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R0002-01512 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0062
Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=7, b=23. 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(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=7, b=23. 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 9}; T(x)=(7x+(23)) mod 10After
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(7, 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, 23)
After
candidate0: 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 10
After
23=2·10+3
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S003
Before
T(0)=3
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, S004
Before
candidate1: a·x := mul(7, 1)
After
candidate1: a·x = 7
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(7, 23)
After
candidate1: a·x+b = 30
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
reduce 30 modulo 10
After
30=3·10+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(7, 2)
After
candidate2: a·x = 14
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(14, 23)
After
candidate2: a·x+b = 37
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
reduce 37 modulo 10
After
37=3·10+7
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S011
Before
T(2)=7
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(7, 3)
After
candidate3: a·x = 21
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(21, 23)
After
candidate3: a·x+b = 44
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
reduce 44 modulo 10
After
44=4·10+4
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S015
Before
T(3)=4
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(7, 4)
After
candidate4: a·x = 28
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(28, 23)
After
candidate4: a·x+b = 51
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
reduce 51 modulo 10
After
51=5·10+1
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S019
Before
T(4)=1
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(7, 5)
After
candidate5: a·x = 35
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(35, 23)
After
candidate5: a·x+b = 58
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
reduce 58 modulo 10
After
58=5·10+8
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S023
Before
T(5)=8
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(7, 6)
After
candidate6: a·x = 42
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(42, 23)
After
candidate6: a·x+b = 65
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
reduce 65 modulo 10
After
65=6·10+5
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S027
Before
T(6)=5
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(7, 7)
After
candidate7: a·x = 49
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(49, 23)
After
candidate7: a·x+b = 72
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
reduce 72 modulo 10
After
72=7·10+2
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S031
Before
T(7)=2
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(7, 8)
After
candidate8: a·x = 56
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(56, 23)
After
candidate8: a·x+b = 79
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
reduce 79 modulo 10
After
79=7·10+9
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S035
Before
T(8)=9
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
candidate9: a·x := mul(7, 9)
After
candidate9: a·x = 63
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
candidate9: a·x+b := sum(63, 23)
After
candidate9: a·x+b = 86
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
reduce 86 modulo 10
After
86=8·10+6
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S039
Before
T(9)=6
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, S040
Before
all 10 carrier elements audited
After
complete fixed set=[]
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, S041
S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S039 · premise_and_result_audit — Recompute candidate9: a·x+b with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute candidate9: a·x with exact rational arithmetic, not rounded decimals.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.(10, 7, 23, [])
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: 685f727f7e55bde3580697e83fbaa4378c2877303aae42a14e4ea9bdc3a2c323candidate record SHA-256: a7efbbeeaa282c9aa8296bf0b53ec46590b97a51a0a04fc123217c9186d48e65semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED