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

This paper develops a mathematical and computational framework for determining which defect characteristics are forced by incomplete and uncertain observations. Its central question is whether every configuration allowed by a declared observation and physical model agrees on a specified feature, such as an isolated-loop count, network connectivity or crystallographic class. The intended application is radiation-defect analysis in body-centered-cubic tungsten. The framework distinguishes three certificate outcomes: identified, when all admissible completions agree; ambiguous, when two explicit admissible completions disagree; and inconsistent, when no completion satisfies the assumptions. A separate unresolved status prevents an unfinished computation from being mistaken for a certificate. Physical vacancies, missing detected primitives and coordinate uncertainty are treated as distinct mechanisms. The mathematical development establishes exact separation criteria for noisy deletion observations, constructs explicit ambiguity witnesses and characterizes the smallest resolving measurement within a declared test library. It also shows why a positive robustness radius for fixed candidate coordinates can disappear when the underlying configurations are allowed to move continuously. A complete reduction for a restricted continuous-coordinate model preserves shared registration errors and supplies exact geometric witnesses. The paper then extends completion analysis to spatially embedded defect networks. Complete enumeration is proved for cell-forest models and broader models with finite crossing and local cycle budgets. These models include hidden circles, cancelling crossings and connected cyclic alternatives. In an executed restricted enrichment of twelve tungsten-derived windows, allowing one previously excluded hidden circle makes every formerly unanimous loop count ambiguous. This demonstrates dependence on the completion assumptions without claiming that every geometric alternative has been realized as a competing tungsten atom configuration. The atomistic investigations use public irradiated-tungsten coordinates and newly constructed, numerically relaxed specimens. They include dislocation extraction, independently checked atomic circulation certificates, coordinate-perturbation experiments and comparisons with a separately implemented reciprocal-order observation method. These experiments distinguish robustness of selected crystallographic information from certification of an entire physical defect population. A further result certifies the complete zero set of a specified smooth Gaussian observation field for one relaxed tungsten specimen. Throughout a declared finite cylinder, the zero set is exactly one smooth embedded circle for every independent atomic displacement of at most 0.0005 angstrom. A separate implementation verifies all 12,766 angular intervals and all 305,444 spatial covering cells. The proof combines transverse contraction with complete exclusion of additional zeros. Its conclusion applies to the specified field, fixed calibration and small uncertainty budget. Matched-population separation tests, a larger-domain repeat and a second-potential comparison strengthen the physical benchmark. The close-pair disagreement between a branched dislocation-extraction network and phase-based circle representations persists under these controls. The narrow-kernel phase count also changes across the two potential and relaxation protocols, exposing a consequential dependence on the chosen representation and modeling assumptions. The accompanying reproducibility archive contains editable manuscript sources, preserved atomistic inputs, benchmark programs, explicit witnesses, interval certificates, separate verification implementations and checksums. A blinded packet is prepared for independent physical topology adjudication; no external reviewer labels are claimed. Complete physical loop-population certification remains open. It requires independently adjudicated topology, a justified relationship between atomistic geometry and physical defect cores, and defensible restrictions on unresolved crossings, hidden components and junctions. The established contribution is a collection of precise, executable certificates for declared observation models, together with concrete evidence showing where physical interpretation remains underdetermined.