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R0002-01726 · q-20261013-advance-20261020-r06-discrete-exponential-growth-0076
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=0/1. 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=0/1. 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.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.
Before
factor q=0≥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.
Rule: fixed_factor_recurrence · Uses: G001
Before
q=0
After
compare q with1: q−1=-1
Conditions: none
Compare exactly with1;the name alone does not imply growth.
Rule: factor_threshold · Uses: G001, S001
Before
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 just0<q.
Rule: qualified_factor_label · Uses: G001, S002
Before
q=0 sends every state to0 after one update
After
q=1 preserves each state; q>1 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.
Rule: factor_not_signed_monotonicity · Uses: G001, S003
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.(0, factor at least 0 and below 1)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 860e9a9f3582025c34336758450e500fc402db57d657e3321c49ce2b4c2b7bbccandidate record SHA-256: 46f3ba8d3e7cc6b0ca688d5469baf02c569ca6bbd479fd5309a35d3cba2e6338semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED