# Finite Orbits: Returns, Least Periods and Transient States

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-7, b=-9. 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=0. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(5, -7, -9, 0, 1, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -7, -9, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0001/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 2. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-6, b=-7. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(6, -6, -7, 1, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -6, -7, 1, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0002/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 3. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-5, b=-5. 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=2. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(7, -5, -5, 2, 6, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -5, -5, 2, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0003/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 4. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-4, b=-3. 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=3. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(8, -4, -3, 3, 1, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -4, -3, 3, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0004/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 5. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-3, b=-1. 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=4. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(9, -3, -1, 4, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -3, -1, 4, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0005/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 6. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-2, b=1. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(10, -2, 1, 5, 1, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, -2, 1, 5, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0006/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 7. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-1, b=3. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(5, -1, 3, 1, 2, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -1, 3, 1, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0007/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 8. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=0, b=5. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(6, 0, 5, 1, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 0, 5, 1, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0008/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 9. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=1, b=7. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(7, 1, 7, 1, 1, Yes)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 1, 7, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0009/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 10. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=2, b=9. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(8, 2, 9, 1, 3, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 2, 9, 1, 3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0010/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 11. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=3, b=11. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(9, 3, 11, 1, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 3, 11, 1, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0011/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 12. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=4, 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=1. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(10, 4, 13, 1, 7, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 4, 13, 1, 7, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0012/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 13. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=5, b=15. 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=2. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(5, 5, 15, 2, 0, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 5, 15, 2, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0013/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 14. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=6, b=17. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(6, 6, 17, 1, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 6, 17, 1, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0014/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 15. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=7, b=19. 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=0. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(7, 7, 19, 0, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 7, 19, 0, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0015/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 16. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=8, b=21. 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=7. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(8, 8, 21, 7, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 8, 21, 7, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0016/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 17. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=9, b=23. 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=7. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(9, 9, 23, 7, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 9, 23, 7, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0017/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 18. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=10, b=25. 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=7. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(10, 10, 25, 7, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 10, 25, 7, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0018/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 19. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=11, b=27. 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=3. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(5, 11, 27, 3, 0, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 11, 27, 3, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0019/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 20. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=12, b=29. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(6, 12, 29, 1, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 12, 29, 1, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0020/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 21. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=13, b=31. 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=6. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(7, 13, 31, 6, 4, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 13, 31, 6, 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0021/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 22. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=14, b=33. 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. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(8, 14, 33, 5, 7, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 14, 33, 5, 7, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0022/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 23. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=15, b=35. 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=4. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(9, 15, 35, 4, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 15, 35, 4, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0023/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 24. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=16, b=37. 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=3. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(10, 16, 37, 3, 5, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 16, 37, 3, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0024/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 25. One State’s Fixed-Point Condition

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=17, b=39. 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=4. Give (modulus,a,b,x,T(x),T(x)=x?).

View answer and explanation(5, 17, 39, 4, 2, No)

Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 17, 39, 4, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0025/)

Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate





## 26. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-3, b=-8. 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=0, proposed positive return time p=1. 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?).

View answer and explanation(5, -3, -8, 0, 1, 2, No, 4, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -3, -8, 0, 1, 2, No, 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0026/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 27. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-2, b=-7. 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=2. 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?).

View answer and explanation(6, -2, -7, 1, 2, 5, No, 3, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -2, -7, 1, 2, 5, No, 3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0027/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 28. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. 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=2, proposed positive return time p=3. 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?).

View answer and explanation(7, -1, -6, 2, 3, 6, No, 2, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -1, -6, 2, 3, 6, No, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0028/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 29. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=0, b=-5. 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=3, proposed positive return time p=4. 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?).

View answer and explanation(8, 0, -5, 3, 4, 3, Yes, 1, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 0, -5, 3, 4, 3, Yes, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0029/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 30. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, 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. x=4, proposed positive return time p=5. 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?).

View answer and explanation(9, 1, -4, 4, 5, 2, No, 9, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 1, -4, 4, 5, 2, No, 9, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0030/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 31. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=2, b=-3. 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?).

View answer and explanation(10, 2, -3, 5, 6, 1, No, 4, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 2, -3, 5, 6, 1, No, 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0031/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 32. A Return Time Need Not Be the Least Positive Period

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?).

View answer and explanation(5, 3, -2, 1, 7, 1, Yes, 1, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 3, -2, 1, 7, 1, Yes, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0032/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 33. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-3, b=-1. 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=8. 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?).

View answer and explanation(6, -3, -1, 1, 8, 5, No, null, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -3, -1, 1, 8, 5, No, null, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0033/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 34. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-2, b=0. 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=1. 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?).

View answer and explanation(7, -2, 0, 1, 1, 5, No, 6, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -2, 0, 1, 1, 5, No, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0034/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 35. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-1, b=1. 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=2. 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?).

View answer and explanation(8, -1, 1, 1, 2, 1, Yes, 2, Yes)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -1, 1, 1, 2, 1, Yes, 2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0035/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 36. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=0, 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=3. 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?).

View answer and explanation(9, 0, 2, 1, 3, 2, No, null, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 0, 2, 1, 3, 2, No, null, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0036/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 37. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=1, b=3. 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=4. 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?).

View answer and explanation(10, 1, 3, 1, 4, 3, No, 10, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 1, 3, 1, 4, 3, No, 10, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0037/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 38. A Return Time Need Not Be the Least Positive Period

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. x=2, proposed positive return time p=5. 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?).

View answer and explanation(5, 2, 4, 2, 5, 3, No, 4, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 2, 4, 2, 5, 3, No, 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0038/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 39. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=3, b=5. 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=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?).

View answer and explanation(6, 3, 5, 1, 6, 5, No, null, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 3, 5, 1, 6, 5, No, null, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0039/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 40. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-3, b=6. 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=0, 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?).

View answer and explanation(7, -3, 6, 0, 7, 6, No, 3, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -3, 6, 0, 7, 6, No, 3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0040/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 41. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-2, b=7. 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=7, proposed positive return time p=8. 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?).

View answer and explanation(8, -2, 7, 7, 8, 5, No, null, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -2, 7, 7, 8, 5, No, null, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0041/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 42. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-1, b=8. 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=7, proposed positive return time p=1. 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?).

View answer and explanation(9, -1, 8, 7, 1, 1, No, 2, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -1, 8, 7, 1, 1, No, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0042/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 43. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=0, b=9. 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=7, proposed positive return time p=2. 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?).

View answer and explanation(10, 0, 9, 7, 2, 9, No, null, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 0, 9, 7, 2, 9, No, null, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0043/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 44. A Return Time Need Not Be the Least Positive Period

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. x=3, proposed positive return time p=3. 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?).

View answer and explanation(5, 1, 10, 3, 3, 3, Yes, 1, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 1, 10, 3, 3, 3, Yes, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0044/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 45. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=2, b=11. 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=4. 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?).

View answer and explanation(6, 2, 11, 1, 4, 1, Yes, 1, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 2, 11, 1, 4, 1, Yes, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0045/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 46. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. 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=6, proposed positive return time p=5. 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?).

View answer and explanation(7, 3, 12, 6, 5, 5, No, 6, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 3, 12, 6, 5, 5, No, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0046/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 47. A Return Time Need Not Be the Least Positive Period

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?).

View answer and explanation(8, -3, 13, 5, 6, 7, No, 8, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -3, 13, 5, 6, 7, No, 8, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0047/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 48. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-2, b=14. 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=4, 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?).

View answer and explanation(9, -2, 14, 4, 7, 0, No, 9, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -2, 14, 4, 7, 0, No, 9, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0048/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 49. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-1, b=15. 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=3, proposed positive return time p=8. 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?).

View answer and explanation(10, -1, 15, 3, 8, 3, Yes, 2, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, -1, 15, 3, 8, 3, Yes, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0049/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 50. A Return Time Need Not Be the Least Positive Period

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=0, b=16. 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=4, proposed positive return time p=1. 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?).

View answer and explanation(5, 0, 16, 4, 1, 1, No, null, No)

Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 0, 16, 4, 1, 1, No, null, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0050/)

Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate





## 51. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-3, b=-8. 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. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(5, -3, -8, 1, [1, 4, 0, 2], 1, 0, 4)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -3, -8, 1, [1, 4, 0, 2], 1, 0, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0051/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 52. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-2, b=-7. 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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(6, -2, -7, 3, [3, 5, 1], 3, 0, 3)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -2, -7, 3, [3, 5, 1], 3, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0052/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 53. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. 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. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(7, -1, -6, 5, [5, 3], 5, 0, 2)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -1, -6, 5, [5, 3], 5, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0053/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 54. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=0, b=-5. 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=7. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(8, 0, -5, 7, [7, 3], 3, 1, nonperiodic-point)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 0, -5, 7, [7, 3], 3, 1, nonperiodic-point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0054/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 55. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, 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. x=0. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(9, 1, -4, 0, [0, 5, 1, 6, 2, 7, 3, 8, 4], 0, 0, 9)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 1, -4, 0, [0, 5, 1, 6, 2, 7, 3, 8, 4], 0, 0, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0055/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 56. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=2, b=-3. 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. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(10, 2, -3, 1, [1, 9, 5, 7], 1, 0, 4)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 2, -3, 1, [1, 9, 5, 7], 1, 0, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0056/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 57. First Repeat, Transient Length and the Starting Point’s Period

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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(5, 3, -2, 3, [3, 2, 4, 0], 3, 0, 4)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 3, -2, 3, [3, 2, 4, 0], 3, 0, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0057/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 58. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-3, b=-1. 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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(6, -3, -1, 3, [3, 2, 5], 2, 1, nonperiodic-point)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -3, -1, 3, [3, 2, 5], 2, 1, nonperiodic-point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0058/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 59. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-2, b=0. 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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(7, -2, 0, 3, [3, 1, 5, 4, 6, 2], 3, 0, 6)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -2, 0, 3, [3, 1, 5, 4, 6, 2], 3, 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0059/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 60. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-1, b=1. 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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(8, -1, 1, 3, [3, 6], 3, 0, 2)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -1, 1, 3, [3, 6], 3, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0060/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 61. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=0, 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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(9, 0, 2, 3, [3, 2], 2, 1, nonperiodic-point)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 0, 2, 3, [3, 2], 2, 1, nonperiodic-point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0061/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 62. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=1, b=3. 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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(10, 1, 3, 3, [3, 6, 9, 2, 5, 8, 1, 4, 7, 0], 3, 0, 10)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 1, 3, 3, [3, 6, 9, 2, 5, 8, 1, 4, 7, 0], 3, 0, 10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0062/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 63. First Repeat, Transient Length and the Starting Point’s Period

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. x=0. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(5, 2, 4, 0, [0, 4, 2, 3], 0, 0, 4)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 2, 4, 0, [0, 4, 2, 3], 0, 0, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0063/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 64. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=3, b=5. 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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(6, 3, 5, 3, [3, 2, 5], 2, 1, nonperiodic-point)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 3, 5, 3, [3, 2, 5], 2, 1, nonperiodic-point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0064/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 65. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-3, b=6. 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. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(7, -3, 6, 1, [1, 3, 4], 1, 0, 3)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -3, 6, 1, [1, 3, 4], 1, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0065/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 66. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-2, b=7. 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=7. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(8, -2, 7, 7, [7, 1, 5], 5, 2, nonperiodic-point)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -2, 7, 7, [7, 1, 5], 5, 2, nonperiodic-point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0066/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 67. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-1, b=8. 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=6. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(9, -1, 8, 6, [6, 2], 6, 0, 2)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -1, 8, 6, [6, 2], 6, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0067/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 68. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=0, b=9. 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. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(10, 0, 9, 5, [5, 9], 9, 1, nonperiodic-point)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 0, 9, 5, [5, 9], 9, 1, nonperiodic-point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0068/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 69. First Repeat, Transient Length and the Starting Point’s Period

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. x=2. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(5, 1, 10, 2, [2], 2, 0, 1)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 1, 10, 2, [2], 2, 0, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0069/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 70. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=2, b=11. 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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(6, 2, 11, 3, [3, 5], 3, 0, 2)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 2, 11, 3, [3, 5], 3, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0070/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 71. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. 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=6. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(7, 3, 12, 6, [6, 2, 4, 3, 0, 5], 6, 0, 6)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 3, 12, 6, [6, 2, 4, 3, 0, 5], 6, 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0071/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 72. First Repeat, Transient Length and the Starting Point’s Period

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=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(8, -3, 13, 3, [3, 4, 1, 2, 7, 0, 5, 6], 3, 0, 8)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -3, 13, 3, [3, 4, 1, 2, 7, 0, 5, 6], 3, 0, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0072/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 73. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-2, b=14. 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=0. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(9, -2, 14, 0, [0, 5, 4, 6, 2, 1, 3, 8, 7], 0, 0, 9)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -2, 14, 0, [0, 5, 4, 6, 2, 1, 3, 8, 7], 0, 0, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0073/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 74. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-1, b=15. 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=7. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(10, -1, 15, 7, [7, 8], 7, 0, 2)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, -1, 15, 7, [7, 8], 7, 0, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0074/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 75. First Repeat, Transient Length and the Starting Point’s Period

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=0, b=16. 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=4. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).

View answer and explanation(5, 0, 16, 4, [4, 1], 1, 1, nonperiodic-point)

Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 0, 16, 4, [4, 1], 1, 1, nonperiodic-point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0075/)

Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate





## 76. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-3, b=-8. 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).

View answer and explanation(5, -3, -8, 1, [[0, 4], [1, 4], [2, 4], [3, 1], [4, 4]], [3])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -3, -8, 1, [[0, 4], [1, 4], [2, 4], [3, 1], [4, 4]], [3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0076/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 77. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-2, b=-7. 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=2. 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).

View answer and explanation(6, -2, -7, 2, [[0, nonperiodic-point], [1, 3], [2, nonperiodic-point], [3, 3], [4, nonperiodic-point], [5, 3]], [])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -2, -7, 2, [[0, nonperiodic-point], [1, 3], [2, nonperiodic-point], [3, 3], [4, nonperiodic-point], [5, 3]], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0077/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 78. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. 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=3. 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).

View answer and explanation(7, -1, -6, 3, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2]], [0, 1, 2, 3, 4, 5, 6])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -1, -6, 3, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2]], [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0078/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 79. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=0, b=-5. 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=4. 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).

View answer and explanation(8, 0, -5, 4, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point]], [3])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 0, -5, 4, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point]], [3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0079/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 80. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, 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=5. 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).

View answer and explanation(9, 1, -4, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], [])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 1, -4, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0080/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 81. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=2, b=-3. 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=6. 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).

View answer and explanation(10, 2, -3, 6, [[0, nonperiodic-point], [1, 4], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, 4], [6, nonperiodic-point], [7, 4], [8, nonperiodic-point], [9, 4]], [1, 3, 5, 7, 9])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 2, -3, 6, [[0, nonperiodic-point], [1, 4], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, 4], [6, nonperiodic-point], [7, 4], [8, nonperiodic-point], [9, 4]], [1, 3, 5, 7, 9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0081/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 82. Every Point’s Least Period in the Complete Finite Model

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. 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).

View answer and explanation(5, 3, -2, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 3, -2, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0082/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 83. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-3, b=-1. 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=2. 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).

View answer and explanation(6, -3, -1, 2, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 2], [3, nonperiodic-point], [4, nonperiodic-point], [5, 2]], [2, 5])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -3, -1, 2, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 2], [3, nonperiodic-point], [4, nonperiodic-point], [5, 2]], [2, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0083/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 84. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-2, b=0. 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=3. 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).

View answer and explanation(7, -2, 0, 3, [[0, 1], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [0])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -2, 0, 3, [[0, 1], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0084/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 85. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-1, b=1. 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=4. 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).

View answer and explanation(8, -1, 1, 4, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], [0, 1, 2, 3, 4, 5, 6, 7])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -1, 1, 4, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], [0, 1, 2, 3, 4, 5, 6, 7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0085/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 86. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=0, 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. Positive period bound=5. 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).

View answer and explanation(9, 0, 2, 5, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 1], [3, nonperiodic-point], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point], [8, nonperiodic-point]], [2])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 0, 2, 5, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 1], [3, nonperiodic-point], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point], [8, nonperiodic-point]], [2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0086/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 87. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=1, b=3. 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=6. 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).

View answer and explanation(10, 1, 3, 6, [[0, 10], [1, 10], [2, 10], [3, 10], [4, 10], [5, 10], [6, 10], [7, 10], [8, 10], [9, 10]], [])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 1, 3, 6, [[0, 10], [1, 10], [2, 10], [3, 10], [4, 10], [5, 10], [6, 10], [7, 10], [8, 10], [9, 10]], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0087/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 88. Every Point’s Least Period in the Complete Finite Model

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).

View answer and explanation(5, 2, 4, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 2, 4, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0088/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 89. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=3, b=5. 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=2. 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).

View answer and explanation(6, 3, 5, 2, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 2], [3, nonperiodic-point], [4, nonperiodic-point], [5, 2]], [2, 5])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 3, 5, 2, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 2], [3, nonperiodic-point], [4, nonperiodic-point], [5, 2]], [2, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0089/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 90. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-3, b=6. 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=3. 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).

View answer and explanation(7, -3, 6, 3, [[0, 3], [1, 3], [2, 3], [3, 3], [4, 3], [5, 1], [6, 3]], [0, 1, 2, 3, 4, 5, 6])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -3, 6, 3, [[0, 3], [1, 3], [2, 3], [3, 3], [4, 3], [5, 1], [6, 3]], [0, 1, 2, 3, 4, 5, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0090/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 91. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-2, b=7. 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=4. 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).

View answer and explanation(8, -2, 7, 4, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point], [5, 1], [6, nonperiodic-point], [7, nonperiodic-point]], [5])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -2, 7, 4, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point], [5, 1], [6, nonperiodic-point], [7, nonperiodic-point]], [5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0091/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 92. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-1, b=8. 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=5. 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).

View answer and explanation(9, -1, 8, 5, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -1, 8, 5, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0092/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 93. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=0, b=9. 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=6. 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).

View answer and explanation(10, 0, 9, 6, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point], [8, nonperiodic-point], [9, 1]], [9])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 0, 9, 6, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point], [8, nonperiodic-point], [9, 1]], [9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0093/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 94. Every Point’s Least Period in the Complete Finite Model

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).

View answer and explanation(5, 1, 10, 1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1]], [0, 1, 2, 3, 4])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 1, 10, 1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1]], [0, 1, 2, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0094/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 95. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=2, b=11. 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=2. 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).

View answer and explanation(6, 2, 11, 2, [[0, nonperiodic-point], [1, 1], [2, nonperiodic-point], [3, 2], [4, nonperiodic-point], [5, 2]], [1, 3, 5])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 2, 11, 2, [[0, nonperiodic-point], [1, 1], [2, nonperiodic-point], [3, 2], [4, nonperiodic-point], [5, 2]], [1, 3, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0095/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 96. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. 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=3. 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).

View answer and explanation(7, 3, 12, 3, [[0, 6], [1, 1], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [1])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 3, 12, 3, [[0, 6], [1, 1], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0096/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 97. Every Point’s Least Period in the Complete Finite Model

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. Positive period bound=4. 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).

View answer and explanation(8, -3, 13, 4, [[0, 8], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8]], [])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -3, 13, 4, [[0, 8], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8]], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0097/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 98. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-2, b=14. 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=5. 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).

View answer and explanation(9, -2, 14, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], [])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -2, 14, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0098/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 99. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-1, b=15. 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=6. 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).

View answer and explanation(10, -1, 15, 6, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, -1, 15, 6, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0099/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate





## 100. Every Point’s Least Period in the Complete Finite Model

Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=0, b=16. 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).

View answer and explanation(5, 0, 16, 1, [[0, nonperiodic-point], [1, 1], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point]], [1])

Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 0, 16, 1, [[0, nonperiodic-point], [1, 1], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point]], [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0100/)

Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations/validation-report.json)