0, 12)
After
point3 step2 affine value = 12
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
Euclidean division of 12 by 7
After
12=1·7+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S052
Before
point3 step3 product := mul(3, 5)
After
point3 step3 product = 15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
point3 step3 affine value := sum(15, 12)
After
point3 step3 affine value = 27
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
Euclidean division of 27 by 7
After
27=3·7+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S055
Before
point3 step4 product := mul(3, 6)
After
point3 step4 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
point3 step4 affine value := sum(18, 12)
After
point3 step4 affine value = 30
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
Euclidean division of 30 by 7
After
30=4·7+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S058
Before
point3 step5 product := mul(3, 2)
After
point3 step5 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
point3 step5 affine value := sum(6, 12)
After
point3 step5 affine value = 18
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
Euclidean division of 18 by 7
After
18=2·7+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S061
Before
point3 step6 product := mul(3, 4)
After
point3 step6 product = 12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
point3 step6 affine value := sum(12, 12)
After
point3 step6 affine value = 24
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
Euclidean division of 24 by 7
After
24=3·7+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S064
Before
distinct trace=[3, 0, 5, 6, 2, 4]; first repeated state=3
After
cycle_start=0; starting-point least period=6
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, S065
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, S066
Before
point4 step1 product := mul(3, 4)
After
point4 step1 product = 12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
point4 step1 affine value := sum(12, 12)
After
point4 step1 affine value = 24
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S068
Before
Euclidean division of 24 by 7
After
24=3·7+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S069
Before
point4 step2 product := mul(3, 3)
After
point4 step2 product = 9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
point4 step2 affine value := sum(9, 12)
After
point4 step2 affine value = 21
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
Euclidean division of 21 by 7
After
21=3·7+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S072
Before
point4 step3 product := mul(3, 0)
After
point4 step3 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
point4 step3 affine value := sum(0, 12)
After
point4 step3 affine value = 12
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S074
Before
Euclidean division of 12 by 7
After
12=1·7+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S075
Before
point4 step4 product := mul(3, 5)
After
point4 step4 product = 15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
point4 step4 affine value := sum(15, 12)
After
point4 step4 affine value = 27
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
Euclidean division of 27 by 7
After
27=3·7+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S078
Before
point4 step5 product := mul(3, 6)
After
point4 step5 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
point4 step5 affine value := sum(18, 12)
After
point4 step5 affine value = 30
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S080
Before
Euclidean division of 30 by 7
After
30=4·7+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S081
Before
point4 step6 product := mul(3, 2)
After
point4 step6 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
point4 step6 affine value := sum(6, 12)
After
point4 step6 affine value = 18
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
Euclidean division of 18 by 7
After
18=2·7+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S084
Before
distinct trace=[4, 3, 0, 5, 6, 2]; first repeated state=4
After
cycle_start=0; starting-point least period=6
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, S085
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, S086
Before
point5 step1 product := mul(3, 5)
After
point5 step1 product = 15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
point5 step1 affine value := sum(15, 12)
After
point5 step1 affine value = 27
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
Euclidean division of 27 by 7
After
27=3·7+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S089
Before
point5 step2 product := mul(3, 6)
After
point5 step2 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S090
Before
point5 step2 affine value := sum(18, 12)
After
point5 step2 affine value = 30
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
Euclidean division of 30 by 7
After
30=4·7+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S092
Before
point5 step3 product := mul(3, 2)
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, S093
Before
point5 step3 affine value := sum(6, 12)
After
point5 step3 affine value = 18
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S094
Before
Euclidean division of 18 by 7
After
18=2·7+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S095
Before
point5 step4 product := mul(3, 4)
After
point5 step4 product = 12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
point5 step4 affine value := sum(12, 12)
After
point5 step4 affine value = 24
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
Euclidean division of 24 by 7
After
24=3·7+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S098
Before
point5 step5 product := mul(3, 3)
After
point5 step5 product = 9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S099
Before
point5 step5 affine value := sum(9, 12)
After
point5 step5 affine value = 21
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S100
Before
Euclidean division of 21 by 7
After
21=3·7+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S101
Before
point5 step6 product := mul(3, 0)
After
point5 step6 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S102
Before
point5 step6 affine value := sum(0, 12)
After
point5 step6 affine value = 12
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
Euclidean division of 12 by 7
After
12=1·7+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S104
Before
distinct trace=[5, 6, 2, 4, 3, 0]; first repeated state=5
After
cycle_start=0; starting-point least period=6
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, S105
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, S106
Before
point6 step1 product := mul(3, 6)
After
point6 step1 product = 18
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
point6 step1 affine value := sum(18, 12)
After
point6 step1 affine value = 30
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 30 by 7
After
30=4·7+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
point6 step2 product := mul(3, 2)
After
point6 step2 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S110
Before
point6 step2 affine value := sum(6, 12)
After
point6 step2 affine value = 18
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 18 by 7
After
18=2·7+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S112
Before
point6 step3 product := mul(3, 4)
After
point6 step3 product = 12
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
point6 step3 affine value := sum(12, 12)
After
point6 step3 affine value = 24
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 24 by 7
After
24=3·7+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
point6 step4 product := mul(3, 3)
After
point6 step4 product = 9
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S116
Before
point6 step4 affine value := sum(9, 12)
After
point6 step4 affine value = 21
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S117
Before
Euclidean division of 21 by 7
After
21=3·7+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S118
Before
point6 step5 product := mul(3, 0)
After
point6 step5 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S119
Before
point6 step5 affine value := sum(0, 12)
After
point6 step5 affine value = 12
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S120
Before
Euclidean division of 12 by 7
After
12=1·7+5
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S121
Before
point6 step6 product := mul(3, 5)
After
point6 step6 product = 15
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S122
Before
point6 step6 affine value := sum(15, 12)
After
point6 step6 affine value = 27
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S123
Before
Euclidean division of 27 by 7
After
27=3·7+6
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S124
Before
distinct trace=[6, 2, 4, 3, 0, 5]; first repeated state=6
After
cycle_start=0; starting-point least period=6
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, S125
Before
[[0, 6], [1, 1], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]]
After
least positive period≤3: [1]
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, S126
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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S124 · premise_and_result_audit — Recompute point6 step6 affine value with exact rational arithmetic, not rounded decimals.S123 · premise_and_result_audit — Recompute point6 step6 product with exact rational arithmetic, not rounded decimals.S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S121 · premise_and_result_audit — Recompute point6 step5 affine value with exact rational arithmetic, not rounded decimals.S120 · premise_and_result_audit — Recompute point6 step5 product with exact rational arithmetic, not rounded decimals.S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S118 · premise_and_result_audit — Recompute point6 step4 affine value with exact rational arithmetic, not rounded decimals.S117 · premise_and_result_audit — Recompute point6 step4 product with exact rational arithmetic, not rounded decimals.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 point6 step3 affine value with exact rational arithmetic, not rounded decimals.S114 · premise_and_result_audit — Recompute point6 step3 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 point6 step2 affine value with exact rational arithmetic, not rounded decimals.S111 · premise_and_result_audit — Recompute point6 step2 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 point6 step1 affine value with exact rational arithmetic, not rounded decimals.S108 · premise_and_result_audit — Recompute point6 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S104 · premise_and_result_audit — Recompute point5 step6 affine value with exact rational arithmetic, not rounded decimals.S103 · premise_and_result_audit — Recompute point5 step6 product with exact rational arithmetic, not rounded decimals.S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S101 · premise_and_result_audit — Recompute point5 step5 affine value with exact rational arithmetic, not rounded decimals.S100 · premise_and_result_audit — Recompute point5 step5 product with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S098 · premise_and_result_audit — Recompute point5 step4 affine value with exact rational arithmetic, not rounded decimals.S097 · premise_and_result_audit — Recompute point5 step4 product with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S095 · premise_and_result_audit — Recompute point5 step3 affine value with exact rational arithmetic, not rounded decimals.S094 · premise_and_result_audit — Recompute point5 step3 product with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S092 · premise_and_result_audit — Recompute point5 step2 affine value with exact rational arithmetic, not rounded decimals.S091 · premise_and_result_audit — Recompute point5 step2 product with exact rational arithmetic, not rounded decimals.S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S089 · premise_and_result_audit — Recompute point5 step1 affine value with exact rational arithmetic, not rounded decimals.S088 · premise_and_result_audit — Recompute point5 step1 product with exact rational arithmetic, not rounded decimals.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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S084 · premise_and_result_audit — Recompute point4 step6 affine value with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Recompute point4 step6 product with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S081 · premise_and_result_audit — Recompute point4 step5 affine value with exact rational arithmetic, not rounded decimals.S080 · premise_and_result_audit — Recompute point4 step5 product with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S078 · premise_and_result_audit — Recompute point4 step4 affine value with exact rational arithmetic, not rounded decimals.S077 · premise_and_result_audit — Recompute point4 step4 product with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S075 · premise_and_result_audit — Recompute point4 step3 affine value with exact rational arithmetic, not rounded decimals.S074 · premise_and_result_audit — Recompute point4 step3 product with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S072 · premise_and_result_audit — Recompute point4 step2 affine value with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Recompute point4 step2 product with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S069 · premise_and_result_audit — Recompute point4 step1 affine value with exact rational arithmetic, not rounded decimals.S068 · premise_and_result_audit — Recompute point4 step1 product with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S066 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S064 · premise_and_result_audit — Recompute point3 step6 affine value with exact rational arithmetic, not rounded decimals.S063 · premise_and_result_audit — Recompute point3 step6 product with exact rational arithmetic, not rounded decimals.S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S061 · premise_and_result_audit — Recompute point3 step5 affine value with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Recompute point3 step5 product with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S058 · premise_and_result_audit — Recompute point3 step4 affine value with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Recompute point3 step4 product with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S055 · premise_and_result_audit — Recompute point3 step3 affine value with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Recompute point3 step3 product with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S052 · premise_and_result_audit — Recompute point3 step2 affine value with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Recompute point3 step2 product with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S049 · premise_and_result_audit — Recompute point3 step1 affine value with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute point3 step1 product with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S046 · 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.S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Recompute point2 step6 affine value with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute point2 step6 product with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S041 · premise_and_result_audit — Recompute point2 step5 affine value with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute point2 step5 product with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S038 · premise_and_result_audit — Recompute point2 step4 affine value with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute point2 step4 product with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Recompute point2 step3 affine value with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute point2 step3 product with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S032 · premise_and_result_audit — Recompute point2 step2 affine value with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute point2 step2 product with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Recompute point2 step1 affine value with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute point2 step1 product with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Recompute point1 step1 affine value with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute point1 step1 product with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S021 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.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.(7, 3, 12, 3, [[0, 6], [1, 1], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [1])
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: 2b752827620115646f5ba56bd0c2322a3b94a5a459944083465e0038fcf9d90acandidate record SHA-256: 1329f589c86ad807b17ad0725bc19dee4cab128501d08418fe1e93585e79d0d4semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED