# Question 2546: First Repeat, Transient Length and the Starting Point’s Period [Return to catalogue page 13](https://ocme.evemisslab.com/en/questions/page/13/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations/#q-20261013-advance-20261021-r03-orbit-fixed-periodic-0071) 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. R0004-02546 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0071 ## Original question and givens Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=6. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point). - G001 — Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=6. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point). ## 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-orbit-fixed-periodic](https://ocme.evemisslab.com/en/mko/mko-orbit-fixed-periodic/) ## Step-by-step derivation S001 · impliesBefore n=7; a=3; b=12 After S={0,...,6}; T(z)=(a·z+b) mod n Conditions: none Remainders lie in0,...,n−1,so every state has one successor in the same finite set;negative dividends do not produce negative canonical remainders. Step identifiers and dependenciesRule: finite_self_map · Uses: G001 S002 · impliesBefore start=6 After record each state before its first successor calculation Conditions: none Start with the specified point itself,not its first successor. Step identifiers and dependenciesRule: orbit_initial_state · Uses: G001, S001 S003 · calculationBefore least period step1 product := mul(3, 6) After least period step1 product = 18 Conditions: none Multiply the current state by the original coefficient a,retaining its sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore least period step1 affine value := sum(18, 12) After least period step1 affine value = 30 Conditions: none Then add the original constant b;this integer is not yet the canonical state. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore Euclidean division of 30 by 7 After 30=4·7+2 Conditions: none Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder