# Question 1569: A Fixed-Length Discrete Orbit Prefix

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0002-01569 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0044



## Original question and givens

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].





## Definition and scope

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.


- [mko-discrete-dynamical-system](https://ocme.evemisslab.com/en/mko/mko-discrete-dynamical-system/)





## Step-by-step derivation

S001 · impliesBefore

X={0 through 7}; T(x)=(19x+(47)) mod 8

After

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.

Step identifiers and dependenciesRule: finite_total_map · Uses: G001





S002 · impliesBefore

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.

Step identifiers and dependenciesRule: orbit_initial_state · Uses: G001, S001





S003 · impliesBefore

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.

Step identifiers and dependenciesRule: orbit_prefix_length · Uses: G001, S002





S004 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_prefix_scope · Uses: G001, S003









## Reverse checks


- 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.





## Result

(8, 19, 47, 2, 0, [2])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 7d157e2658dc1b43e8ee4b09c4d78fe3b9048b7fff0ef58be54fb443e33ec947
- candidate record SHA-256: 92533cb8ee0c179d749ca760a15e95169e85ffecd4bcbf2f672b0dc415701ece
- semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861
- question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261020-r01-discrete-dynamical-system-0044.json)