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R0002-01504 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0054
Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=-1, b=7. 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(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=-1, b=7. 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 7}; T(x)=(-1x+(7)) mod 8After
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(-1, 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, 7)
After
candidate0: a·x+b = 7
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
reduce 7 modulo 8
After
7=0·8+7
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S003
Before
T(0)=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, S004
Before
candidate1: a·x := mul(-1, 1)
After
candidate1: a·x = -1
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(-1, 7)
After
candidate1: a·x+b = 6
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
reduce 6 modulo 8
After
6=0·8+6
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S007
Before
T(1)=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, S008
Before
candidate2: a·x := mul(-1, 2)
After
candidate2: a·x = -2
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(-2, 7)
After
candidate2: a·x+b = 5
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
reduce 5 modulo 8
After
5=0·8+5
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S011
Before
T(2)=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, S012
Before
candidate3: a·x := mul(-1, 3)
After
candidate3: a·x = -3
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(-3, 7)
After
candidate3: a·x+b = 4
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
reduce 4 modulo 8
After
4=0·8+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(-1, 4)
After
candidate4: a·x = -4
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(-4, 7)
After
candidate4: a·x+b = 3
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
reduce 3 modulo 8
After
3=0·8+3
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S019
Before
T(4)=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, S020
Before
candidate5: a·x := mul(-1, 5)
After
candidate5: a·x = -5
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(-5, 7)
After
candidate5: a·x+b = 2
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
reduce 2 modulo 8
After
2=0·8+2
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S023
Before
T(5)=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, S024
Before
candidate6: a·x := mul(-1, 6)
After
candidate6: a·x = -6
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(-6, 7)
After
candidate6: a·x+b = 1
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
reduce 1 modulo 8
After
1=0·8+1
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S027
Before
T(6)=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, S028
Before
candidate7: a·x := mul(-1, 7)
After
candidate7: 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, S029
Before
candidate7: a·x+b := sum(-7, 7)
After
candidate7: a·x+b = 0
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
reduce 0 modulo 8
After
0=0·8+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
all 8 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, S033
S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(8, -1, 7, [])
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: 9634ea41baa4cf773dd0affbf835083b12ce37ec95fb83038e54dc74d09fc6eccandidate record SHA-256: 4c942edf241e08278959509ce9b120f5098c8c9133dd1a4595ae0acc14450d1asemantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED