# Question 2367: A Short Exact Orbit of the Specified Logistic Increment Model

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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.

R0003-02367 · q-20261013-advance-20261021-r07-discrete-logistic-growth-0042



## Original question and givens

Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=16/25, rate=1/4, capacity=6/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.


- G001 — Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=16/25, rate=1/4, capacity=6/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.





## 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-logistic-growth](https://ocme.evemisslab.com/en/mko/mko-discrete-logistic-growth/)





## Step-by-step derivation

S001 · impliesBefore

0≤r=1/4≤1; K=6>0; 0≤x=16/25≤K

After

T(x)=x+r·x·(1−x/K)

Conditions: none

This fixed model adds the increment to the old state;it is not the different direct-product map r·x·(1−x/K).

Step identifiers and dependenciesRule: logistic_increment_model · Uses: G001





S002 · impliesBefore

initial=16/25; steps=0

After

retain the initial state before updating

Conditions: none

A zero-transition prefix still contains its initial state.

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





S003 · impliesBefore

orbit=[16/25]

After

entries=1=steps+1

Conditions: none

This is an exact finite orbit,not a stability proof,empirical prediction or model-fit evidence.

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









## Reverse checks


- 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

(16/25, 1/4, 6, 0, [16/25])

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: 0e827e64de01638bcbf547645e9c365a15fcc2faa172784a30d1c41bc418643f
- candidate record SHA-256: ce3dffdd4f8c33644d9d61766d57e77859513d59a16855c29e7a8851b658b0ab
- semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f
- question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r07-discrete-logistic-growth-0042.json)