Question 1555: A Fixed-Length Discrete Orbit Prefix

Original question and givens

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].

• G001 — 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].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-discrete-dynamical-system

Step-by-step derivation

X={0 through 8}; T(x)=(5x+(5)) mod 9

remainder representative always lies in X

x₀=4; transitions=4

initial orbit prefix=[4]

transition1: a·x := mul(5, 4)

transition1: a·x = 20

transition1: a·x+b := sum(20, 5)

transition1: a·x+b = 25

reduce 25 modulo 9

25=2·9+7

transition2: a·x := mul(5, 7)

transition2: a·x = 35

transition2: a·x+b := sum(35, 5)

transition2: a·x+b = 40

reduce 40 modulo 9

40=4·9+4

transition3: a·x := mul(5, 4)

transition3: a·x = 20

transition3: a·x+b := sum(20, 5)

transition3: a·x+b = 25

reduce 25 modulo 9

25=2·9+7

transition4: a·x := mul(5, 7)

transition4: a·x = 35

transition4: a·x+b := sum(35, 5)

transition4: a·x+b = 40

reduce 40 modulo 9

40=4·9+4

computed transitions=4

orbit=[4, 7, 4, 7, 4]; number of entries=5

a finite requested prefix

no unrequested least-period claim

Reverse checks

• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute transition4: a·x+b with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute transition4: a·x with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute transition3: a·x+b with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute transition3: a·x with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute transition2: a·x+b with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute transition2: a·x with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute transition1: a·x+b with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute transition1: a·x with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(9, 5, 5, 4, 4, [4, 7, 4, 7, 4])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.