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 \ₖ\ of its substrate Laplacian L_, produces a naturally stratified set of stabilisable modes. Three conditions are formalised: projectability (dynamic, via a cascade cut-off ₏ₑ₎₉ (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^* () = E₀^ 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 ₏ₑ₎₉ (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 ₏ₑ₎₉ (n) from first principles constitutes the central open problem. Keywords: particle generations; mass hierarchy; spectral stratigraphy; spectral graph theory; representation theory; emergent particle spectra