Definition. A resource-bounded model lineage [ftip-00MJ]

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Fix a specification

Here \(I_0\) contains the initial models, corpora, libraries, prompts, optimizers, evaluators, software and cached results. The nonempty class \(\mathcal C_0\) specifies controller programs and constants available from \(I_0\); it does not grant an arbitrary program chosen after a discovery. An initial generator of additional controllers is allowed, with its generation and selection work charged as explained in § [ftip-00N6]. The semantics \(\Gamma \) specifies executable operations and costs; \(\mathcal E_0\) specifies baseline observations and tool-response laws. The checker \(V\), task law and reveal schedule \(\mathcal D\), allowed final artifacts \(\mathcal F\), campaign cap \(\mathbf B\) and deployment cap \(\mathbf b_{\mathrm {eval}}\) are fixed before evaluation. Include the no-update procedure as a baseline.

A campaign is a causal execution starting from this endowment. Its state contains available checkpoints, contexts, generated code, retained traces, libraries, optimizer state and resource usage. A controller may use only its charged accessible state: discarded history must be reconstructed at cost. Environment state may remain hidden. All random seeds follow their declared laws; a fortunate completed trajectory is not an available program.

Whenever admitted by \(\Gamma \), operations include model inference, proof and program search, tool calls, parallel workers, memory management, candidate comparison, reward and credit assignment, synthetic tasks, parameter training, architecture changes, and controller replacement. Generated controllers execute under the same semantics and caps. Selection, evaluation and the controller's own computation are charged. These operations may recur in any order; learning need not first produce a complete successful trace or an explicitly verbalized idea.

The lineage is closed relative to \((I_0,\mathcal E_0)\) when its only campaign inputs are scheduled tasks, its randomness and admitted environment responses. This allows computation to reveal useful structure. A simulator using independent randomness is different from a tool observing new hidden world state; the latter access belongs in \(\mathcal E_0\).

Controller selection is independent of unrevealed instance seeds, conditional on declared public family information. This alone does not exclude family-wide answer tables: all such constants and preprocessing belong in \(I_0\). An asymptotic claim specifies uniform generation across instance sizes or explicitly accounts for nonuniform advice and preparation.