Bookkeeping state: parameters, memory, runtime, and observations

In the motivating account, x denotes a particular system or its reference, not a value shared by all systems. Two instances may use the same architecture and parameters while possessing different runtime histories. Conversely, two fine-tuning runs may begin at the same checkpoint and later acquire different parameters. These are distinct sources of variation.

Use the following bookkeeping state:

St=θt,mt,ct,ot,

where θt contains trainable parameters, mt persistent memory, ct runtime or conversation state, and ot the system’s observations. This tuple is proposed notation for the study, not a recovered formula from the original research. In a frozen model, θt does not change during an ordinary forward pass; in fine-tuning, an optimizer updates trainable components. A reset conversation need not preserve the previous conversation’s mt or ct.

The identity expression x=x is reflexive equality. It does not, without further specification, establish a temporal invariant, an internal act of self-recognition, or a declaration. For example, xt=xt is compatible with xt+1≠xt. The proposed theory can assign a functional role to a declaration, but that role must be specified as an operation or observation rather than inferred from the tautology alone.

2.2 Reference, representation, and declaration

{S}_{t}=\left({\theta }_{t},{m}_{t},{c}_{t},{o}_{t}\right),