Explore related knowledge and questions →
← Question derivation index · Original question batch
R0002-01580 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0005
Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=-3, b=-3. 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=4. Give (modulus, a, b, x, T(x)).
G001 — Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=-3, b=-3. 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=4. 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 8}; T(x)=(-3x+(-3)) 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
one step: a·x := mul(-3, 4)
After
one step: a·x = -12
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(-12, -3)
After
one step: a·x+b = -15
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
reduce -15 modulo 9
After
-15=-2·9+3
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S003
Before
x=4
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.(9, -3, -3, 4, 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: 0704531be927f688bab2557a9e773ab9a469db7f79be0558a4b788bcdc3f7d04candidate record SHA-256: dc64dec6c6a188f95f71ec2ecef006e7ca5e542fe51e2b46d0d991192c16055bsemantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED