Characterizing Scoring Function Stability under Open-World Taxonomy Growth

Taxonomy expansion and completion represent a query and the available taxonomy, score candidate placements, and select parent links. Knowledge growth can change this attachment problem after training. A parent can become a grandparent, a sibling can become a parent, or another parent can become necessary without changing the query's meaning. Evaluation against a fixed seed taxonomy leaves these consequences untested. In this paper, we develop `TaxoAtlas` as a theoretical framework to analytically study the stability of scoring and selection families under such growth. It records taxonomy relationships and places concepts using calibrated semantic similarities. Related concepts contribute support to an evidence field, a map showing where evidence for a query is concentrated. Under explicit conditions, we establish which structural and semantic distinctions this construction preserves and bound how growth changes its fields. To isolate scorer limitations from representation error, every rule receives exactly the current query and candidate inputs prescribed by the evolving reference. We call this *oracle reference encoding*. The theory characterizes eight recurring scoring-operation classes as mathematical objects, independent of how well an encoder or complete model is trained. The resulting conditions cover retaining valid parents, acquiring newly required parents, rejecting false links, and exact parent-set recovery. They separate the effects of changing evidence, new competitors, normalization, structural constraints, and final selection. Growth elsewhere can suppress an unchanged local match, while a correct ranking can still yield missing or false links. Conversely, query and parent changes can cancel and preserve their score. In a controlled construction, a parent-set score that penalizes larger sets misses a new parent in all forty later states. A Gaussian score ranks both true parents highest but admits false links in thirty-nine. In paired conditional ranking with a parent-complete candidate neighborhood, a structurally gated angular scorer's Hit@1 falls from 49.0% to 21.6% on Environment and from 37.0% to 31.5% on Science after growth. A separate Environment replay and descriptive field profiles for Environment, Science, and Food provide further grounding. Our theoretical and empirical results provide detailed insights into the stability of scoring and selection rules under taxonomy growth.