Question 2597: A Return Time Need Not Be the Least Positive Period

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0004-02597 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0047

Original question and givens

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-3, b=13. 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=5, proposed positive return time p=6. 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(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-3, b=13. 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=5, proposed positive return time p=6. 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

Step-by-step derivation

S001 · implies

Before

n=8; a=-3; b=13

After

S={0,...,7}; 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 dependencies

Rule: finite_self_map · Uses: G001

S002 · implies

Before

start=5

After

record each state before its first successor calculation

Conditions: none

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

Step identifiers and dependencies

Rule: orbit_initial_state · Uses: G001, S001

S003 · calculation

Before

least period step1 product := mul(-3, 5)

After

least period step1 product = -15

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

least period step1 affine value := sum(-15, 13)

After

least period step1 affine value = -2

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

Euclidean division of -2 by 8

After

-2=-1·8+6

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S004

S006 · calculation

Before

least period step2 product := mul(-3, 6)

After

least period step2 product = -18

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

least period step2 affine value := sum(-18, 13)

After

least period step2 affine value = -5

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

Euclidean division of -5 by 8

After

-5=-1·8+3

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S007

S009 · calculation

Before

least period step3 product := mul(-3, 3)

After

least period step3 product = -9

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

least period step3 affine value := sum(-9, 13)

After

least period step3 affine value = 4

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

Euclidean division of 4 by 8

After

4=0·8+4

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S010

S012 · calculation

Before

least period step4 product := mul(-3, 4)

After

least period step4 product = -12

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

least period step4 affine value := sum(-12, 13)

After

least period step4 affine value = 1

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

Euclidean division of 1 by 8

After

1=0·8+1

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S013

S015 · calculation

Before

least period step5 product := mul(-3, 1)

After

least period step5 product = -3

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

least period step5 affine value := sum(-3, 13)

After

least period step5 affine value = 10

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

Euclidean division of 10 by 8

After

10=1·8+2

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S016

S018 · calculation

Before

least period step6 product := mul(-3, 2)

After

least period step6 product = -6

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

least period step6 affine value := sum(-6, 13)

After

least period step6 affine value = 7

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

Euclidean division of 7 by 8

After

7=0·8+7

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S019

S021 · calculation

Before

least period step7 product := mul(-3, 7)

After

least period step7 product = -21

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

least period step7 affine value := sum(-21, 13)

After

least period step7 affine value = -8

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

Euclidean division of -8 by 8

After

-8=-1·8+0

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S022

S024 · calculation

Before

least period step8 product := mul(-3, 0)

After

least period step8 product = 0

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

least period step8 affine value := sum(0, 13)

After

least period step8 affine value = 13

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

Euclidean division of 13 by 8

After

13=1·8+5

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S025

S027 · implies

Before

distinct trace=[5, 6, 3, 4, 1, 2, 7, 0]; first repeated state=5

After

cycle_start=0; starting-point least period=8

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 dependencies

Rule: finite_orbit_certificate · Uses: G001, S026

S028 · calculation

Before

return audit1 product := mul(-3, 5)

After

return audit1 product = -15

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

return audit1 affine value := sum(-15, 13)

After

return audit1 affine value = -2

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

Euclidean division of -2 by 8

After

-2=-1·8+6

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S029

S031 · calculation

Before

return audit2 product := mul(-3, 6)

After

return audit2 product = -18

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

return audit2 affine value := sum(-18, 13)

After

return audit2 affine value = -5

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · implies

Before

Euclidean division of -5 by 8

After

-5=-1·8+3

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S032

S034 · calculation

Before

return audit3 product := mul(-3, 3)

After

return audit3 product = -9

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

return audit3 affine value := sum(-9, 13)

After

return audit3 affine value = 4

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

Euclidean division of 4 by 8

After

4=0·8+4

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S035

S037 · calculation

Before

return audit4 product := mul(-3, 4)

After

return audit4 product = -12

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

return audit4 affine value := sum(-12, 13)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · implies

Before

Euclidean division of 1 by 8

After

1=0·8+1

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S038

S040 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

return audit5 affine value := sum(-3, 13)

After

return audit5 affine value = 10

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

Euclidean division of 10 by 8

After

10=1·8+2

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S041

S043 · calculation

Before

return audit6 product := mul(-3, 2)

After

return audit6 product = -6

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

return audit6 affine value := sum(-6, 13)

After

return audit6 affine value = 7

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

Euclidean division of 7 by 8

After

7=0·8+7

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

Step identifiers and dependencies

Rule: orbit_successor · Uses: G001, S044

S046 · implies

Before

T^6(5)=7; least period=8

After

returns=No; 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 dependencies

Rule: return_vs_least_period · Uses: G001, S045

Reverse checks

S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 return audit6 affine value with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute return audit6 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 return audit5 affine value with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute return audit5 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 return audit4 affine value with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute return audit4 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 return audit3 affine value with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute return audit3 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 return audit2 affine value with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute return audit2 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 return audit1 affine value with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute return audit1 product with exact rational arithmetic, not rounded decimals.
S027 · 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.
S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S025 · premise_and_result_audit — Recompute least period step8 affine value with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute least period step8 product with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S022 · premise_and_result_audit — Recompute least period step7 affine value with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute least period step7 product with exact rational arithmetic, not rounded decimals.
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 least period step6 affine value with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute least period 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 least period step5 affine value with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute least period 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 least period step4 affine value with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute least period 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 least period step3 affine value with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute least period 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 least period step2 affine value with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute least period 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 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

(8, -3, 13, 5, 6, 7, No, 8, 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.

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input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 01a1dad18a44336e95819a7c1a345b3bd19ac27b8ca6aa0330402d02d14bffde
candidate record SHA-256: 9eea0e8b39d1a9fda8274def1c7613205bd576880954530c9855894c288f2075
semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
Lean: NOT_RUN · human review: NOT_PERFORMED

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