# Question 2370: A Short Exact Orbit of the Specified Logistic Increment Model [Return to catalogue page 12](https://ocme.evemisslab.com/en/questions/page/12/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations/#q-20261013-advance-20261021-r07-discrete-logistic-growth-0045) 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. R0003-02370 · q-20261013-advance-20261021-r07-discrete-logistic-growth-0045 ## 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=19/25, rate=1/1, capacity=9/1, updates=1. 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=19/25, rate=1/1, capacity=9/1, updates=1. 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≤1; K=9>0; 0≤x=19/25≤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=19/25; steps=1 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 · calculationBefore x/K at step1 := div(19/25, 9) After x/K at step1 = 19/225 Conditions: none Positive K permits normalization of the state. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore 1−x/K at step1 := sub(1, 19/225) After 1−x/K at step1 = 206/225 Conditions: none This feedback factor is nonnegative in the declared interval. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore x(1−x/K) at step1 := mul(19/25, 206/225) After x(1−x/K) at step1 = 3914/5625 Conditions: none Multiply the current state by the remaining proportion first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore increment at step1 := mul(1, 3914/5625) After increment at step1 = 3914/5625 Conditions: none Then multiply by the increment rate r. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore next state at step1 := sum(19/25, 3914/5625) After next state at step1 = 8189/5625 Conditions: none Add the original x back;the addition must not be omitted. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore T(x)−x=r·x·(1−x/K)≥0; K−T(x)=(K−x)(1−r·x/K)≥0 After 0≤8189/5625≤9 Conditions: none The factor inequalities follow from0≤r≤1 and0≤x≤K,proving interval preservation for this model rather than guessing from samples. Step identifiers and dependenciesRule: qualified_interval_preservation · Uses: G001, S007 S009 · impliesBefore orbit=[19/25, 8189/5625] After entries=2=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, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute next state at step1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute increment at step1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute x(1−x/K) at step1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute 1−x/K at step1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute x/K at step1 with exact rational arithmetic, not rounded decimals. - 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 (19/25, 1, 9, 1, [19/25, 8189/5625]) 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: cbe315829f6ebd7d7a8dc8d0ac9a64383788a61ae65af3ae372beda03d34ab12 - candidate record SHA-256: 3232b126f48d21cd13412a3d18a9d4b0509a3502166ebefe700c2a0fdee3a7a1 - 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-0045.json)