9.4 From infinitesimal evidence to a finite jump

Weil probes diagnose the local behavior of the current representation. They may reveal a null direction that should be visible, a mixed response absent from the declared composition law, a bracket that fails to close, or a comparison map that fails to be invertible. Such signals are valuable because they localize where the present geometry is inadequate.

They do not uniquely determine a replacement theory. Many global extensions can share the same local profile, and a finite conceptual jump need not lie on an integrable path in the old geometry. Frontier models, analogical retrieval, program synthesis, or human investigators must therefore propose candidate extensions. Differential diagnosis constrains this proposal process without pretending to derive the new ontology automatically.

Suppose the old and proposed differential worlds are

Let \(K:\mathcal M\to \mathcal M^+\) be the semantic transport associated with the proposed theory change. A candidate differential theory extension includes comparison maps

The map \(\chi _A\) asks whether an old state probed and then transported agrees with that state transported and then probed in the new theory. Invertibility indicates strong preservation for probe \(A\); a controlled failure may reveal new directions introduced by the extension.

. Infinitesimal evidence constrains a creative jump but does not cause or certify it. Local defects underdetermine global ontology, and a proposed extension must be compared on finite models, interventions, or proofs.