# Question 2555: Every Point’s Least Period in the Complete Finite Model [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-0080) 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. R0004-02555 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0080 ## Original question and givens Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, b=-4. 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. Positive period bound=5. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound). - G001 — Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, b=-4. 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. Positive period bound=5. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound). ## 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=9; a=1; b=-4 After S={0,...,8}; 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=0 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 point0 step1 product := mul(1, 0) After point0 step1 product = 0 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 point0 step1 affine value := sum(0, -4) After point0 step1 affine value = -4 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 -4 by 9 After -4=-1·9+5 Conditions: none Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder0 and0≤remainder