Registration Epistemics Finite Embedded Observers, Quotient Measurement, and the Limits of Self-Certification

A finite observer can be genuinely adequate to a declared task; what it cannot obtain from the same current interface is a final certificate that its registration exhausts every task-relevant finer distinction. This paper develops the framework behind that claim by separating three levels that earlier versions partially conflated. P0 states the constitutive structure of realised distinction: a distinction requires first-term uptake, maintenance across a comparison span, and the independent resolution of a second term; these are not three interchangeable resources but three irreducible aspects of one distinction-making act. P1 states that their physical realisation in any current observer is finite and non-zero; P2 states that every resource actually used in that realisation belongs to the embedded observer. The number two therefore enters before Shannon: one anchor can register presence, whereas two are the minimum arity of a non-trivial relation and first permit same/different and, for live succession, before/after. Measurement is consequently quotient formation rather than cost-free read-out. We distinguish six quotienting mechanisms and isolate a physical self-referential limit based on generative-grain / resolving-grain asymmetry: full-grain current self-resolution cannot be implemented at the same current registration grain. The same minimal relational structure constitutes registered temporal order without purporting to derive substrate time. Epistemically, the framework separates theoretical awareness, instance adjudication, and final certification; it is non-sceptical because refinement can correct earlier blindness even though no current finite interface thereby acquires a final certificate of exhaustive grain. Two domain-general corollaries follow: every registered value is loop-constituted rather than implementation-free read-out (a1), and no quotient value may be asserted as grain-exhaustive (a2). Consequences include the operational locus of probability without substrate randomness, the underdetermination of resolving grain by measured rate, and — at the quantum interface — the claims that the EPR reality inference is not licensed by finite embedded registration alone, that measurement independence is finitely defeasible but not finitely exhaustively confirmable, and that Bell’s theorem remains untouched.