11. What a typed operator language must guarantee
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A program has a source type, target type, admissible domain, evaluation rule and declared approximation status. A nonlinear partial operation is not a total matrix merely because its inputs are stored in a vector. The domain of a composite is
Composition must preserve those types and domains. Coordinate changes must preserve the represented operation as in Proposition 2. Packet refinement must preserve exact transfer semantics as in Proposition 4, or carry an explicit approximation residual. Nonlinear interactions must retain their cross-packet terms as in Proposition 5.
A grammar can be closed under a specified set of constructors without containing every possible physical law. It can also be expressive while computationally intractable. Universal discovery, practical search and semantic closure are distinct claims.
For a known operator, Proposition 8 can build its minimal linear summary. For an unknown operator, learning must address identifiability: which different candidates make identical predictions under the available observations and actions? More coordinate systems alone do not distinguish candidates that agree on the entire observed signal. Additional interventions or data may be needed.
The foundation should therefore treat a learned hypothesis as an operator with an evidence record, not just a matrix with a high normalized weight. Its record includes fitting assumptions, admissible inputs, compared alternatives, unresolved equivalences and the representation in which its residual is measured.