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Theorem · impossibility

FPRD-LRN-T02

Pointwise stabilization does not support universal learning

Exact statement

No learner identifies every rational target in the limit from arbitrary polynomial snapshots PtP_t whose coefficients merely stabilize pointwise; the impossibility already holds using the zero series and single monomials qkq^k.

StatusProved by a diagonal presentation
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Pointwise convergence gives each coefficient a last-change time but does not reveal any finite trustworthy frontier to the learner.

Hypotheses and scope

  • The learner must succeed on every presentation satisfying pointwise coefficient stabilization.
  • The target is promised rational.

Proof or evidence

Choose fresh increasing exponents. Repeat each monomial snapshot long enough that a supposedly successful learner must adopt it, then move to the next exponent. Every fixed coefficient is nonzero in at most one block, so the stream converges coefficientwise to zero while the learner is forced away from zero infinitely often.

Diagonal impossibility theorem →

Verification notes

The construction uses only rational polynomial snapshots and rational targets. The constant-tail extension invoked at each stage is a valid presentation of the current monomial target.

Limitations

  • The theorem does not apply when the data carry a computable finality frontier or another effective validation mechanism.
  • It concerns exact identification, not heuristic prediction or statistical approximation.

Open work

Test the intended approximation semantics for a computable stabilization modulus, finality markers, monotone bounds, or active validation.