9.1 Claim discipline and the theorem ladder

The program separates four layers that must not be collapsed. A construction is categorical when its types and compositions are explicit; universal when it is characterized by a universal property; decision-theoretic when it selects or compares admissible actions; and causal only when an intervention semantics and identification assumptions justify that interpretation. Thus a natural isomorphism need not imply low regret, and a nonzero tangent response need not identify the cause of a failure.

The remainder of the book follows six theorem families:

1. representation: progressive evidence determines a persistent online UDL object;

2. invariance: restriction, localization, and presentation change preserve the declared universal constructions;

3. comparison: differently informed branches admit a typed observer and comparator object;

4. repair: strict, homotopy-coherent, or infinitesimal defects select admissible repairs;

5. stability: tangent lifting preserves well-posedness and, under additional hypotheses, convergence; and

6. transport: learned structure survives a change of task, information pattern, or environment.