2631 — Rosenbrock Zeros Do Not Commute with Completion
For a finite realization, transfer zeros, invariant zeros, unobservable modes, uncontrollable modes, and zero dynamics are distinct typed objects. Transfer and invariant zeros coincide off the state spectrum for a minimal SISO realization, but scalar transfer data do not reconstruct hidden states.
An exact rational family is controllable and observable at every cutoff while both minimality determinants scale as and the weak-mode energy scales as . Its transfers converge pointwise, but the limit acquires a hidden mode. Separate strictly proper families show a pointwise nonzero value becoming zero and a persistent zero disappearing when a pole enters the limiting point.
Grothendieck's terminal amplitude supplies the missing cutoff state coordinate: its shear obeys exact composition. That repairs algebraic domain nesting, not uniform graph-norm, reachability, or observability control.
Scope
This is a finite algebraic taxonomy and hostile suite. It does not construct the adelic completion, prove uniform graph bounds, passivity, conservativity, KYP positivity, an RH orientation, or RH.
Durable verification
- Packet:
research/kitaev/rosenbrock-zeros-do-not-commute-with-completion.md - Checker:
research/kitaev/checkers/check_rosenbrock_completion_obstructions.py - Results:
research/kitaev/results/rosenbrock-completion-obstructions.json - Sequence authority:
seqclaim-ac972123e0f38888d248cf09 - Epistemic graph event:
ev-000000003945-c867e2d4-f809-49f7-9b14-9f32b247bc55
The research state is coherent but uncommitted. No commit or push was authorized.