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R0002-01569 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0044
Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=19, b=47. 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=2, updates=0. 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=19, b=47. 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=2, updates=0. 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)=(19x+(47)) 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=2; transitions=0
After
initial orbit prefix=[2]
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
computed transitions=0
After
orbit=[2]; number of entries=1
Conditions: none
Apply the same T to the preceding result; k transitions give k+1 states.
Rule: orbit_prefix_length · Uses: G001, S002
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, S003
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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, 19, 47, 2, 0, [2])
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: 7d157e2658dc1b43e8ee4b09c4d78fe3b9048b7fff0ef58be54fb443e33ec947candidate record SHA-256: 92533cb8ee0c179d749ca760a15e95169e85ffecd4bcbf2f672b0dc415701ecesemantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED