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.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
Read the surrounding source passage
6.2. Persistence is not the same as irreversibility The saved effective updates remain nonzero after checkpoint 1. That establishes persistence on the observed grid. It does not determine behavior at unsaved checkpoints or under an intervention that was not performed. Permanent difference from an initialization is also mathematically broad. Consider yn+1=1+a(yn−1) with y0=2 and 0<a<1. Its exact solution is yn=1+an, which differs from two at every finite n≥1. A routine could repeatedly report that value. This example does not prove the recurrence lacks experience. It shows the breadth of a criterion based only on displacement from an origin. A theory accepting that whole class must state that consequence rather than present the inequality as independent evidence for consciousness. Internal consistency is a different relation again. An estimate and a prediction can agree while both differ from a measured target. Repeated coherent declarations therefore do not establish reference accuracy. Correcting an error also does not, by itself, remove cognition. The hypothesis must specify whether its endpoint concerns construction, accuracy, persistence, or some combination, and retain that specification when the result is inconvenient. 6.3. Relation to existing machine evidence [Context excerpt; full source linked.]
Exact original notation
{y}_{n}=1+{a}^{n}