Information, Minimal Closure, and Structural Resolution Bounds

The most basic precondition of knowledge is rarely articulated as a question: how is distinction itself possible? Aristotle took this question to its limit when he identified the principle of non-contradiction as the most absolute presupposition of thought (Metaphysics IV.4) — A cannot at once be X and not-X. But the principle itself rests on a prior condition: X and not-X must be Aristotle distinguishable. Without distinguishability, the principle is empty. Distinguishability is therefore prior to thought, and the conditions under which distinguishability can be achieved are the conditions under which knowledge can take any form at all. This paper develops a minimal structural framework addressing that condition, under the constraint that the entity attempting to distinguish is itself part of what it distinguishes. Two assumptions are introduced: (i) information consists in the registration of a difference between two events, and (ii) all observation is performed by subsystems embedded within the system they observe. From these assumptions, together with the existence of minimal temporal and spatial steps t and l, a structural resolution bound r/2 is derived, where r = l/t. Since registering a difference requires at least two events, and each event occupies at least one minimal step, the minimal closure time is 2t and the maximal resolvable rate is r/2. This bound does not depend on measurement precision or specific physical implementations. It follows from the minimal requirements for observational closure under resource constraints, and is therefore structurally distinct from familiar resolution limits in signal processing, which arise from external sampling considerations rather than from the structure of distinguishability itself. Observation is characterised as a coarse-grained process: multiple configurations map to indistinguishable outcomes, and differences that cannot be registered do not enter any internal description. The result is a structural form of epistemic incompleteness — not a deficit of knowledge that better instruments might remove, but a feature of what registration can be when the registering subsystem operates within the system it registers. Possible connections to statistical structure, observation under varying conditions, and the formation of knowledge from accumulated regularities are briefly indicated, without commitment to specific physical interpretations. Where Aristotle established that thought presupposes the principle of non-contradiction, the present framework articulates the prior condition: the structural circumstances under which two events can be registered as distinct, and the bound on what can be so registered when the observer is embedded. form at all.