ora-0048
3.4 The finite-transcript obstruction
Suppose a finite transcript \(T\) admits models \(\mathcal C,\mathcal D\in \mathsf H(T)\) with \(\mathcal C\not\simeq _{\mathcal Q}\mathcal D\). No learner using only \(T\) can be guaranteed to answer every query in \(\mathcal Q\) correctly.
Choose a query on which the two models disagree. The learner receives the same transcript whether the target is \(\mathcal C\) or \(\mathcal D\), and therefore returns the same answer in both cases. That answer is wrong for at least one admissible target.
The proposition is elementary but fixes the burden of every positive ORACLE theorem. A presentation must eventually separate competing models. Positive texts often fail because one can add a hidden object, arrow, equation, or universal witness without contradicting any finite positive evidence.
The prior changes the obstruction but does not remove it. It can exclude an extension that violates solidity, approximate magnitude, viewpoint coherence, goal-directedness, social indexing, or linguistic admissibility. But whenever two core-preserving extensions survive the same transcript and disagree on an admissible query, the argument applies unchanged.
Fix a core doctrine \(\mathbb T_{\mathrm{core}}\). If a finite core-typed transcript has two models in \(\mathbf{Cat}_{\mathcal U}^{\mathrm{core}}\) that agree on all exposed fragment and overlap data but differ on \(\mathcal Q_{\mathrm{core}}\), then the doctrine and transcript do not yet identify the target.
Thus core knowledge earns its keep by shrinking the fiber \(\mathsf H(T)\), not by turning finite observation into omniscience. Its empirical and mathematical value is measured by which otherwise indistinguishable extensions it rules out.