Question 2329: A Fixed Candidate, Including the Zero-Rate Degeneracy

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. candidate=0/1, rate=1/1, capacity=8/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.

• 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. candidate=0/1, rate=1/1, capacity=8/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.

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

Step-by-step derivation

0≤r=1≤1; K=8>0; 0≤x=0≤K

T(x)=x+r·x·(1−x/K)

x/K at step1 := div(0, 8)

x/K at step1 = 0

1−x/K at step1 := sub(1, 0)

1−x/K at step1 = 1

x(1−x/K) at step1 := mul(0, 1)

x(1−x/K) at step1 = 0

increment at step1 := mul(1, 0)

increment at step1 = 0

next state at step1 := sum(0, 0)

next state at step1 = 0

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

0≤0≤8

T(candidate)=0; candidate=0

fixed=Yes

T(x)−x=r·x·(1−x/K)

r=0:every allowed x is fixed; r>0:fixed points are0 andK

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 — 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

(0, 1, 8, 0, Yes)

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.