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R0002-01559 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0034
Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=9, b=17. 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. x0=0, updates=8. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].
G001 — Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=9, b=17. 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. x0=0, updates=8. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].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)=(9x+(17)) 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
x₀=0; transitions=8
After
initial orbit prefix=[0]
Conditions: none
Retain the initial state;steps=0 still has one known state,not an empty list.
Rule: orbit_initial_state · Uses: G001, S001
Before
transition1: a·x := mul(9, 0)
After
transition1: 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, S002
Before
transition1: a·x+b := sum(0, 17)
After
transition1: a·x+b = 17
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
reduce 17 modulo 8
After
17=2·8+1
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S004
Before
transition2: a·x := mul(9, 1)
After
transition2: a·x = 9
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
transition2: a·x+b := sum(9, 17)
After
transition2: a·x+b = 26
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
reduce 26 modulo 8
After
26=3·8+2
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S007
Before
transition3: a·x := mul(9, 2)
After
transition3: 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, S008
Before
transition3: a·x+b := sum(18, 17)
After
transition3: a·x+b = 35
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
reduce 35 modulo 8
After
35=4·8+3
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S010
Before
transition4: a·x := mul(9, 3)
After
transition4: a·x = 27
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
transition4: a·x+b := sum(27, 17)
After
transition4: a·x+b = 44
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
reduce 44 modulo 8
After
44=5·8+4
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S013
Before
transition5: a·x := mul(9, 4)
After
transition5: 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, S014
Before
transition5: a·x+b := sum(36, 17)
After
transition5: a·x+b = 53
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
reduce 53 modulo 8
After
53=6·8+5
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S016
Before
transition6: a·x := mul(9, 5)
After
transition6: a·x = 45
Conditions: none
Multiply the original coefficient by the current state;do not clip negative values to0.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
transition6: a·x+b := sum(45, 17)
After
transition6: a·x+b = 62
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
reduce 62 modulo 8
After
62=7·8+6
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S019
Before
transition7: a·x := mul(9, 6)
After
transition7: 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, S020
Before
transition7: a·x+b := sum(54, 17)
After
transition7: a·x+b = 71
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
reduce 71 modulo 8
After
71=8·8+7
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S022
Before
transition8: a·x := mul(9, 7)
After
transition8: 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, S023
Before
transition8: a·x+b := sum(63, 17)
After
transition8: a·x+b = 80
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
reduce 80 modulo 8
After
80=10·8+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S025
Before
computed transitions=8
After
orbit=[0, 1, 2, 3, 4, 5, 6, 7, 0]; number of entries=9
Conditions: none
Apply the same T to the preceding result; k transitions give k+1 states.
Rule: orbit_prefix_length · Uses: G001, S026
Before
a finite requested prefix
After
no unrequested least-period claim
Conditions: none
This output is the requested prefix;it is not an unverified least-period proof from a few orbit values.
Rule: finite_prefix_scope · Uses: G001, S027
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Recompute transition8: a·x+b with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute transition8: a·x with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Recompute transition7: a·x+b with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute transition7: a·x with exact rational arithmetic, not rounded decimals.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 transition6: a·x+b with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute transition6: 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 — Recompute transition5: a·x+b with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute transition5: a·x with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Recompute transition4: a·x+b with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute transition4: a·x with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute transition3: a·x+b with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute transition3: a·x with exact rational arithmetic, not rounded decimals.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 transition2: a·x+b with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute transition2: 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 transition1: a·x+b with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute transition1: 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.(8, 9, 17, 0, 8, [0, 1, 2, 3, 4, 5, 6, 7, 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: 0dfdfe94eab51a1a5555beed283bf31b7713f6a9b8c2c4ddd72d0814fc673fffcandidate record SHA-256: bc77b14b23481062fa90061f381a5b07536a1f72d18038c048114787352d38absemantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED