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R0002-01523 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0073
Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=18, b=45. 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. Give (modulus, a, b, every x with T(x)=x in increasing order).
G001 — Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=18, b=45. 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. Give (modulus, a, b, every x with T(x)=x in increasing order).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)=(18x+(45)) 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
candidate0: a·x := mul(18, 0)
After
candidate0: a·x = 0
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
candidate0: a·x+b := sum(0, 45)
After
candidate0: a·x+b = 45
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
reduce 45 modulo 9
After
45=5·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S003
Before
T(0)=0
After
fixed-point condition T(x)=x: Yes
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S004
Before
candidate1: a·x := mul(18, 1)
After
candidate1: a·x = 18
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
candidate1: a·x+b := sum(18, 45)
After
candidate1: a·x+b = 63
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
reduce 63 modulo 9
After
63=7·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S007
Before
T(1)=0
After
fixed-point condition T(x)=x: No
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S008
Before
candidate2: a·x := mul(18, 2)
After
candidate2: a·x = 36
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
candidate2: a·x+b := sum(36, 45)
After
candidate2: a·x+b = 81
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
reduce 81 modulo 9
After
81=9·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S011
Before
T(2)=0
After
fixed-point condition T(x)=x: No
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S012
Before
candidate3: a·x := mul(18, 3)
After
candidate3: a·x = 54
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
candidate3: a·x+b := sum(54, 45)
After
candidate3: a·x+b = 99
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
reduce 99 modulo 9
After
99=11·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S015
Before
T(3)=0
After
fixed-point condition T(x)=x: No
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S016
Before
candidate4: a·x := mul(18, 4)
After
candidate4: a·x = 72
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
candidate4: a·x+b := sum(72, 45)
After
candidate4: a·x+b = 117
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
reduce 117 modulo 9
After
117=13·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S019
Before
T(4)=0
After
fixed-point condition T(x)=x: No
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S020
Before
candidate5: a·x := mul(18, 5)
After
candidate5: a·x = 90
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
candidate5: a·x+b := sum(90, 45)
After
candidate5: a·x+b = 135
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
reduce 135 modulo 9
After
135=15·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S023
Before
T(5)=0
After
fixed-point condition T(x)=x: No
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S024
Before
candidate6: a·x := mul(18, 6)
After
candidate6: a·x = 108
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
candidate6: a·x+b := sum(108, 45)
After
candidate6: a·x+b = 153
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
reduce 153 modulo 9
After
153=17·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S027
Before
T(6)=0
After
fixed-point condition T(x)=x: No
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S028
Before
candidate7: a·x := mul(18, 7)
After
candidate7: a·x = 126
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
candidate7: a·x+b := sum(126, 45)
After
candidate7: a·x+b = 171
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
reduce 171 modulo 9
After
171=19·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S031
Before
T(7)=0
After
fixed-point condition T(x)=x: No
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S032
Before
candidate8: a·x := mul(18, 8)
After
candidate8: a·x = 144
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
candidate8: a·x+b := sum(144, 45)
After
candidate8: a·x+b = 189
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
reduce 189 modulo 9
After
189=21·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S035
Before
T(8)=0
After
fixed-point condition T(x)=x: No
Conditions: none
A fixed point must remain the same state after updating;merely remaining in the carrier is insufficient.
Rule: fixed_point_equality · Uses: G001, S036
Before
all 9 carrier elements audited
After
complete fixed set=[0]
Conditions: none
The full finite carrier was exhausted,so no fixed point is omitted;an empty list means this T has none.
Rule: exhaustive_fixed_inventory · Uses: G001, S037
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Recompute candidate8: a·x+b with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute candidate8: a·x with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S031 · premise_and_result_audit — Recompute candidate7: a·x+b with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute candidate7: a·x with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Recompute candidate6: a·x+b with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute candidate6: a·x with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Recompute candidate5: a·x+b with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute candidate5: a·x with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Recompute candidate4: a·x+b with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute candidate4: a·x with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Recompute candidate3: a·x+b with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute candidate3: a·x with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute candidate2: a·x+b with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute candidate2: a·x with exact rational arithmetic, not rounded decimals.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 candidate1: a·x+b with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute candidate1: 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute candidate0: a·x+b with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute candidate0: 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, 18, 45, [0])
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: 52f227478e4f6b216816551d136c93562a4116933f2c85debd732b06f7af17f6candidate record SHA-256: fe31787ee599f8d5cde3dcdf4ded869604ead33052d7d92b68588c89982d7236semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED