# Question 2502: One State’s Fixed-Point Condition

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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.

R0004-02502 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0002



## Original question and givens

Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-6, b=-7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).


- G001 — Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-6, b=-7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=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-orbit-fixed-periodic](https://ocme.evemisslab.com/en/mko/mko-orbit-fixed-periodic/)





## Step-by-step derivation

S001 · impliesBefore

n=6; a=-6; b=-7

After

S={0,...,5}; T(z)=(a·z+b) mod n

Conditions: none

Remainders lie in0,...,n−1,so every state has one successor in the same finite set;negative dividends do not produce negative canonical remainders.

Step identifiers and dependenciesRule: finite_self_map · Uses: G001





S002 · calculationBefore

fixed audit product := mul(-6, 1)

After

fixed audit product = -6

Conditions: none

Multiply the current state by the original coefficient a,retaining its sign.

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





S003 · calculationBefore

fixed audit affine value := sum(-6, -7)

After

fixed audit affine value = -13

Conditions: none

Then add the original constant b;this integer is not yet the canonical state.

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





S004 · impliesBefore

Euclidean division of -13 by 6

After

-13=-3·6+5

Conditions: none

Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.

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





S005 · impliesBefore

T(1)=5

After

T(x)=x: No

Conditions: none

A fixed point returns in one step;this is equivalent to least positive period1,not merely a later return.

Step identifiers and dependenciesRule: fixed_point_definition · 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 fixed audit affine value with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute fixed audit product 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

(6, -6, -7, 1, 5, No)

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: 23554f262901efe8d8cb6e24b4d37e22bf070f94815fe96e99c4bc144eb5cbfb
- candidate record SHA-256: a365b0c11bdc1bdcd55bab003426d0b99a7faeaa9dc554c87393ce80db7a1bdc
- semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
- question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r03-orbit-fixed-periodic-0002.json)