# Question 1593: A Typed Finite State and Its Canonical Residue Update

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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.

R0002-01593 · q-20261013-advance-20261020-r01-discrete-dynamical-system-0018



## Original question and givens

Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=10, b=23. 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. x=7. Give (modulus, a, b, x, T(x)).


- G001 — Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=10, b=23. 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. x=7. Give (modulus, a, b, x, T(x)).





## 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 9}; T(x)=(10x+(23)) mod 10

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 · calculationBefore

one step: a·x := mul(10, 7)

After

one step: a·x = 70

Conditions: none

Multiply the original coefficient by the current state;do not clip negative values to0.

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





S003 · calculationBefore

one step: a·x+b := sum(70, 23)

After

one step: a·x+b = 93

Conditions: none

Add the same offset before reducing modulo n.

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





S004 · impliesBefore

reduce 93 modulo 10

After

93=9·10+3

Conditions: none

The integer identity together with0≤remainder<n determines the next state.

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





S005 · impliesBefore

x=7

After

T(x)=3∈X

Conditions: none

This computes one update,not a whole orbit or a fixed-point classification.

Step identifiers and dependenciesRule: state_membership · Uses: G001, S004









## Reverse checks


- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute one step: a·x+b with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute one step: a·x with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(10, 10, 23, 7, 3)

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: 5c77f26c2197afbbcae99b22005f7c7547dc93b23d6427d0908d6445ce019476
- candidate record SHA-256: 9bda62537d2e18e3711ba5bf9faa2b5b9123c3b1c227fe60d34c07b7ff70859d
- 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-0018.json)