Definition. Finite controller class with hard resource admission [ftip-00M5]

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Fix a horizon \(H\in \mathbb N\), finite nonempty observed-history sets \(\mathcal H_t\) for \(0\leq t\leq H\), an initial law \(\mu \) on \(\mathcal H_0\), and finite nonempty permitted-action sets \(A_t(h)\) for \(t<H\). An environment kernel \(K_t(\cdot \mid h,a)\) is a probability law on histories extending \(h\) by action \(a\) and its observed outcome. The initial law and all kernels are fixed before choosing a controller.

A controller \(\pi \) assigns a probability law \(\pi _t(\cdot \mid h)\) on \(A_t(h)\). It may depend on every observed component of \(h\), including retained traces, generated programs, and previous updates. Random choices may be recorded in the history. The class \(\Pi (\mathbf B)\) consists of all such controllers under the declared hard resource vector \(\mathbf B\in \mathbb R_+^m\). It does not require a fixed prompt, a memoryless policy, or independent worker outputs. Full-history access is allowed in this mathematical class; it may enlarge the class of executable controllers with limited memory or computation.

Each history records accumulated nonnegative cost \(\mathbf c(h)\preceq \mathbf B\). Every permitted non-stop action has a declared worst-case increment \(\bar {\mathbf c}_t(h,a)\) satisfying

Every successor in the kernel's support must have realized increment between zero and this declared bound, componentwise. Proposal work, controller computation, worker calls, communication, verification, failed attempts, and persistent updates must be included in whichever resource coordinates are bounded; the units remain those of Definition [ftip-005H]. This is the enforced-contract assumption of Theorem [ftip-00CK].

A stop action is always permitted: it commits the current terminal artifact and pads the remaining steps with no further cost or change in its evaluated quality. At the horizon a fixed evaluator assigns \(q(h_H)\in [0,1]\) to the committed artifact recorded in the terminal history. If the evaluator has random outcomes, include them in the history law. Evaluation work must be charged before the zero-cost padding begins. Define

The artifact identity, evaluator, observations, kernels, and action sets are part of the mathematical problem. A training procedure or new tool belongs to this frontier only if it is among the permitted actions and its effects and costs are represented. Finiteness is an explicit restriction on histories, representations, and horizon; a theorem for this class does not bound an unrestricted agent that can extend them.

For an executable system, an upper bound applies only after its observations, actions, outcomes, and charged costs are represented by this model. Conversely, a mathematical policy supplies an executable lower bound only when it has an implementation respecting the stated resource cap; an arbitrary history-to-action table does not establish that fact.