Question 1570: 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=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].

• G001 — 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].

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)=(20x+(50)) mod 9

remainder representative always lies in X

x₀=1; transitions=1

initial orbit prefix=[1]

transition1: a·x := mul(20, 1)

transition1: a·x = 20

transition1: a·x+b := sum(20, 50)

transition1: a·x+b = 70

reduce 70 modulo 9

70=7·9+7

computed transitions=1

orbit=[1, 7]; number of entries=2

a finite requested prefix

no unrequested least-period claim

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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, 20, 50, 1, 1, [1, 7])

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.