## 9 Paid inference and reusable organisation ### 9.1 Equal evidence can support unequal bounded achievement A finite rule example makes the epistemic distinction precise. Two reasoners receive $p$, the rules $p\to q$ and $q\to r$, and eight irrelevant implications whose antecedents are absent. Both use a scanning procedure that charges one unit per inspected rule. One ordering places the useful rules first; the other places them after the irrelevant rules. Within a two-inspection budget, the first derives $r$ and the second does not. Their premises and unrestricted deductive consequences agree. Their realised bounded access differs. The improvement belongs to the ordering and procedure under the declared access model. If preprocessing chose that ordering, its computation and storage must be charged. If a recipient obtains the useful ordering from a teacher, the relevant transfer includes that organisational information. If the rules are provided in an indexed structure, the baseline has changed. The example proves a bounded separation, not a lower bound for every possible reasoner. A breadth-first graph search similarly requires a real algorithmic contract. On a finite reachable graph, a queue with duplicate detection that expands each discovered vertex once will eventually expand every reachable vertex, provided each finite expansion terminates and the queue continues to run. Induction on shortest-path distance proves the claim. An unqualified instruction to be fair, without defining scheduling or excluding endless duplicate work, is weaker. Infinite graphs, infinite branching and unbounded expansion costs need separate assumptions. ### 9.2 Compilation, reuse and break-even analysis Let a baseline solve a repeated task at cost $c_0$ per use. Suppose learning a reusable procedure costs $P$, maintaining or storing it over the considered run costs $S$, and each later use costs $c_1\frac{P+S}{c_0-c_1}.$$ This elementary inequality gives a useful research obligation: an observed improvement in test-time latency may merely move cost into preparation. Reuse can make that trade worthwhile, but the number and distribution of future uses matter. If storage or maintenance scales with time, $S$ must be replaced by the appropriate function. If the compiled procedure has different error, the comparison needs an explicit accuracy constraint or loss. Acquiring a useful abstraction and recognizing when to invoke it are separate achievements. Recognition cost, false invocation and selection failures belong in the comparison. A macro is therefore not a free edge. It can reduce search depth by making a previously expensive transformation directly callable, while its execution still incurs the cost of its physical implementation. Treating every learned macro as a unit step is legitimate only in a model whose unit is explicitly a macro call and whose comparison does not silently claim equal physical time. The philosophical claim is retained: incorporation can make a consequence more accessible. Its magnitude depends on the resource model. Lower mean cost does not imply better success at each deadline. For example, an always-correct method that always finishes at cost two has greater mean cost than a method finishing at cost one with probability $0.9$ and cost ten with probability $0.1$. The latter has mean $1.9$, yet by budget two its success is $0.9$ instead of one. A full budget curve can reveal this distinction, but task averaging can still hide opposite specializations. A serious comparison reports the task-resolved curves or justifies the aggregation. ### 9.3 Retained relations as intervention-sensitive organisation Correlation between a learned representation and performance is insufficient for incorporation. A stronger study identifies a physical mediator, manipulates it lawfully, tests persistence after the original interaction ends, and attempts a specific rescue. The retained object may be a register, parameter set, data structure or policy. The intervention must act on its implementation, not on an abstract label while holding all of its determining physical variables fixed. For a derived relation $R=f(A,B)$, an imagined intervention changing $R$ while fixing the complete realising $A,B$ and the deterministic computation can be impossible. An actual separately stored register can be overwritten after computation, because its physical state is no longer logically constrained to agree with the old inputs under every allowed intervention. This difference is central to experimental design. Sham interventions, off-manifold states, resource disruption and direct readout changes must be distinguished from removal of the claimed mediator.