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R0002-01535 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0085
Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=10, b=6. 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. Candidate subset=[2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).
G001 — Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=10, b=6. 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. Candidate subset=[2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).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 8}; T(x)=(10x+(6)) mod 9After
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
listed subset=[2, 3]
After
set S=[2, 3]
Conditions: none
Deduplication and sorting give a consistent set presentation without adding states.
Rule: subset_normalization · Uses: G001, S001
Before
member2: a·x := mul(10, 2)
After
member2: a·x = 20
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
member2: a·x+b := sum(20, 6)
After
member2: a·x+b = 26
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
reduce 26 modulo 9
After
26=2·9+8
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S004
Before
member3: a·x := mul(10, 3)
After
member3: a·x = 30
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
member3: a·x+b := sum(30, 6)
After
member3: a·x+b = 36
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
reduce 36 modulo 9
After
36=4·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S007
Before
T(S)=[0, 8]
After
T(S)⊆S: No; exiting witnesses=[[2, 8], [3, 0]]
Conditions: none
Every image must remain in S;one explicit exit refutes invariance.The empty S satisfies the condition vacuously.
Rule: forward_invariance · Uses: G001, S008
Before
x₀∈S and T(S)⊆S
After
induction keeps every later iterate in S
Conditions: none
When the condition holds,the initial state and every update stay in S,so induction gives all later iterates,not merely one arbitrary short orbit.
Rule: invariance_induction · Uses: G001, S009
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 member3: a·x+b with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute member3: a·x 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 member2: a·x+b with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute member2: a·x 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.(9, 10, 6, [2, 3], [0, 8], No)
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: ac03ee20fc9ecf5ef3f4a2dc0c51f3efe8b0ac0f1b81765bb0723ae6360c069acandidate record SHA-256: 93648cbc62e914f9fd2085ae64f9db1995e94b8791e0de87dbf7084ee4611f56semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED