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
zn+1=azn+1−a, z0=2. The exact solution is zn=1+an. For every integer n≥1, 1<zn<2. The trajectory never returns exactly to the origin two, although it converges to one. A routine can repeatedly output zn=zn at every step. This satisfies the proposed weak numerical pattern of an internally computed change, permanent origin difference, and repeated declarations. The audit instantiated 100 different values of a from 0.05 to 0.95. All 100 satisfy that property analytically. This is not evidence that 100 consciousness events were generated. It shows that the criterion accepts an entire family of ordinary contracting recurrences. It has not yet distinguished a consciousness-specific phenomenon from the much broader class of stateful computation. A one-bit latch is an even smaller illustration: after a trigger it stores one instead of zero and repeatedly reports one. Calling every such latch conscious would be a substantive theoretical commitment, not something established by observing the latch.
Exact original notation
{z}_{n}=1+{a}^{n}.