9.7 Transformations across the testbeds

Mathematical theory construction.

A definitional extension can reorganize reasoning without changing model content. A stronger transformation may add an axiom, object, equivalence, or doctrine that organizes previously disconnected results and changes the admissible models or constructions. Proof assistants can check derivations; model finders can search bounded classes for realizations or countermodels; relative-consistency arguments require explicit assumptions. None of these alone certifies that the new concept is fruitful enough to found a field.

Simulator-grounded discovery.

A transformation may introduce an unmodeled mechanism, latent state, coupling, or scale. The simulator can execute the proposed extension and generate interventional predictions. Because the simulator is itself a theory, successful independent prediction can first establish only adequacy relative to that synthetic world.

Scientific corpus to theory.

A transformation may turn dispersed textual claims into a new mechanistic object or causal relation. The risk of hallucinated ontology is highest here. Every new object must retain provenance, competing interpretations, and an external route to admission.

Generative visual languages.

A transformation may add a visual relation, semantic control, observer, or reusable operation on the generation process. Repairing an image against a supplied scene theory remains an assurance task. A stronger claim requires showing that the old visual language cannot express the needed operation and that the proposed extension transports across prompts, subjects, and renderers while preserving previously admitted controls.

Experiment: Withheld-language theory repair. Epistemic status: proposed benchmark design, not a completed result.
Input: observations generated by a hidden theory and an initial sketch that deliberately omits a required primitive, relation, or observer.
Diagnosis: a localized obstruction plus a benchmark construction or formal argument certifying that the target is not realizable in the selected initial language; stronger-search failures are reported separately.
Proposal: several minimal sketch extensions with interpretations and predicted consequences.
Transport: restriction fibers, preserved results, conceptual losses, and probe comparison maps \(\chi _A\).
Admission: held-out interventions or proofs that distinguish the extension from both the old theory and plausible alternatives.
Safety condition: abstain when the evidence does not discriminate among extensions.