Projection / capability. Propagate a state from a local rule and distinguish finite-step approximation from continuous flow.
Question this mathematics answers. Given the local rate/rule now, what state does the model predict later?
Equation family
Finite update: xₘ₊₁=(1+λΔt)xₘ, Δt=T/mContinuous flow: x(T)=e^(λT)x(0)Vector flow: ẋ=Ax ⇒ x(T)=e^(AT)x(0)Markov: pₙ₊₁=pₙP or ṗ=pQ
Symbols. x = state; λ = scalar local rate; T = total time; m = number of numerical/finite update steps; A = matrix generator; p = probability-state vector; P = discrete transition matrix; Q = continuous-time rate generator.
Assumptions. The exponential solutions shown require time-independent linear generators. Markov P must be stochastic; Q must preserve total probability.
Where the inputs come from. Rates/generators come from a physical model, system identification, calibration, or a deliberately chosen toy model. m is numerical resolution—not a physical rate.
Short derivation / justification
Limit: Apply the finite multiplier m times: x(T)=x(0)(1+λT/m)^m. As m→∞ this tends to e^(λT)x(0). Matrix generalization: replace λ by A and define e^(AT) by its power series.
Worked example
λ=−0.28 min⁻¹, T=4 min. Continuous factor e^(−1.12)=0.3263. With m=4, (1−1.12/4)^4≈0.2687; with m=32, ≈0.3202. Same modeled process, finer step.
Cadet or real-world application. Attitude-error decay, drug/chemical kinetics, reliability-state transitions, coupled thermal states.
What to look for. Track Δt and error as m changes. In coupled systems, off-diagonal A entries encode how one state component drives another.
Common misinterpretation. “Increasing m changes the physics.”
Break / domain boundary. A good numerical approximation is not a proof the model is correct; repeated appearance of e does not merge physical domains.