Every equally useful acquired solution [ftip-00N7]
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    Fix a difficulty \(n\) and task-family parameter, the truth contract, fresh evaluation law and deployment cap of Definition [ftip-00MM]. For an allowed frozen artifact \(f\), including the failure artifact \(\bot \), let \(q_n(f)\) be its expected fresh-task score under that evaluation. The expectation includes fresh tasks and deployment randomness; set \(q_n(\bot )=0\). Evaluation begins with exactly the retained state permitted by the specification. Define

    \[\mathcal U_{n,\tau }=\{f\in \mathcal F_n\cup \{\bot \}:q_n(f)\geq \tau \}.\]

    If the comparison randomizes a shared family parameter, apply this definition conditionally on that parameter and then average the resulting scores. This class is defined for mathematical analysis; membership need not be decidable or cheaply recognizable during research. It includes any artifact with the required performance: explicit lemmas, learned libraries, new architectures, implicit weights, and methods that avoid the contributor's representation entirely, whenever those artifacts are allowed.

    For a random final artifact \(F_P\), the acquisition score is \(Q_{\mathrm {acq}}(P)=\mathbb E[q_n(F_P)]\). The desired lower bound must control this expectation for every admitted closed campaign. It cannot merely show that one named representation is unlikely to appear. Nor is a bound on reaching \(\mathcal U_{n,\tau }\) automatically a bound below \(\tau \): many outcomes just below the threshold can still have high mean score. A proposed proof must connect its event or structural quantity to the complete score distribution, as the explicit hypotheses of proposition [ftip-00MP] illustrate.

    The contributor need not transmit an entire solution. A question, analogy or example may change which experiments the recipient attempts. The resulting interpretation, validation and learning remain charged. Acquisition is established only by the frozen recipient's performance on fresh instances; an insightful exchange by itself establishes neither that performance nor a lower bound on unaided discovery.