Certifying What Radiation-Defect Observations Determine: Completion certificates, tungsten data and continuous topology models

This paper develops a framework for determining which conclusions about radiation defects are justified by incomplete and uncertain observations. Its central question is whether a specified feature is shared by every defect configuration admitted by a declared observation and completion model. The work combines exact combinatorial analysis, continuous geometric models, atomistic tungsten benchmarks and independently replayed computational certificates. It addresses reliable defect identification and the assumptions behind population estimates. The framework distinguishes three substantive outcomes. A feature is identified when the admissible completion set is nonempty and every completion agrees on that feature. Ambiguity is established by two explicit admissible completions with different feature values. Inconsistency means that no completion satisfies the stated assumptions. A separate unresolved outcome records incomplete searches, timeouts and unsuccessful sufficient numerical bounds. The formulation preserves unlabeled measurement occurrences, multiplicities, shared coordinate registration and crystallographic restrictions. Missing detected primitives are treated as observation failures and are distinguished from physical vacancies or the deletion of material atoms. The mathematical development gives exact matching conditions for compatibility under bounded coordinate error and missing detections, together with feature-dependent separation thresholds and local robustness certificates. Counterexamples show how a missing bridge or an unseen component can change a population conclusion, and why a positive robustness radius in a fixed-site catalog may disappear when latent coordinates can move. A finite test-library formulation characterizes which additional measurements distinguish competing feature values and supports exact minimization within that declared library. Complete reductions also cover continuous coordinate boxes while retaining correlations imposed by a shared registration variable. Reproducible synthetic benchmarks illustrate the difference between selecting a plausible explanation and certifying what the observations determine. A finite catalog contains 899 admissible states. Among 480 designed observations generated within that model, the certificate procedure identifies 349 cases correctly and returns explicit ambiguities for 131; it also certifies inconsistency for 24 deliberate model violations. A continuous-geometry experiment produces 194 correct identifications and 94 ambiguities across 288 cases. These are results for specified surrogate models and designed uncertainty channels, with independently checked witnesses and clearly stated comparison denominators. The tungsten component begins with eight coordinate crops from published irradiation simulations. It preserves atomistic geometry, uses the Dislocation Extraction Algorithm to propose candidate structures, and supplies sixteen independently checked reference-lattice circulation witnesses around selected candidates. Coordinate and crop sensitivity studies examine the stability of these diagnostics and the extracted representations. A separate finite-bank experiment tests observations against stored atomistic hypotheses, while explicitly recognizing that agreement within that bank does not establish coverage of every physically admissible defect configuration. The paper then extends completion analysis beyond stored configurations. It derives complete finite reductions for continuously embedded cell forests and for declared models with bounded crossing multiplicity and local cycle complexity. These models distinguish isolated circles, connected networks and graph cycle rank, and permit explicit competing connectivity witnesses. On twelve atom-derived tungsten windows, the restricted forest analysis includes an example with different network topologies but an identical isolated-circle count. Allowing one additional unseen circle makes all twelve previously unanimous counts ambiguous within a specified model enrichment. This demonstrates dependence on structural assumptions; the inserted geometric alternatives are not claimed to be independently realized tungsten atom configurations. Controlled atomistic experiments construct and numerically relax tungsten specimens, compare matched populations at different separations, repeat a close-pair case in a larger domain, and examine two interatomic-potential protocols. They expose a consequential disagreement between a branched dislocation network and resolution-dependent phase-field circles. The experiments separate sensitivity to extraction, smoothing, boundary conditions and relaxation protocol from the question of which representation correctly describes physical core topology. The strongest continuous certificates concern a specified untruncated Gaussian field constructed from atomic positions. For one isolated specimen, independent interval replay verifies that its complete zero set inside a declared cylinder consists of exactly one smooth embedded circle for every choice of atomic positions within independent Euclidean balls of radius 0.0005 angstrom around the stored coordinates. Verification covers 12,766 angular intervals and 305,444 spatial leaves. For the disputed 24,155-atom close pair, a smaller separating-plane certificate establishes a two-component lower bound, and complete replay of 17,685 angular intervals strengthens this to at least two isolated smooth circular zero components of the narrower field at the same atomic uncertainty bound. Additional components have not been excluded in that case, so its exact field census remains open. Both field experiments hold atom identities, atom count, lattice parameter, crystal orientation and field parameters fixed; no missing-atom or missing-line-detection channel is included. These results establish properties of explicit observation models. Complete physical loop-population certification still requires independently adjudicated core topology, a physically justified completion domain covering hidden structures and uncertain connectivity, and a supported correspondence between the certified representation and physical defects throughout the admissible uncertainty family. A nominal physical label alone would not establish that uniform correspondence. The small atomic error bound is not a calibrated microscopy tolerance or a thermal-stability guarantee, and the work does not predict irradiation hardening or component lifetime. The accompanying materials provide the manuscript, source code, benchmark inputs, explicit completion witnesses, interval certificates, separate verification implementations, checksums and instructions for replaying the computations. They also include a blinded physical adjudication protocol and clearly marked incomplete exterior searches that can be resumed. The paper offers an auditable basis for distinguishing established defect features, demonstrated ambiguities and conclusions that still depend on unvalidated physical assumptions.