# Question 1738: Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity [Return to catalogue page 9](https://ocme.evemisslab.com/en/questions/page/9/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations/#q-20261013-advance-20261020-r06-discrete-exponential-growth-0088) 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. R0002-01738 · q-20261013-advance-20261020-r06-discrete-exponential-growth-0088 ## Original question and givens Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=3/6. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1. - G001 — Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=3/6. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1. ## 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-exponential-growth](https://ocme.evemisslab.com/en/mko/mko-discrete-exponential-growth/) ## Step-by-step derivation S001 · impliesBefore factor q=1/2≥0 After X_(k+1)=q·X_k Conditions: none Use the same factor in the declared discrete model,rather than infer a general law from a few values. Step identifiers and dependenciesRule: fixed_factor_recurrence · Uses: G001 S002 · impliesBefore q=1/2 After compare q with1: q−1=-1/2 Conditions: none Compare exactly with1;the name alone does not imply growth. Step identifiers and dependenciesRule: factor_threshold · Uses: G001, S001 S003 · impliesBefore qualified interval: 0≤q<1 After preserved source label=factor between 0 and 1 Conditions: none Retain the source label while making explicit that this contract's low-factor branch includes q=0,not just01 need not increase a negative initial value Conditions: none Factor classification is not a claim that every initial value increases;distinguish zero or negative initial values and a zero factor. Step identifiers and dependenciesRule: factor_not_signed_monotonicity · Uses: G001, S003 ## Reverse checks - 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 — 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 (1/2, factor at least 0 and below 1) 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: 73eba36b28572f230334b392ed2339abbdb1b681835dbe082f511ec10781f8eb - candidate record SHA-256: 2ce18cba45c32386832ed6f0bbaea813be114532e235fef9e27c4ad6107cb4a0 - semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861 - question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261020-r06-discrete-exponential-growth-0088.json)