Unified Embeddings Extended

We construct formal embeddings of eight algebraic structures into a universal type \(\mathcal{U}\) equipped with a canonical set of observables. For each algebra \(A_i\) (Generalized Spectral Algebra, Resonance, Jet, Tropical Spectrum, Tropical Path, Causal, Self-Reference, and M-Tree), we define injection–projection pairs \((\iota_i, \pi_i)\) satisfying the retraction identity \(\pi_i \circ \iota_i = \mathrm{id}\) on key observables. We prove 80 theorems establishing these retractions and coherence relations, showing that domain-specific observables are preserved through the universal representation. As a byproduct, we establish cross-embedding coherence theorems demonstrating that physical quantities (tension, depth, order, flow) are consistently represented across all embedded algebras. The entire development comprises 179 verified statements with 42 hypotheses and 57 axioms, formalized in the Platonic proof kernel.