# Question 1707: Normalize a Candidate before a Fixed-Factor Closed-Form Audit [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-0057) 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. R0002-01707 · q-20261013-advance-20261020-r06-discrete-exponential-growth-0057 ## 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. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-1/1, factor=3/2, n=1, candidate=-2/2. Give (initial, factor, n, exact term, normalized candidate, equal?). - 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. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-1/1, factor=3/2, n=1, candidate=-2/2. Give (initial, factor, n, exact term, normalized candidate, equal?). ## 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=3/2≥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 X₀=-1; n=1 After n=0 uses the initial value; an empty product equals1 Conditions: none n counts updates;n=0 performs none.No division by q is required,so q=0 remains admissible. Step identifiers and dependenciesRule: initial_and_empty_product · Uses: G001, S001 S003 · calculationBefore X_1 := mul(-1, 3/2) After X_1 = -3/2 Conditions: none Apply the same recurrence stepwise and retain each intermediate value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore X_k=X₀q^k ⇒ X_(k+1)=qX_k=X₀q^(k+1) After X_n=X₀q^n for every nonnegative integer n Conditions: none The initial case holds and multiplication by q gives the induction step;this is not a guess from a finite table. Step identifiers and dependenciesRule: recurrence_induction · Uses: G001, S003 S005 · impliesBefore (3/2)^1 After q^n = 3/2 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S004 S006 · calculationBefore closed-form X_n := mul(-1, 3/2) After closed-form X_n = -3/2 Conditions: none Substitute into the power formula as a separate calculation;it must equal the stepwise update. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore computed X_1=-3/2; candidate=-1 After candidate correct at this n: No Conditions: none Check the candidate value at this index;one agreement would not prove an arbitrary candidate formula globally. Step identifiers and dependenciesRule: candidate_at_index · 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 closed-form X_n with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute X_1 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 (-1, 3/2, 1, -3/2, -1, No) 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: 6301f8675a4a4890e9c3ea3f2daaba3f09844ea77a69407e83a4f7c0b849a471 - candidate record SHA-256: d27082d82f8bce2862f3ec0fed5f43f9acbb08af88ffb673ad56ee103fc4e778 - 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-0057.json)