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R0002-01588 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0013
Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=5, b=13. 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. x=2. Give (modulus, a, b, x, T(x)).
G001 — Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=5, b=13. 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. x=2. Give (modulus, a, b, x, T(x)).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.
Before
X={0 through 4}; T(x)=(5x+(13)) mod 5After
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.
Rule: finite_total_map · Uses: G001
Before
one step: a·x := mul(5, 2)
After
one step: a·x = 10
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
one step: a·x+b := sum(10, 13)
After
one step: a·x+b = 23
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
reduce 23 modulo 5
After
23=4·5+3
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S003
Before
x=2
After
T(x)=3∈X
Conditions: none
This computes one update,not a whole orbit or a fixed-point classification.
Rule: state_membership · Uses: G001, S004
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute one step: a·x+b with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute one step: a·x with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(5, 5, 13, 2, 3)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: e36edd4357af808c19d2c8a28a9862b876695b46f6ed85868cfd877c6f2520fbcandidate record SHA-256: 581e7e79f14bb1588984ac2cc36d02acce1637f838927fb9a775148d685414f4semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED