The Gate Between Constraint and Identification

Abstract

Abstract The Gate Between Constraint and Identification studies how a temporally extended bounded observer can fail to identify what generated its evidence even after substantial constraint has been achieved. It distinguishes two structurally different mechanisms. A source-side gate arises when accessible evidence never discriminates candidate generators, either because generated histories coincide or because the access channel merges them. A retention gate arises when evidence once discriminated relevant histories but the observer’s retention channel later merges them. These mechanisms compose, but they have different provenance and require different repairs. The formal framework represents inquiry as an evidence channel whose serial stages can preserve or merge distinctions but cannot recreate distinctions already lost upstream, while jointly available channels can refine distinguishability. This yields a retention quotient, a Retention-Indistinguishability Lemma, a source/access/retention kernel hierarchy, and an Indistinguishability Provenance result that assigns every finally merged pair in a finite nested kernel chain a unique first-collapse stage. External artifacts are thereby treated as declared retention channels for selected verdict-relevant distinctions: they can preserve distinctions they independently encode, but cannot recover information solely from a representation that has already collapsed it. The paper then develops the dual repair problem. An added channel is verdict-sufficient exactly when its information, combined with the retained state, separates every pair that some admissible future verdict still requires distinguished. In the finite unrestricted-map class, the minimum repairing alphabet is determined by the largest number of verdict-equivalence classes occurring within any single retained-state fiber. The same quantity is the chromatic number of the associated repair-conflict graph. Minimum repair capacity is therefore canonical even though optimal repairing representations need not be: incomparable maximal repairs can exist without any greatest repairing representation. A Branch-Before-Collapse Principle shows that a channel derived only from an already-collapsed representation cannot restore the lost distinction; repair requires access to some surviving information path that did not factor through that collapse. The paper also distinguishes object-target persistence, evaluator persistence, and evidence-lineage persistence. Evaluator persistence is defined behaviorally through agreement of future verdict traces under admissible continuations, yielding a verdict-sufficiency quotient: the coarsest retained state sufficient to preserve every admissible future verdict. Finally, the paper proves a refinement law. Enlarging a declared verdict family can only refine verdict equivalence, increase or preserve repair obligations and minimum repair capacity, and reduce or preserve what may safely be forgotten. It separately distinguishes newly introduced obligations from the actual deficiency of an existing artifact, since an artifact may already preserve distinctions that its earlier declaration did not require. Multi-channel systems compose by kernel intersection, giving a deterministic criterion for resilience to channel loss. The resulting framework separates constraint from identification, generator ambiguity from access and retention loss, loss provenance from repair obligation, and regime-relative sufficiency from stronger claims of epistemic closure. It does not claim that identification is impossible. It specifies when identification is licensed, where relevant distinguishability can be lost, and what additional information channels are sufficient to preserve distinctions required by future evaluation. Version note: Version 2.4.3 fully supersedes the original Gate Between Constraint and Identification. The original paper established the finite-evidence boundary between constraint and unique identification, but treated temporal retention largely narratively. Version 2.4.3 resolves that limitation by formalizing an independent retention gate and adding the source/access decomposition, indistinguishability provenance, exact repair conditions and minimum repair capacity, behavioral evaluator persistence, verdict-sufficient retention, branch-before-collapse, multi-channel resilience, and monotonicity under declared verdict-family refinement. The original version should therefore be read as the conceptual precursor to the present theory rather than as a coequal statement of it.