Read the equation
The solution is one plus aⁿ because the initial error is one and each step multiplies it by a. For 0<a<1 it approaches one but never equals one at a finite integer n in exact arithmetic. For a fixed positive tolerance ε, the discrepancy eventually falls below that tolerance, showing why bare non-equality and thresholded persistence differ.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| a | Contraction factor |
| n | Step index |
| y_n | State; zₙ in another version |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
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Original context
Read the surrounding source passage
Appendix C. Why the bare divergence condition needs specificity Consider the scalar recurrence zn+1=azn+1−a with z0=2 and 0<a<1. Its solution is zn=1+an. It changes internally, differs from its original value at every subsequent finite step, and never returns to 2. A program could repeatedly report its current value without changing this algebra. Thus permanent difference from an initialization is mathematically broad, not peculiar to the present model. This example does not prove the recurrence lacks experience. It establishes what the proposed criterion accepts. A theory identifying all such systems with primitive cognition may adopt that consequence, but should derive additional observable constraints rather than presenting the same inequality as independent evidence for its interpretation. Similarly, merely placing a loop near a node is not a specified causal mechanism. If an accumulation law is An+1=λAn+q with 0≤λ<1, q>0, and A0=0, then An=q1−λn/1−λ. A threshold strictly above q/1−λ is never crossed despite infinitely many updates. With no leakage and fixed positive increments, a finite threshold is crossed. These are different models. A general “enough speed or enough circulation” law requires assumptions about coupling, cancellation, and dissipation. [Context excerpt; full source linked.]
Exact original notation
{z}_{n}=1+{a}^{n}