# Question 2569: Every Point’s Least Period in the Complete Finite Model

[Return to catalogue page 13](https://ocme.evemisslab.com/en/questions/page/13/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations/#q-20261013-advance-20261021-r03-orbit-fixed-periodic-0094)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0004-02569 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0094



## Original question and givens

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).





## Definition and scope

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.


- [mko-orbit-fixed-periodic](https://ocme.evemisslab.com/en/mko/mko-orbit-fixed-periodic/)





## Step-by-step derivation

S001 · impliesBefore

n=5; a=1; b=10

After

S={0,...,4}; T(z)=(a·z+b) mod n

Conditions: 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.

Step identifiers and dependenciesRule: finite_self_map · Uses: G001





S002 · impliesBefore

start=0

After

record each state before its first successor calculation

Conditions: none

Start with the specified point itself,not its first successor.

Step identifiers and dependenciesRule: orbit_initial_state · Uses: G001, S001





S003 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

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.

Step identifiers and dependenciesRule: orbit_successor · Uses: G001, S004





S006 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_orbit_certificate · Uses: G001, S005





S007 · impliesBefore

start=1

After

record each state before its first successor calculation

Conditions: none

Start with the specified point itself,not its first successor.

Step identifiers and dependenciesRule: orbit_initial_state · Uses: G001, S006





S008 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

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.

Step identifiers and dependenciesRule: orbit_successor · Uses: G001, S009





S011 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_orbit_certificate · Uses: G001, S010





S012 · impliesBefore

start=2

After

record each state before its first successor calculation

Conditions: none

Start with the specified point itself,not its first successor.

Step identifiers and dependenciesRule: orbit_initial_state · Uses: G001, S011





S013 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

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.

Step identifiers and dependenciesRule: orbit_successor · Uses: G001, S014





S016 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_orbit_certificate · Uses: G001, S015





S017 · impliesBefore

start=3

After

record each state before its first successor calculation

Conditions: none

Start with the specified point itself,not its first successor.

Step identifiers and dependenciesRule: orbit_initial_state · Uses: G001, S016





S018 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · impliesBefore

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.

Step identifiers and dependenciesRule: orbit_successor · Uses: G001, S019





S021 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_orbit_certificate · Uses: G001, S020





S022 · impliesBefore

start=4

After

record each state before its first successor calculation

Conditions: none

Start with the specified point itself,not its first successor.

Step identifiers and dependenciesRule: orbit_initial_state · Uses: G001, S021





S023 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · impliesBefore

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.

Step identifiers and dependenciesRule: orbit_successor · Uses: G001, S024





S026 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_orbit_certificate · Uses: G001, S025





S027 · impliesBefore

[[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.

Step identifiers and dependenciesRule: least_period_bound_filter · Uses: G001, S026









## Reverse checks


- 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.





## Result

(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.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 60d138d85a48383f16e78c860d87b322363239ee66468c431b2a9425a2b3ed5c
- candidate record SHA-256: 1b8c3bd40947c2bccc3d30e04fd9650097f5aa5a1ee594c0ab95ff2aae81aba9
- semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
- question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0094.json)