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R0002-01558 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0033
Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=8, b=14. 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=7. 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(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=8, b=14. 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=7. 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 6}; T(x)=(8x+(14)) mod 7After
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=7
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(8, 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, 14)
After
transition1: a·x+b = 14
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
reduce 14 modulo 7
After
14=2·7+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S004
Before
transition2: a·x := mul(8, 0)
After
transition2: 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, S005
Before
transition2: a·x+b := sum(0, 14)
After
transition2: a·x+b = 14
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
reduce 14 modulo 7
After
14=2·7+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S007
Before
transition3: a·x := mul(8, 0)
After
transition3: 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, S008
Before
transition3: a·x+b := sum(0, 14)
After
transition3: a·x+b = 14
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
reduce 14 modulo 7
After
14=2·7+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S010
Before
transition4: a·x := mul(8, 0)
After
transition4: 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, S011
Before
transition4: a·x+b := sum(0, 14)
After
transition4: a·x+b = 14
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
reduce 14 modulo 7
After
14=2·7+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S013
Before
transition5: a·x := mul(8, 0)
After
transition5: 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, S014
Before
transition5: a·x+b := sum(0, 14)
After
transition5: a·x+b = 14
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
reduce 14 modulo 7
After
14=2·7+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S016
Before
transition6: a·x := mul(8, 0)
After
transition6: 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, S017
Before
transition6: a·x+b := sum(0, 14)
After
transition6: a·x+b = 14
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
reduce 14 modulo 7
After
14=2·7+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S019
Before
transition7: a·x := mul(8, 0)
After
transition7: 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, S020
Before
transition7: a·x+b := sum(0, 14)
After
transition7: a·x+b = 14
Conditions: none
Add the same offset before reducing modulo n.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
reduce 14 modulo 7
After
14=2·7+0
Conditions: none
The integer identity together with0≤remainder<n determines the next state.
Rule: euclidean_remainder · Uses: G001, S022
Before
computed transitions=7
After
orbit=[0, 0, 0, 0, 0, 0, 0, 0]; number of entries=8
Conditions: none
Apply the same T to the preceding result; k transitions give k+1 states.
Rule: orbit_prefix_length · Uses: G001, S023
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, S024
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 — 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.(7, 8, 14, 0, 7, [0, 0, 0, 0, 0, 0, 0, 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: 5ca6c765817a677003fa5314dbc714fca99d691963c45dd145009e81d7af9428candidate record SHA-256: 786a6310cee06ac89356edc6ee7a91875d864c3d7a6aaca4c1b2316be41756d4semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED