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Counterexample · failed abstraction

FPRD-CPA-C08

Injective observation does not imply polynomial recovery

Exact statement

Injectivity of a polynomial observation map does not imply polynomial recovery of the hidden coordinate: the map t ↦ (t²,t³) is injective, but t does not belong to Q[t²,t³].

StatusProved by the cusp coordinate ring
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

A finite family of observations may distinguish hidden states without generating their coordinate functions polynomially.

Proof or evidence

The map is injective because its second and third powers determine the parameter, while the semigroup of exponents generated by 2 and 3 does not contain 1.

Polynomial-recovery boundary →

Verification notes

The coordinate-ring calculation was checked directly.

Limitations

  • The example rules out a general inference from injectivity; it does not preclude recovery under stronger hypotheses.

Open work

Require explicit polynomial recovery identities rather than finite separation alone.