The Distinguishability Foundation: From a Registered Difference to the Reflexive Limits of Finite Embedded Observers

This paper takes a single registered difference as its starting point, works back to the minimal conditions it requires, and derives classification, re-identification over time and questions of identity only where further premises are explicitly declared. Proposition A states three necessary conditions of a registered difference: two relational positions, resolution relative to the configuration and the task, and joint availability within a single operation. Finitude and embeddedness are treated as restrictions on the type of system studied: relational, operationally constitutive and index-local. Every proposition is marked for its argumentative status and for its standpoint. The marking separates the modeller's statements, events on the observer's side, and bridging claims that need a model–implementation condition. On this basis the paper introduces symmetric negation. Under a comparison condition and an agreement premise, a negative registration licenses only "at least one side has judged incorrectly", without saying which. Where the declared realization class contains candidates compatible with the registration under both attributions of failure, the negation is symmetric, and breaking the symmetry requires further discriminating evidence or an explicit asymmetric restriction. A symmetric negation may put in question not only the content of a claim but also the claimant's qualification to produce it: the correctness of a judgement and the conformity of the operation that produces it are two different propositions. Registered blindness can then be characterized as a configuration's negative registration that calls its own attribution of qualification into question. A difference of direction follows: a registration can call qualification into question, while positive qualification can be certified only conditionally under declared realization conditions, and cannot be established without presupposition from the outputs of the operation being certified. The paper further proves that, at a symmetric registration of disagreement, every sound single attribution of failure rests on a declared restriction the registration cannot certify; adding sources in separated dependency classes cannot remove this restriction; it can only transform the restriction from an asymmetric premise about identity into a symmetric structural premise requiring at least 2f + 1 dependency classes. The partition into dependency classes is itself a modeller's declaration: agreement cannot certify it, and disagreement can negate it only symmetrically. A finite embedded observer can approach the partition asymptotically, with no terminal certification. The paper does not claim that distinction creates underlying existence, nor that description achieves ultimate self-grounding. The declarations it uses but does not establish are collected in the D list (§9.2). By the paper's own classification, that list records the author's registered blindness about the paper's premises, and cannot claim to be complete.