17. Why finite agreement is not unrestricted discovery
On this page · Permalink · Source line 664
Proposition 17: Finite-prefix indistinguishability
For every integer \(T\geq0\), there are two finite-dimensional autonomous rational linear systems with identical scalar observations at times \(0,\ldots,T\) and different observations at time \(T+1\). Both transition matrices can be chosen nilpotent.
Read proof · Recorded Lean scope · Visual guide: What four observations identify · Permalink · Source line 666
Proof
One system has scalar output identically zero. For the other, take dimension \(T+2\), initial state \(x_0=e_{T+2}\), observation row \(C=e_1^\top\), and shift \(Le_j=e_{j-1}\) for \(j\geq2\), \(Le_1=0\). Then \(CL^tx_0=0\) for \(0\leq t\leq T\) and \(CL^{T+1}x_0=1\). The shift is rational and nilpotent. \(\square\)
An invisible delayed component of this kind can also be adjoined to any fixed base model. The combined output agrees with the base output on the entire inspected prefix and then differs. This is not an argument against learning. It identifies why a claimed guarantee must state a dimension bound, a model class, a coverage condition, an intervention protocol or a probabilistic assumption.
Even nilpotent operators have only zero eigenvalues, so stability language alone does not remove this ambiguity. A finite rank plateau cannot certify the absence of a delayed mode when arbitrarily large hidden dimensions are admitted.