# Explicit Finite Discrete Dynamics and Invariant Subsets

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-20261020-r01-discrete-dynamical-system-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. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=-7, b=-11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=0. Give (modulus, a, b, x, T(x)).

View answer and explanation(5, -7, -11, 0, 4)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, -7, -11, 0, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0001/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 2. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=-6, b=-9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(6, -6, -9, 1, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, -6, -9, 1, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0002/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 3. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=-5, b=-7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=2. Give (modulus, a, b, x, T(x)).

View answer and explanation(7, -5, -7, 2, 4)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, -5, -7, 2, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0003/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 4. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=-4, b=-5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=3. Give (modulus, a, b, x, T(x)).

View answer and explanation(8, -4, -5, 3, 7)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, -4, -5, 3, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0004/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 5. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=-3, b=-3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=4. Give (modulus, a, b, x, T(x)).

View answer and explanation(9, -3, -3, 4, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, -3, -3, 4, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0005/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 6. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=-2, b=-1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=5. Give (modulus, a, b, x, T(x)).

View answer and explanation(10, -2, -1, 5, 9)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, -2, -1, 5, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0006/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 7. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=-1, b=1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(5, -1, 1, 1, 0)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, -1, 1, 1, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0007/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 8. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=0, b=3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(6, 0, 3, 1, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 0, 3, 1, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0008/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 9. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=1, b=5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(7, 1, 5, 1, 6)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 1, 5, 1, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0009/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 10. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=2, b=7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(8, 2, 7, 1, 1)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 2, 7, 1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0010/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 11. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=3, b=9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(9, 3, 9, 1, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 3, 9, 1, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0011/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 12. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=4, b=11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(10, 4, 11, 1, 5)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 4, 11, 1, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0012/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 13. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=5, b=13. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=2. Give (modulus, a, b, x, T(x)).

View answer and explanation(5, 5, 13, 2, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 5, 13, 2, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0013/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 14. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=6, b=15. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(6, 6, 15, 1, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 6, 15, 1, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0014/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 15. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=7, b=17. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=0. Give (modulus, a, b, x, T(x)).

View answer and explanation(7, 7, 17, 0, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 7, 17, 0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0015/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 16. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=8, b=19. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=7. Give (modulus, a, b, x, T(x)).

View answer and explanation(8, 8, 19, 7, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 8, 19, 7, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0016/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 17. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=9, b=21. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=7. Give (modulus, a, b, x, T(x)).

View answer and explanation(9, 9, 21, 7, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 9, 21, 7, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0017/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 18. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=10, b=23. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=7. Give (modulus, a, b, x, T(x)).

View answer and explanation(10, 10, 23, 7, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 10, 23, 7, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0018/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 19. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=11, b=25. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=3. Give (modulus, a, b, x, T(x)).

View answer and explanation(5, 11, 25, 3, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 11, 25, 3, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0019/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 20. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=12, b=27. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).

View answer and explanation(6, 12, 27, 1, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 12, 27, 1, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0020/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 21. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=13, b=29. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=6. Give (modulus, a, b, x, T(x)).

View answer and explanation(7, 13, 29, 6, 2)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 13, 29, 6, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0021/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 22. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=14, b=31. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=5. Give (modulus, a, b, x, T(x)).

View answer and explanation(8, 14, 31, 5, 5)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 14, 31, 5, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0022/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 23. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=15, b=33. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=4. Give (modulus, a, b, x, T(x)).

View answer and explanation(9, 15, 33, 4, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 15, 33, 4, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0023/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 24. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=16, b=35. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=3. Give (modulus, a, b, x, T(x)).

View answer and explanation(10, 16, 35, 3, 3)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 16, 35, 3, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0024/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 25. A Typed Finite State and Its Canonical Residue Update

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=17, b=37. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=4. Give (modulus, a, b, x, T(x)).

View answer and explanation(5, 17, 37, 4, 0)

Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 17, 37, 4, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0025/)

Discrete-time dynamical system · qf-app20-dynamics-state-update · candidate





## 26. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=1, b=-7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=0. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

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

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 1, -7, 0, 0, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0026/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 27. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=2, b=-4. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=1, updates=1. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(6, 2, -4, 1, 1, [1, 4])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 2, -4, 1, 1, [1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0027/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 28. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=3, b=-1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=2, updates=2. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

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

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 3, -1, 2, 2, [2, 5, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0028/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 29. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=4, b=2. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=3, updates=3. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(8, 4, 2, 3, 3, [3, 6, 2, 2])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 4, 2, 3, 3, [3, 6, 2, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0029/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 30. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=5, b=5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=4, updates=4. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(9, 5, 5, 4, 4, [4, 7, 4, 7, 4])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 5, 5, 4, 4, [4, 7, 4, 7, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0030/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 31. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=6, b=8. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=5. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

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

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 6, 8, 0, 5, [0, 3, 1, 4, 2, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0031/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 32. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=7, b=11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=6. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(6, 7, 11, 0, 6, [0, 5, 4, 3, 2, 1, 0])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 7, 11, 0, 6, [0, 5, 4, 3, 2, 1, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0032/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 33. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=8, b=14. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=7. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(7, 8, 14, 0, 7, [0, 0, 0, 0, 0, 0, 0, 0])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 8, 14, 0, 7, [0, 0, 0, 0, 0, 0, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0033/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 34. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=9, b=17. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=8. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(8, 9, 17, 0, 8, [0, 1, 2, 3, 4, 5, 6, 7, 0])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 9, 17, 0, 8, [0, 1, 2, 3, 4, 5, 6, 7, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0034/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 35. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=10, b=20. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=0. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(9, 10, 20, 0, 0, [0])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 10, 20, 0, 0, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0035/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 36. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=11, b=23. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=1. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(5, 11, 23, 0, 1, [0, 3])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 11, 23, 0, 1, [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0036/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 37. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=12, b=26. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=5, updates=2. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(6, 12, 26, 5, 2, [5, 2, 2])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 12, 26, 5, 2, [5, 2, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0037/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 38. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=13, b=29. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=5, updates=3. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(7, 13, 29, 5, 3, [5, 3, 5, 3])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 13, 29, 5, 3, [5, 3, 5, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0038/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 39. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=14, b=32. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=5, updates=4. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(8, 14, 32, 5, 4, [5, 6, 4, 0, 0])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 14, 32, 5, 4, [5, 6, 4, 0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0039/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 40. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=15, b=35. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=5, updates=5. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(9, 15, 35, 5, 5, [5, 2, 2, 2, 2, 2])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 15, 35, 5, 5, [5, 2, 2, 2, 2, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0040/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 41. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=16, b=38. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=6. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(5, 16, 38, 0, 6, [0, 3, 1, 4, 2, 0, 3])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 16, 38, 0, 6, [0, 3, 1, 4, 2, 0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0041/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 42. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=17, b=41. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=4, updates=7. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(6, 17, 41, 4, 7, [4, 1, 4, 1, 4, 1, 4, 1])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 17, 41, 4, 7, [4, 1, 4, 1, 4, 1, 4, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0042/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 43. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=18, b=44. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=3, updates=8. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(7, 18, 44, 3, 8, [3, 0, 2, 3, 0, 2, 3, 0, 2])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 18, 44, 3, 8, [3, 0, 2, 3, 0, 2, 3, 0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0043/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 44. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=19, b=47. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=2, updates=0. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(8, 19, 47, 2, 0, [2])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 19, 47, 2, 0, [2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0044/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 45. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=20, b=50. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=1, updates=1. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(9, 20, 50, 1, 1, [1, 7])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 20, 50, 1, 1, [1, 7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0045/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 46. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=21, b=53. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=2. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(5, 21, 53, 0, 2, [0, 3, 1])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 21, 53, 0, 2, [0, 3, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0046/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 47. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=22, b=56. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=3, updates=3. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(6, 22, 56, 3, 3, [3, 2, 4, 0])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 22, 56, 3, 3, [3, 2, 4, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0047/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 48. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=23, b=59. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=1, updates=4. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(7, 23, 59, 1, 4, [1, 5, 6, 1, 5])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 23, 59, 1, 4, [1, 5, 6, 1, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0048/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 49. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=24, b=62. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=7, updates=5. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(8, 24, 62, 7, 5, [7, 6, 6, 6, 6, 6])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 24, 62, 7, 5, [7, 6, 6, 6, 6, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0049/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 50. A Fixed-Length Discrete Orbit Prefix

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=25, b=65. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=6, updates=6. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

View answer and explanation(9, 25, 65, 6, 6, [6, 8, 4, 3, 5, 1, 0])

Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 25, 65, 6, 6, [6, 8, 4, 3, 5, 1, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0050/)

Discrete-time dynamical system · qf-app20-dynamics-orbit-prefix · candidate





## 51. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=-4, b=1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

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

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, -4, 1, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0051/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 52. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=-3, b=3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(6, -3, 3, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, -3, 3, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0052/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 53. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=-2, b=5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(7, -2, 5, [4])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, -2, 5, [4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0053/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 54. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=-1, b=7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(8, -1, 7, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, -1, 7, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0054/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 55. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=0, b=9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(9, 0, 9, [0])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 0, 9, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0055/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 56. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=1, b=11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(10, 1, 11, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 1, 11, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0056/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 57. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=2, b=13. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(5, 2, 13, [2])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 2, 13, [2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0057/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 58. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=3, b=15. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(6, 3, 15, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 3, 15, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0058/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 59. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=4, b=17. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(7, 4, 17, [6])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 4, 17, [6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0059/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 60. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=5, b=19. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(8, 5, 19, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 5, 19, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0060/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 61. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=6, b=21. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(9, 6, 21, [3])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 6, 21, [3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0061/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 62. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=7, b=23. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(10, 7, 23, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 7, 23, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0062/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 63. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=8, b=25. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(5, 8, 25, [0])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 8, 25, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0063/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 64. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=9, b=27. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(6, 9, 27, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 9, 27, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0064/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 65. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=10, b=29. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(7, 10, 29, [3])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 10, 29, [3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0065/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 66. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=11, b=31. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(8, 11, 31, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 11, 31, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0066/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 67. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=12, b=33. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(9, 12, 33, [6])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 12, 33, [6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0067/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 68. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=13, b=35. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(10, 13, 35, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 13, 35, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0068/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 69. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=14, b=37. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(5, 14, 37, [1])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 14, 37, [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0069/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 70. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=15, b=39. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(6, 15, 39, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 15, 39, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0070/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 71. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=16, b=41. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(7, 16, 41, [1])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 16, 41, [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0071/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 72. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=17, b=43. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(8, 17, 43, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 17, 43, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0072/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 73. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=18, b=45. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(9, 18, 45, [0])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 18, 45, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0073/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 74. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=19, b=47. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(10, 19, 47, [])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 19, 47, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0074/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 75. Every Fixed Point of the Given Finite Self-Map

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=20, b=49. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

View answer and explanation(5, 20, 49, [4])

Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 20, 49, [4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0075/)

Discrete-time dynamical system · qf-app20-dynamics-fixed-point-inventory · candidate





## 76. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=1, b=-3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(5, 1, -3, [0, 1], [2, 3], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 1, -3, [0, 1], [2, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0076/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 77. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=2, b=-2. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[2], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(6, 2, -2, [2], [2], Yes)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 2, -2, [2], [2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0077/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 78. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=3, b=-1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 2], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(7, 3, -1, [0, 2], [5, 6], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 3, -1, [0, 2], [5, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0078/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 79. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=4, b=0. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 2], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(8, 4, 0, [1, 2], [0, 4], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 4, 0, [1, 2], [0, 4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0079/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 80. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=5, b=1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 2], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(9, 5, 1, [0, 1, 2], [1, 2, 6], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 5, 1, [0, 1, 2], [1, 2, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0080/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 81. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=6, b=2. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(5, 6, 2, [3], [0], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 6, 2, [3], [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0081/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 82. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=7, b=3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(6, 7, 3, [0, 3], [0, 3], Yes)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 7, 3, [0, 3], [0, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0082/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 83. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=8, b=4. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(7, 8, 4, [1, 3], [0, 5], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 8, 4, [1, 3], [0, 5], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0083/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 84. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=9, b=5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(8, 9, 5, [0, 1, 3], [0, 5, 6], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 9, 5, [0, 1, 3], [0, 5, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0084/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 85. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=10, b=6. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(9, 10, 6, [2, 3], [0, 8], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 10, 6, [2, 3], [0, 8], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0085/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 86. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=11, b=7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(5, 11, 7, [0, 2, 3], [0, 2, 4], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 11, 7, [0, 2, 3], [0, 2, 4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0086/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 87. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=12, b=8. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(6, 12, 8, [1, 2, 3], [2], Yes)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 12, 8, [1, 2, 3], [2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0087/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 88. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=13, b=9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(7, 13, 9, [0, 1, 2, 3], [0, 1, 2, 6], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 13, 9, [0, 1, 2, 3], [0, 1, 2, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0088/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 89. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=14, b=10. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(8, 14, 10, [4], [2], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 14, 10, [4], [2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0089/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 90. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=15, b=11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(9, 15, 11, [0, 4], [2, 8], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 15, 11, [0, 4], [2, 8], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0090/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 91. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=16, b=12. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(5, 16, 12, [1, 4], [1, 3], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 16, 12, [1, 4], [1, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0091/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 92. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=17, b=13. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(6, 17, 13, [0, 1, 4], [0, 1, 3], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 17, 13, [0, 1, 4], [0, 1, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0092/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 93. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=18, b=14. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[2, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(7, 18, 14, [2, 4], [1, 2], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 18, 14, [2, 4], [1, 2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0093/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 94. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=19, b=15. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 2, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(8, 19, 15, [0, 2, 4], [3, 5, 7], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 19, 15, [0, 2, 4], [3, 5, 7], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0094/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 95. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=20, b=16. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 2, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(9, 20, 16, [1, 2, 4], [0, 2, 6], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 20, 16, [1, 2, 4], [0, 2, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0095/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 96. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=21, b=17. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 2, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(5, 21, 17, [0, 1, 2, 4], [1, 2, 3, 4], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 21, 17, [0, 1, 2, 4], [1, 2, 3, 4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0096/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 97. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=22, b=18. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(6, 22, 18, [3, 4], [0, 4], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 22, 18, [3, 4], [0, 4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0097/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 98. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=23, b=19. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(7, 23, 19, [0, 3, 4], [4, 5, 6], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 23, 19, [0, 3, 4], [4, 5, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0098/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 99. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=24, b=20. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(8, 24, 20, [1, 3, 4], [4], Yes)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 24, 20, [1, 3, 4], [4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0099/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate





## 100. Forward Invariance versus Pointwise Fixing

Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=25, b=21. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).

View answer and explanation(9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], No)

Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0100/)

Discrete-time dynamical system · qf-app20-dynamics-invariant-subset-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations/validation-report.json)