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R0002-01575 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0050
Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=25, b=65. 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=6, updates=6. 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(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=25, b=65. 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=6, updates=6. 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 8}; T(x)=(25x+(65)) 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
x₀=6; transitions=6
After
initial orbit prefix=[6]
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(25, 6)
After
transition1: a·x = 150
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(150, 65)
After
transition1: a·x+b = 215
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
reduce 215 modulo 9
After
215=23·9+8
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S004
Before
transition2: a·x := mul(25, 8)
After
transition2: a·x = 200
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(200, 65)
After
transition2: a·x+b = 265
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
reduce 265 modulo 9
After
265=29·9+4
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S007
Before
transition3: a·x := mul(25, 4)
After
transition3: a·x = 100
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(100, 65)
After
transition3: a·x+b = 165
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
reduce 165 modulo 9
After
165=18·9+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(25, 3)
After
transition4: a·x = 75
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(75, 65)
After
transition4: a·x+b = 140
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
reduce 140 modulo 9
After
140=15·9+5
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S013
Before
transition5: a·x := mul(25, 5)
After
transition5: a·x = 125
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(125, 65)
After
transition5: a·x+b = 190
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
reduce 190 modulo 9
After
190=21·9+1
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S016
Before
transition6: a·x := mul(25, 1)
After
transition6: a·x = 25
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(25, 65)
After
transition6: a·x+b = 90
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
reduce 90 modulo 9
After
90=10·9+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S019
Before
computed transitions=6
After
orbit=[6, 8, 4, 3, 5, 1, 0]; number of entries=7
Conditions: none
Apply the same T to the preceding result; k transitions give k+1 states.
Rule: orbit_prefix_length · Uses: G001, S020
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, S021
S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 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.(9, 25, 65, 6, 6, [6, 8, 4, 3, 5, 1, 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: 23cc4118f038fdc8d53e66f058bc841586354c0f63ce7521dbb4d2e7e07a0ccecandidate record SHA-256: 66aff0e7cf08c7c774797e191425cef78c537583111e12226a92796d092300d6semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED