Determinacy Under Quotient Representations

Every measurement, model, and computation proceeds through representation: a full state space is replaced by a compressed description that retains some structure and discards the rest. Such compressions are indispensable. They are also geometrically consequential in a way that modern practice rarely makes explicit.

This paper develops a structural audit for representational pipelines based on a single classical criterion: a scalar query descends through a representation if and only if it is constant on the fibers of that representation. When this condition fails, no function of the compressed data can recover the query's value without introducing additional structure. The criterion is classical; but what is not yet canonized in practice is the discipline of enforcing it.

Three consequences are formalized. The post-quotient repertoire establishes that when scalar determinacy fails, four responses are available: retain set-valued fiber information, extract extremal bounds, introduce an explicit closure rule, or refine the representation by restoring the discarded coordinate. The first three responses accept the quotient and manage its consequences from within; the fourth undoes it. An irreversibility principle shows that once a query is fiber-sensitive at any stage of a pipeline, no downstream compression restores determinacy without added structure. A closure tax bounds the unavoidable cost: any scalar rule applied to a fiber-sensitive query incurs worst-case error of at least half the fiber variation, independent of the rule chosen.

These results are instantiated across five domains: electromagnetism, nonlinear queries under linear projection, probability, harmonic analysis, and digital computation. The cross-domain range is deliberate. Within any single field, the obstruction can appear to be a domain-specific limitation or a modeling convention. Seen across fields, it reveals itself as a single structural pattern that arises wherever non-injective representations are used and fiber geometry is not tracked.

The paper concludes with a four-question audit applicable to any pipeline, regardless of domain or computing substrate: What is the representation? What are its fibers? Does the query descend? If not, what closure is being used — or can the discarded coordinate be restored?

Version 1.1 update note

This version corrects a structural inconsistency between the Introduction and Section 3. The Introduction listed four available responses to a fiber-sensitive query — including representation refinement (restoring the discarded coordinate) — but the formal section named only three, calling them the "post-quotient trilemma." The fourth response has been added as a formally distinct type (§3, item iv), the section has been renamed "The post-quotient repertoire," and a new remark explicitly identifies the companion papers of this series as the instantiation of response (iv) at the level of the electromagnetic vacuum. The abstract contributions list and two inline references have been updated accordingly. No theorems, proofs, empirical results, or conclusions have been changed.