# Question 1575: A Fixed-Length Discrete Orbit Prefix [Return to catalogue page 8](https://ocme.evemisslab.com/en/questions/page/8/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations/#q-20261013-advance-20261020-r01-discrete-dynamical-system-0050) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0002-01575 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0050 ## 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=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]. - G001 — 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]. ## 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](https://ocme.evemisslab.com/en/mko/mko-discrete-dynamical-system/) ## Step-by-step derivation S001 · impliesBefore X={0 through 8}; T(x)=(25x+(65)) mod 9 After remainder representative always lies in X Conditions: none Fix the full finite state carrier and the same update rule;negative integers also use the standard remainder from0 through n−1. Step identifiers and dependenciesRule: finite_total_map · Uses: G001 S002 · impliesBefore x₀=6; transitions=6 After initial orbit prefix=[6] Conditions: none Retain the initial state;steps=0 still has one known state,not an empty list. Step identifiers and dependenciesRule: orbit_initial_state · Uses: G001, S001 S003 · calculationBefore transition1: a·x := mul(25, 6) After transition1: a·x = 150 Conditions: none Multiply the original coefficient by the current state;do not clip negative values to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore transition1: a·x+b := sum(150, 65) After transition1: a·x+b = 215 Conditions: none Add the same offset before reducing modulo n. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore reduce 215 modulo 9 After 215=23·9+8 Conditions: none The integer identity together with0≤remainder