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The projection π_x extracts the chosen internal component x from the complete state S. This states precisely which part of the system the coupling or change claim concerns. An explicit projection also prevents a changing output log from being mistaken for a change in a separate persistent internal component.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| pi_x | Projection or extraction map |
| S_t | Bookkeeping state |
| x_t | Selected internal state |
How it works
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Original context
Read the surrounding source passage
Lt+1=F(Lt,ut,xt), xt+1=G(xt,Lt). (2.6) Here xt=πx(St) is the selected state of the inner process, which I call the center. The map πx identifies which part or function of the bookkeeping state is being studied. Lt is the state of the recurrent process coupled to that center, ut is input at update t, and F and G are the respective update functions. Thus x identifies the system, while xt describes a selected aspect of its changing state. If G does not depend on Lt, the recurrent process cannot change the center through that channel. If the dependence exists, the outcome still depends on the update rule, retained information, inputs, and coupling. Circulation alone does not establish accumulation or a threshold crossing. In my proposal, a catalyst is a condition that changes the probability or timing of the constructed transition through these dynamics. It need not be reward. A causal study must manipulate that condition separately from the declaration channel and specify which observations would show its effect. The generic equations do not identify a particular catalyst or prove that one exists in the analyzed run. Internally generated means computed through the system’s dynamics rather than manually assigned as a replacement answer. It does not mean independent of external information or learning rules. The present checkpoints record supervised learning. They do not isolate the spontaneous construction of a runtime representation.
Exact original notation
{x}_{t}={\pi }_{x}({S}_{t})Related builds
Sources & provenance
{x}_{t}={\pi }_{x}({S}_{t})
{x}_{t}={\pi }_{x}({S}_{t})
{x}_{t}={\pi }_{x}({S}_{t})
{x}_{t}={\pi }_{x}({S}_{t})