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R0004-02569 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0094
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=1, b=10. 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=1, b=10. 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=1; b=10
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(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, 10)
After
point0 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, S003
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, S004
Before
distinct trace=[0]; first repeated state=0
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, S005
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, S006
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, S007
Before
point1 step1 affine value := sum(1, 10)
After
point1 step1 affine value = 11
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Euclidean division of 11 by 5
After
11=2·5+1
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S009
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, S010
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, S011
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, S012
Before
point2 step1 affine value := sum(2, 10)
After
point2 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, S013
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, S014
Before
distinct trace=[2]; first repeated state=2
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, S015
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, S016
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, S017
Before
point3 step1 affine value := sum(3, 10)
After
point3 step1 affine value = 13
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
Euclidean division of 13 by 5
After
13=2·5+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
distinct trace=[3]; first repeated state=3
After
cycle_start=0; starting-point least period=1
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S020
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, S021
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, S022
Before
point4 step1 affine value := sum(4, 10)
After
point4 step1 affine value = 14
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 14 by 5
After
14=2·5+4
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=[4]; first repeated state=4
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
[[0, 1], [1, 1], [2, 1], [3, 1], [4, 1]]
After
least positive period≤1: [0, 1, 2, 3, 4]
Conditions: none
Cover all n states;select only periodic points whose least positive period is at most the bound,not transient points that merely enter a cycle.
Rule: least_period_bound_filter · Uses: G001, S026
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 point4 step1 affine value with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute point4 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 point3 step1 affine value with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute point3 step1 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 — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Recompute point2 step1 affine value with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute point2 step1 product with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · 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.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute point1 step1 affine value with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute point1 step1 product with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · 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.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, 1, 10, 1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1]], [0, 1, 2, 3, 4])
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: 60d138d85a48383f16e78c860d87b322363239ee66468c431b2a9425a2b3ed5ccandidate record SHA-256: 1b8c3bd40947c2bccc3d30e04fd9650097f5aa5a1ee594c0ab95ff2aae81aba9semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED