Context
Finite observation is not enough by itself. The certificate requires explicit polynomial recovery of each hidden control from output derivatives.
Definitions
- D(f) is the Hadamard algebra generated by all left derivatives of f.
- A recovery certificate writes each control series as a polynomial in finitely many selected derivatives of f.
Hypotheses and scope
- The output series f is commutative.
- For every control, an explicit polynomial recovery identity is supplied.
Proof or evidence
Derivatives of a commutative series remain commutative, and rational linear combinations and Hadamard products preserve commutativity. Proposed identities can be checked by the known equivalence procedure.
Verification notes
The closure argument was rechecked. The site separately records the external equality-decidability dependency rather than treating it as an internally reproved theorem.
Limitations
- The criterion is sufficient, not necessary.
- Injective observation maps do not imply polynomial recovery; the cusp map is a guardrail.