Explore related knowledge and questions →
← Question derivation index · Original question batch
R0004-02563 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0088
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=2, 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=1. 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(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=2, 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=1. 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=5; a=2; b=4
After
S={0,...,4}; T(z)=(a·z+b) mod nConditions: none
Remainders lie in0,...,n−1,so every state has one successor in the same finite set;negative dividends do not produce negative canonical remainders.
Rule: finite_self_map · Uses: G001
Before
start=0
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S001
Before
point0 step1 product := mul(2, 0)
After
point0 step1 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
point0 step1 affine value := sum(0, 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 5
After
4=0·5+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S004
Before
point0 step2 product := mul(2, 4)
After
point0 step2 product = 8
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(8, 4)
After
point0 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, S006
Before
Euclidean division of 12 by 5
After
12=2·5+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S007
Before
point0 step3 product := mul(2, 2)
After
point0 step3 product = 4
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(4, 4)
After
point0 step3 affine value = 8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Euclidean division of 8 by 5
After
8=1·5+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S010
Before
point0 step4 product := mul(2, 3)
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 = 10
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 10 by 5
After
10=2·5+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S013
Before
distinct trace=[0, 4, 2, 3]; first repeated state=0
After
cycle_start=0; starting-point least period=4
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S014
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, S015
Before
point1 step1 product := mul(2, 1)
After
point1 step1 product = 2
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
point1 step1 affine value := sum(2, 4)
After
point1 step1 affine value = 6
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
Euclidean division of 6 by 5
After
6=1·5+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S018
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, S019
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, S020
Before
point2 step1 product := mul(2, 2)
After
point2 step1 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
point2 step1 affine value := sum(4, 4)
After
point2 step1 affine value = 8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
Euclidean division of 8 by 5
After
8=1·5+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S023
Before
point2 step2 product := mul(2, 3)
After
point2 step2 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
point2 step2 affine value := sum(6, 4)
After
point2 step2 affine value = 10
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
Euclidean division of 10 by 5
After
10=2·5+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S026
Before
point2 step3 product := mul(2, 0)
After
point2 step3 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
point2 step3 affine value := sum(0, 4)
After
point2 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, S028
Before
Euclidean division of 4 by 5
After
4=0·5+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 step4 product := mul(2, 4)
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, S030
Before
point2 step4 affine value := sum(8, 4)
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, S031
Before
Euclidean division of 12 by 5
After
12=2·5+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S032
Before
distinct trace=[2, 3, 0, 4]; first repeated state=2
After
cycle_start=0; starting-point least period=4
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S033
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, S034
Before
point3 step1 product := mul(2, 3)
After
point3 step1 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
point3 step1 affine value := sum(6, 4)
After
point3 step1 affine value = 10
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
Euclidean division of 10 by 5
After
10=2·5+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S037
Before
point3 step2 product := mul(2, 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, S038
Before
point3 step2 affine value := sum(0, 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, S039
Before
Euclidean division of 4 by 5
After
4=0·5+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S040
Before
point3 step3 product := mul(2, 4)
After
point3 step3 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
point3 step3 affine value := sum(8, 4)
After
point3 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, S042
Before
Euclidean division of 12 by 5
After
12=2·5+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S043
Before
point3 step4 product := mul(2, 2)
After
point3 step4 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
point3 step4 affine value := sum(4, 4)
After
point3 step4 affine value = 8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
Euclidean division of 8 by 5
After
8=1·5+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S046
Before
distinct trace=[3, 0, 4, 2]; first repeated state=3
After
cycle_start=0; starting-point least period=4
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S047
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, S048
Before
point4 step1 product := mul(2, 4)
After
point4 step1 product = 8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
point4 step1 affine value := sum(8, 4)
After
point4 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, S050
Before
Euclidean division of 12 by 5
After
12=2·5+2
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S051
Before
point4 step2 product := mul(2, 2)
After
point4 step2 product = 4
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
point4 step2 affine value := sum(4, 4)
After
point4 step2 affine value = 8
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
Euclidean division of 8 by 5
After
8=1·5+3
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S054
Before
point4 step3 product := mul(2, 3)
After
point4 step3 product = 6
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
point4 step3 affine value := sum(6, 4)
After
point4 step3 affine value = 10
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
Euclidean division of 10 by 5
After
10=2·5+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S057
Before
point4 step4 product := mul(2, 0)
After
point4 step4 product = 0
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
point4 step4 affine value := sum(0, 4)
After
point4 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, S059
Before
Euclidean division of 4 by 5
After
4=0·5+4
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S060
Before
distinct trace=[4, 2, 3, 0]; first repeated state=4
After
cycle_start=0; starting-point least period=4
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S061
Before
[[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]]
After
least positive period≤1: [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, S062
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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S060 · premise_and_result_audit — Recompute point4 step4 affine value with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute point4 step4 product with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S057 · premise_and_result_audit — Recompute point4 step3 affine value with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Recompute point4 step3 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 point4 step2 affine value with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute point4 step2 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 point4 step1 affine value with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute point4 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.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 — Recompute point3 step4 affine value with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Recompute point3 step4 product with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S043 · premise_and_result_audit — Recompute point3 step3 affine value with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute point3 step3 product with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S040 · premise_and_result_audit — Recompute point3 step2 affine value with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute point3 step2 product with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S037 · premise_and_result_audit — Recompute point3 step1 affine value with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute point3 step1 product with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S034 · 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.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 step4 affine value with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute point2 step4 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 step3 affine value with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute point2 step3 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 — Recompute point2 step2 affine value with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute point2 step2 product with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Recompute point2 step1 affine value with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute point2 step1 product with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Recompute point1 step1 affine value with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute point1 step1 product with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · 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.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.(5, 2, 4, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [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: e5236dd304b417a1c8a46a70b74c7f5b9ea394e878796cac5b81a70e3f64abf7candidate record SHA-256: ff9f1b1fa75829ba6e3a047c91e4397a57425499e3ecea6445eb3e1f5bc94011semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED