Project T Volume XXXII : The Mass Ladder Theorem Emergent Mass from Cayley Graph Embeddings in the Symmetric Group Chain

We prove that the graph Laplacian of the Cayley graph of SK+1, when restricted to the diagonal block corresponding to the canonical subgroup SK, equals L(SK) + I exactly. This Mass Ladder Theorem holds for the full chain S3 ⊂ S4 ⊂ S5 and is verified computationally to machine precision (< 10⁻¹⁶). The identity is equivalent to the Klein-Gordon heat kernel relation KAA(t) = e⁻mt Kspatial(t), where m equals the number of collapsed dimensions. We establish that the temporal generator σK+1 distributes elements of SK uniformly across all external cosets, producing the complete graph KK+1 as quotient structure. Mass generation, temporal isotropy, and the Klein-Gordon propagator all emerge from pure algebraic embedding with zero free parameters.