# Question 2582: A Return Time Need Not Be the Least Positive Period

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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.

R0004-02582 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0032



## Original question and givens

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=3, b=-2. 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. x=1, proposed positive return time p=7. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).


- G001 — Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=3, b=-2. 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. x=1, proposed positive return time p=7. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).





## 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=3; b=-2

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=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, S001





S003 · calculationBefore

least period step1 product := mul(3, 1)

After

least period 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, S002





S004 · calculationBefore

least period step1 affine value := sum(3, -2)

After

least period step1 affine value = 1

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 1 by 5

After

1=0·5+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, S004





S006 · 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, S005





S007 · calculationBefore

return audit1 product := mul(3, 1)

After

return audit1 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, S006





S008 · calculationBefore

return audit1 affine value := sum(3, -2)

After

return audit1 affine value = 1

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, S007





S009 · impliesBefore

Euclidean division of 1 by 5

After

1=0·5+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, S008





S010 · calculationBefore

return audit2 product := mul(3, 1)

After

return audit2 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, S009





S011 · calculationBefore

return audit2 affine value := sum(3, -2)

After

return audit2 affine value = 1

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, S010





S012 · impliesBefore

Euclidean division of 1 by 5

After

1=0·5+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, S011





S013 · calculationBefore

return audit3 product := mul(3, 1)

After

return audit3 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, S012





S014 · calculationBefore

return audit3 affine value := sum(3, -2)

After

return audit3 affine value = 1

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 1 by 5

After

1=0·5+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, S014





S016 · calculationBefore

return audit4 product := mul(3, 1)

After

return audit4 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, S015





S017 · calculationBefore

return audit4 affine value := sum(3, -2)

After

return audit4 affine value = 1

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, S016





S018 · impliesBefore

Euclidean division of 1 by 5

After

1=0·5+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, S017





S019 · calculationBefore

return audit5 product := mul(3, 1)

After

return audit5 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, S018





S020 · calculationBefore

return audit5 affine value := sum(3, -2)

After

return audit5 affine value = 1

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, S019





S021 · impliesBefore

Euclidean division of 1 by 5

After

1=0·5+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, S020





S022 · calculationBefore

return audit6 product := mul(3, 1)

After

return audit6 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, S021





S023 · calculationBefore

return audit6 affine value := sum(3, -2)

After

return audit6 affine value = 1

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, S022





S024 · impliesBefore

Euclidean division of 1 by 5

After

1=0·5+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, S023





S025 · calculationBefore

return audit7 product := mul(3, 1)

After

return audit7 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, S024





S026 · calculationBefore

return audit7 affine value := sum(3, -2)

After

return audit7 affine value = 1

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, S025





S027 · impliesBefore

Euclidean division of 1 by 5

After

1=0·5+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, S026





S028 · impliesBefore

T^7(1)=1; least period=1

After

returns=Yes; period exactly p=No

Conditions: none

T^p(x)=x records a return.If the least positive period is d,return times are positive multiples of d;not every return time p is the least period.

Step identifiers and dependenciesRule: return_vs_least_period · Uses: G001, S027









## Reverse checks


- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 return audit7 affine value with exact rational arithmetic, not rounded decimals.
- S025 · premise_and_result_audit — Recompute return audit7 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 return audit6 affine value with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Recompute return audit6 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 — Recompute return audit5 affine value with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Recompute return audit5 product with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Recompute return audit4 affine value with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute return audit4 product with exact rational arithmetic, not rounded decimals.
- 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 return audit3 affine value with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute return audit3 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 — Recompute return audit2 affine value with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute return audit2 product with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Recompute return audit1 affine value with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute return audit1 product with exact rational arithmetic, not rounded decimals.
- 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 least period step1 affine value with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute least period 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, 3, -2, 1, 7, 1, Yes, 1, No)

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: dacb154fe5d12838e3910d80fefb6a50d88cbc20f05b79bc3e7620b0511f956f
- candidate record SHA-256: f6a2cba36f2603355bba2aaf892f244647101622820c44c2f318934573feef14
- 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-0032.json)