# Discrete exponential growth model

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

With a fixed time step and fixed nonnegative factor r, the discrete recurrence x_(n+1)=r*x_n gives x_n=x_0*r^n.

mko-discrete-exponential-growth · definition · v0.10.0



## Mathematical statement

Fix a time step and a nonnegative constant r. The discrete exponential model is x_(n+1)=r*x_n. From initial x_0, repeated substitution or induction gives x_n=x_0*r^n. This is an algebraic result within the specified model, not proof that observed data fits it.





## Representative formula (exact TeX)

x(n+1)=M(r,x(n))





## Assumptions


- Time steps are fixed, n starts at 0, and r remains constant throughout the modeled period.
- Exercises use exact rational initial values and nonnegative rational factors, not a continuous-time model or empirical biological population forecast.





## Symbol scope


- x — modeled quantity
- n — time-step index
- r — fixed per-step factor





## Intuition, definition, and misconceptions

A factor r>1 scales each step by more than 1, r=1 leaves the factor neutral, and 0≤r<1 scales down. A zero initial value remains zero.

The recurrence and closed form can be compared step by step and proved equivalent by natural-number induction.

Real biological systems can involve limited resources, randomness, time variation and measurement error. A fixed-r model alone establishes no empirical fit.

Do not infer an eternally fixed factor from a few growing periods or identify a discrete multiplier directly with a continuous growth rate.





## Adopted derivation guide

[Discrete exponential growth: proving x_n=x_0·r^n from x_(n+1)=r·x_n by induction](https://ocme.evemisslab.com/en/derivations/derivation-mko-discrete-exponential-growth-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — Discrete states are ordered by natural-number time.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — The closed form uses natural powers of r.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Each recurrence step multiplies by the same factor.





## Verification boundary

This slice checks only the exact fixed-step, fixed-nonnegative-factor model; it is not an empirical prediction.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Develop logistic models, time-varying factors, calibration and residuals.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/discrete-exponential-growth.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-discrete-exponential-growth/index.md)