Hankel rank and minimal linear realizations
Field-valued semantics admit linear state representations that can be much smaller than a discrete observer storing exact future behavior.
For a weighted series f : Sigma* -> K, define its residual after prefix u by
f_u(v) = f(uv).
A discrete future-equivalence observer identifies prefixes only when the full
residual functions are equal. A linear realization instead stores a vector in
the span of the residual functions. Over a field, the dimension of this span is
the rank of the Hankel matrix H[u,v] = f(uv) and equals the state dimension of
a minimal weighted finite automaton. This is the classical weighted-automata
and rational-series correspondence.
A polyphase example
Take the polyphase route packet for the two-tap filter h=(1,1),
polyphase order N, and evaluate the coarse-lag polynomial variable at z=2.
The resulting route matrix is
M_N = I + S, with S^N = 2 I.
For the scalar downstream query
f_N(n) = e_0^T M_N^n e_0,
the N x N Hankel block has full rank N. Thus the existing N-phase linear
route state is already minimal for this query.
For N=2,
M = [[1,2],[1,1]]
and
f(n) = 1,1,3,7,17,41,...,
which satisfies
f(n+2)=2 f(n+1)+f(n).
All discrete residual functions k -> f(n+k) are distinct, so a deterministic
Moore observer storing exact future behavior requires infinitely many states.
The linear/Hankel representation requires dimension two.
This example illustrates why weighted automata and linear state-space models can compress numerical series more effectively than deterministic state equality.
Architectural consequence
The continuation quotient must be typed by the observation algebra:
- set/Boolean semantics: future-equivalence classes / Myhill--Nerode;
- field-linear semantics: residual vector-space quotient / Hankel rank;
- other semirings: the appropriate semimodule/congruence notion, when it is well behaved.
API boundary
The polyphase route packet supplies the finite linear realization used here. The recurrence subsystem supplies companion-state coordinates. A generic weighted-automaton constructor is outside these APIs. Minimality depends on the observation algebra, as described in carrier-change obstructions.