Abstract

Abstract

Let $S$ be a source-realization space obtained after removal of declared gauge and coordinate redundancy, and let $p\colon S\to T_1$ be an exact surjective readout onto an observable tier. We establish equivalent criteria for source completeness of $p$; we combine fiber variation with target dependence and canonical minimal completion; and we prove that any model containing an essential non-gauge distinction inside a $p$-fiber , two physically inequivalent source states with the same readout but different correct answers to a nominated target , is, relative to that model and target, a non-source projection.

The dynamical consequences divide sharply. If a deterministic source evolution preserves readout fibers, it descends uniquely to an autonomous law on $T_1$ even when $p$ is noninjective; autonomy of the projected law therefore does not imply source completeness. If the evolution breaks fibers, no deterministic law on the instantaneous $T_1$ state reproduces every source trajectory. What this establishes is a negative: the instantaneous readout is insufficient. It does not by itself supply a replacement , a completion variable, an unresolved-state distribution, or a memory representation each requires an additional construction, and memory in particular need not encode the missing coordinate at all.

We then bound the inference available to an observer confined to the readout. A family of probes identifies the source exactly when its joint map separates source states; the maximally identifiable object is otherwise the quotient by the intersection of the probe equivalence relations. Under the explicit hypothesis that every admissible internal probe factors through $p$, no probe value, derived statistic, or deterministic selector distinguishes members of a common fiber. The statistical counterpart replaces probe values by complete outcome laws, and the finite-state stochastic counterpart is a strong lumpability theorem giving the exact condition under which a projected Markov source process remains Markov.

The synthesis is a boundary theorem. The mathematics determines when a readout is a non-source quotient, what a completion must restore, when projected laws close, and what internal observation can identify. It does not by itself supply a source domain, a physical equivalence, or a witness; those must be instantiated and tested by a physical model, and we mark that boundary explicitly throughout.

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