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Explicit Finite Discrete Dynamics and Invariant Subsets

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Canonical question JSONL · English JSONL · Validation report