This paper develops a unified matrix–sequence–filter framework for deterministic observation. The central thesis is that sequence structure is not intrinsic to a bare mathematical object; it appears only after a presentation datum has been chosen: a transition, an enumeration, a quotient tower, a recurrence, a group action, or an observable orbit. Once such a datum is fixed, the object supplies a Moore source (or a cyclic source in the invertible case). The source is then linearized by the free-vector-space construction: every deterministic map becomes a basis-endomorphism matrix, and every scalar filter becomes a matrix coefficient. Thus every observed sequence takes the universal form an=L(Tϕney0),an=L(Tϕney0), where TϕTϕ is the linearized transition. This single normal form unifies finite fields, recurrences, p-adic towers, Diophantine solution systems, arithmetic-geometric coefficient packets (curves, Jacobians, elliptic curves, modular forms), representation-theoretic matrix coefficients, group cohomology, symbolic dynamics, automata, and signal-processing windows. The paper develops a categorical calculus in which products become tensor products, disjoint unions become direct sums, quotients become intertwiners, and compatible finite towers yield period spectra. The category of cyclic sources is shown to be equivalent to the category of pointed permutation-matrix systems; the category of Moore sources is equivalent to the category of pointed basis-endomorphism systems. Reachable–observable quotients give minimal matrix realizations, and Hankel-rank criteria provide finite-dimensional compression. A distinctive feature of the framework is its rigorous separation of internal statements—finite periods, stabilizers, spectra, quotient towers, matrix recurrences, and finite reconstructibility—from external comparison theorems that require arithmetic, analytic, geometric, probabilistic, or representation-theoretic input. This separation is formalized through the notion of a frontier conjecture probe: a finite or profinite source–filter package that organizes evidence without claiming to prove global conjectures by finite testing alone. Applications span finite-field dynamics (Frobenius orbits, polynomial maps, normal bases, linearized recurrences), p-adic odometers and Diophantine solution towers, irrationality-measure profiles (all-denominator sources, continued-fraction spectra, selected Padé sequences, and probabilistic tail laws), arithmetic geometry (height windows, curve zeta functions, elliptic and Hecke companion recurrences, cohomological trace packets, and finite descent data), representation theory (finite-group matrix coefficients, induced representations, group cohomology, and Hecke algebras), operator-algebraic and topological-dynamical envelopes of the bilateral shift, and finite-state reconstruction, control, and automata theory. The framework is both constructive and restrictive: it shows how to build matrix–sequence systems from chosen data, and it marks precisely where external input remains necessary for global conclusions. Keywords Moore source; cyclic source; matrix–sequence–filter system; free-vector-space linearization; permutation-matrix representation; basis-endomorphism representation; finite fields; Frobenius orbits; p-adic odometers; Diophantine solution towers; irrationality measure; continued fractions; Padé approximation; selected denominators; probabilistic tail laws; arithmetic geometry; elliptic curves; modular forms; Hecke recurrences; cohomological traces; representation theory; finite groups; group cohomology; Hecke algebras; symbolic dynamics; finite automata; frontier conjecture probes; comparison theorems; categorical dynamics; shift operators
Jianming Wang (Fri,) studied this question.