# Question 2398: One Exact Capacity-Limited Discrete Increment

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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.

R0003-02398 · q-20261013-advance-20261021-r07-discrete-logistic-growth-0023



## 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. current=22/5, rate=1/2, capacity=7/1. Give (current,rate,capacity,next).


- 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. current=22/5, rate=1/2, capacity=7/1. Give (current,rate,capacity,next).





## 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=1/2≤1; K=7>0; 0≤x=22/5≤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 · calculationBefore

x/K at step1 := div(22/5, 7)

After

x/K at step1 = 22/35

Conditions: none

Positive K permits normalization of the state.

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





S003 · calculationBefore

1−x/K at step1 := sub(1, 22/35)

After

1−x/K at step1 = 13/35

Conditions: none

This feedback factor is nonnegative in the declared interval.

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





S004 · calculationBefore

x(1−x/K) at step1 := mul(22/5, 13/35)

After

x(1−x/K) at step1 = 286/175

Conditions: none

Multiply the current state by the remaining proportion first.

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





S005 · calculationBefore

increment at step1 := mul(1/2, 286/175)

After

increment at step1 = 143/175

Conditions: none

Then multiply by the increment rate r.

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





S006 · calculationBefore

next state at step1 := sum(22/5, 143/175)

After

next state at step1 = 913/175

Conditions: none

Add the original x back;the addition must not be omitted.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

T(x)−x=r·x·(1−x/K)≥0; K−T(x)=(K−x)(1−r·x/K)≥0

After

0≤913/175≤7

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, S006









## Reverse checks


- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Recompute next state at step1 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute increment at step1 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute x(1−x/K) at step1 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute 1−x/K at step1 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute x/K at step1 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

(22/5, 1/2, 7, 913/175)

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: 4e00e181416e9ccf4e5d55b7697d6a2d1418ff88d7b0b973d54a1ca9130268fc
- candidate record SHA-256: 45dc0ffdca41df78e725b617cc297c800a2ea8b62a5014405190f2aaea316e29
- semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f
- question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa
- Lean: NOT_RUN · human review: NOT_PERFORMED

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