The Goldstone Diamond: A Hodge-Theoretic Recasting of Spontaneous Symmetry Breaking

We recast Goldstone's theorem, that spontaneous breaking of a continuous symmetry produces massless scalar excitations, in the language of the Hodge--de~Rham complex, Clifford bundles, and exceptional geometry developed in~{JanikHodge}. The standard theorem counts broken generators but does not articulate the manifold nature of the symmetries involved: the coset space $G/H$ is not merely a set of broken generators but carries its own Hodge--de~Rham complex, Clifford structure, and potentially exceptional geometry. We develop four complementary perspectives on symmetry breaking: Homotopy Type Theory (SSB as truncation of a higher groupoid to a torsor), Category Theory (Goldstone modes as sections of a natural bundle over the moduli stack [*/G] → [*/H]), Noncommutative Geometry (the spectral action on the broken vacuum generating the Goldstone effective Lagrangian), and Quantum Information Theory (SSB as a decoherence channel projecting the symmetric vacuum onto a symmetry-broken sector, with Goldstone modes as the zero-entropy directions of the channel). These perspectives expose the cohomological, spectral, and information-theoretic structure that the classical Nambu--Goldstone theorem leaves implicit, and connect naturally to the E₈ breaking chain E₈ ⊃ (16) ⊃ (10) × (6) ⊃ (3) × (2) × (1).