Worked case studies

Piano. The concept piano is represented as \mathit{piano} = \mathit{instrument}(\mathit{music}). Because both instrument and music are nominal, the genus--differentiae pattern yields \mathit{piano} ⊆ \mathit{instrument}; this follows from the compounding rule rather than manual assertion. From piano, a radius of 2 reaches music along the path \mathit{piano} → \mathit{instrument}(\mathit{music}) → \mathit{music}. For “Max is playing piano”, 3A-LLM predicts the implicit concept music at rank 1 without co-occurrence data---the inference is a structural consequence of the concept's construction history. Cross-lingually, piano (EN), Klavier (DE), piano (FR) map to the same conceptual structure whenever they share this definitional decomposition.

Air pressure. \mathit{air_pressure} = \mathit{pressure}(\mathit{air}), with subsumption \mathit{air_pressure} ⊆ \mathit{pressure}. Applying the instrumental vertical transformation yields \mathit{instrument}(\mathit{pressure}(\mathit{air})), corresponding to barometer or altimeter. The full derivation chain is \mathit{air} → \mathit{pressure}(\mathit{air}) → \mathit{instrument}(\mathit{pressure}(\mathit{air})); each step is a definitional or transformational edge, making the inference fully interpretable. Query expansion from air_pressure yields barometer, altimeter, \mathit{opposite}(\mathit{air_pressure}) (vacuum), and related compounds such as \mathit{pressure}(\mathit{water}).

Validation in hybrid pipelines. In a hybrid KG--LLM pipeline, an LLM-generated statement can be validated by attempting to reconstruct its 3A-LLM derivation. Successful reconstruction yields a human-readable explanation path; failure provides a principled signal that the statement is semantically unsupported in the axiomatic framework. For example, if an LLM claims that “a barometer measures wind speed”, 3A-LLM can trace back from \mathit{barometer} = \mathit{instrument}(\mathit{pressure}(\mathit{air})) to show that the instrument is structurally linked to pressure, not to wind speed, flagging a potential hallucination. This enables practical XAI-style explanations and quality monitoring without requiring access to opaque model internals.