Constraint-Governed Compression: What Should Determine the Resolution of a Finite Representation

Every finite intelligence, biological or artificial, must compress. No system with bounded representational capacity can preserve its observations in full; it must summarize a high-dimensional world into a lower-dimensional internal representation. The debates that dominate contemporary machine learning—how large the model, how long the context, how much data—implicitly assume that the central problem is the *quantity* of compression. This paper argues that the central problem is not how much a system compresses but what *governs* the compression: which distinctions it is permitted to discard and which it must preserve. We distinguish two fundamentally different classes of information. Compressible information—wording, phrasing, ordering, redundant illustration, incidental detail—may be discarded without altering the validity of what remains. Constraint information—logical consistency, arithmetic identity, causal dependency, provenance, conservation law, type structure, contradiction relations—cannot be discarded without changing correctness. Constraints are valuable not because they are detailed but because they are invariant: they remain true regardless of observational scale. This invariance does not make them self-preserving—ungoverned compression destroys them routinely—but it makes them naturally compatible with compression, cheap and safe to retain, so that a well-governed process can preserve them while compressing scale-dependent detail aggressively. Ungoverned compression fails precisely because it does not distinguish invariant constraint information from compressible detail. From this we derive our central thesis. Constraint recursion is not, at bottom, a verification architecture that checks outputs after abstraction. It is a theory of *constraint-governed compression*: the domain's constraints should determine, in advance, what a compressor is permitted to discard. The correct architecture is therefore constraints → admissible compression → abstraction → reasoning, not compression → later verification. We redefine admissible compression accordingly: a compression process is admissible only if the constraints governing the domain determine what may be discarded during compression. We then show that the familiar pathologies of large language models—structural drift, the holographic mirage, the collapse of exclusion boundaries, the impossibility of recursive falsification—are not intrinsic costs of compression but the specific downstream signature of *ungoverned* compression. A larger context window does not repair them, because adding context adds further compressed representations rather than restoring the resolution that abstraction already discarded. Keywords: finite representation, representational resolution, compression, invariants, constraint-governed compression, admissible compression, structural fidelity, neurosymbolic AI