Closed form of the leaky accumulator

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.

Finally, xt=xt does not imply xt+1=xt, nor does xt+1≠xt imply a false representation of the present. These distinctions keep the motivating conceptual vocabulary compatible with ordinary mathematical types and temporal indexing.
[Context excerpt; full source linked.]

{A}_{n}=q\left(1-{\lambda }^{n}\right)/\left(1-\lambda \right)