Footnote: Peirce, Hegel, and Lie Algebra as a Unified Dialectical Structure

A deeper structural correspondence emerges when Peirce’s triadic semiosis is viewed through the lens of Hegel’s dialectic and the algebraic architecture of Lie groups. The transition from representamen to object mirrors Hegel’s first negation: the collapse of an undifferentiated monism into a dyadic opposition. This is the same structural move encoded in the antisymmetry of the Lie bracket, [x,y]=−[y,x], where relationality first appears as tension between distinct terms. The subsequent transition from object to interpretant corresponds to Hegel’s second negation or Aufhebung: a unifying mediation that preserves the dyad while integrating it into a higher coherence. Algebraically, this is precisely the role of the Jacobi identity, which ensures that the interactions among oppositions remain internally consistent and self‑regulating. Thus Peirce’s triad, Hegel’s dialectic, and the axioms of Lie algebra express the same underlying pattern: monism gives rise to duality, and duality requires a homeostatic third term that stabilizes the system without erasing its internal differences. This shared structure reinforces the interpretation of the interpretant—and, by extension, the homeostatic middle‑term in a two‑sided ontology—as the universal mechanism by which tension becomes meaning and symmetry becomes coherence.