Where Structure Ends - A boundary problem for objectivity-as-invariance and a diagnostic for structural theories of consciousness

This preprint develops a structural boundary problem for “objectivity-as-invariance” within the Observer-Equivariance (OE) framework. We model the space of observer perspectives as a groupoid O and treat physical structure S as what descends under the relevant equivalence (written informally as S ≃ O/G). A central constraint is forgetfulness: distinct perspectives may project to the same structural description, so the projection Proj: O -> S is not expected to be injective on objects in any OE-relevant sense.

We then introduce an internal notion of a limit of structure. Fixing an OE-relevant invariant signature I, we order structural descriptions by comparative “differentiation” and define a minimal regime S_min in which the essential image of Proj collapses to a single OE-relevant isomorphism class. This isolates a principled sense in which structural description can become maximally symmetric and correspondingly insensitive to phenomenological distinction.

On the metaphysical side, we formulate an Ontological Closure Principle (OCP) as a defeasible constraint against strong emergence “solely by rearrangement of non-phenomenal facts”. This blocks appeals to brute psycho-physical bridging laws as explanatory solutions (they may label correlations but do not bridge the ontological gap).

Finally, we derive a Quotient Test for theories of consciousness: any candidate “public” quantity Q proposed to capture phenomenality faces a dilemma. If Q is invariant under admissible representational transformations, it descends to S and is therefore fiber-blind; if it fails to descend, it depends on perspective and cannot function as an objective physical observable in the usual sense. The test is applied as a diagnostic to structural proposals in the vicinity of Integrated Information Theory and functionalist/AI approaches, emphasizing that admissible transformations must be stated explicitly and defended as OE-relevant.

The paper is intended both (i) to isolate an internal boundary problem for OE-style objectivity, and (ii) to provide an operational diagnostic for structural theories that attempt to identify public structure with private phenomenality.