# Question 1787: One Exact Fixed-Factor Discrete Update

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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.

R0002-01787 · q-20261013-advance-20261020-r06-discrete-exponential-growth-0012



## 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. current=3/2, factor=5/7. Give (current, factor, next=current*factor).


- 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. current=3/2, factor=5/7. Give (current, factor, next=current*factor).





## 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=5/7≥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

current=3/2; q=5/7

After

substitute into one recurrence step

Conditions: none

The current value is given;do not assume it is positive or the initial value.

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





S003 · calculationBefore

next value := mul(3/2, 5/7)

After

next value = 15/14

Conditions: none

Multiply by the same full rational factor.

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





S004 · impliesBefore

X_k=3/2

After

X_(k+1)=15/14

Conditions: none

This single equality is not a global-growth or convergence claim about an unspecified model.

Step identifiers and dependenciesRule: single_step_scope · 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 — Recompute next value 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

(3/2, 5/7, 15/14)

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: 22c1e03c52c7ec84ab8c248944efeedb96d418b107b4824f03a8b49b1d5bb298
- candidate record SHA-256: 5273027424e0be8b835e54180f336120a69c0098db00b842082759f9aabbfc19
- 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-0012.json)