Three Particle Generations and Mass Hierarchy from Spectral Stratigraphy

We introduce a concrete computational framework for testing whether a candidate relational graph, characterised by the spectrum \λₖ\ of its substrate Laplacian L_χ, produces a naturally stratified set of stabilisable modes. Three conditions are formalised: projectability (dynamic, via a cascade cut-off Λₚᵣₒⱼ(n)); Born–Infeld admissibility (static spectral filter); and saturability (existence of a non-empty stabilisation window). The note establishes two complementary results. Negative: a scalar power-law stabilisation threshold E^*(λ) = E₀λ^β with two-anchor calibration produces a strictly featureless stratigraphic profile (Proposition 2); no discrete generational structure can emerge from such an ansatz. Positive: discrete three-level stratigraphies arise naturally when the threshold acts representation by representation on ADE-type binary Cayley graphs; four pairs $(G, S)$ in the family \2T, 2O, 2I\ are shown to yield exactly three non-zero spectral classes (Proposition 3, Corollary 1). The results suggest a separation of roles: the group–generator pair $(G, S)$ fixes the number of stratigraphic levels (hence the generation count), while the dynamics of projective-resolution growth Λₚᵣₒⱼ(n) determines the inter-level separations (hence the mass ratios). Imposing the neutral (order-four) generator convention of the admissibility programme singles out the binary icosahedral group $2I$ at the group level, though it does not fix the generator set within $2I$. Deriving Λₚᵣₒⱼ(n) from first principles constitutes the central open problem.