Question 2528: First Repeat, Transient Length and the Starting Point’s Period

Original question and givens

Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. 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=5. 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=-1, b=-6. 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=5. 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

Step-by-step derivation

n=7; a=-1; b=-6

S={0,...,6}; T(z)=(a·z+b) mod n

start=5

record each state before its first successor calculation

least period step1 product := mul(-1, 5)

least period step1 product = -5

least period step1 affine value := sum(-5, -6)

least period step1 affine value = -11

Euclidean division of -11 by 7

-11=-2·7+3

least period step2 product := mul(-1, 3)

least period step2 product = -3

least period step2 affine value := sum(-3, -6)

least period step2 affine value = -9

Euclidean division of -9 by 7

-9=-2·7+5

distinct trace=[5, 3]; first repeated state=5

cycle_start=0; starting-point least period=2

Reverse checks

• S009 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.
• 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 least period step2 affine value with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute least period step2 product 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 least period step1 affine value with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute least period step1 product 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

(7, -1, -6, 5, [5, 3], 5, 0, 2)

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.