Overview
This article investigates how discrete generational structures can emerge from the spectral organisation of relational systems. In many physical theories the existence of particle generations and large mass hierarchies appears as an empirical fact rather than a structural necessity.
The analysis introduces the notion of spectral stratigraphy: a hierarchy of stabilisation depths associated with eigenmodes of a relational Laplacian. When spectral sectors become stabilisable at different stages of the relaxation cascade, the resulting profile naturally forms discrete stratified levels.
Within this framework, particle generations correspond to clusters of modes that stabilise at comparable cascade depths. The number of generations is therefore controlled by the structure of the underlying relational spectrum.
Core contributions
- Three conditions for stabilisability: projectability (dynamic, via a cascade cut-off $\Lambda_{\mathrm{proj}}(n)$), Born–Infeld admissibility (static spectral filter), and saturability (existence of a non-empty stabilisation window) — formalised as the stratigraphic function $n_{\min}(\lambda_k)$.
- No-go (Proposition 2): a scalar power-law stabilisation threshold $E^{*}(\lambda) = E_{0}\lambda^{\beta}$ with two-anchor calibration yields a strictly featureless stratigraphic profile — no discrete generational structure can emerge from such an ansatz.
- Positive result (Proposition 3, Corollary 1): when the threshold acts representation by representation on ADE-type binary Cayley graphs, four pairs $(G,S)$ in the family $\{2T, 2O, 2I\}$ yield exactly three non-zero spectral classes — discrete three-level stratigraphies arise structurally.
- Separation of roles: the group–generator pair $(G,S)$ fixes the number of stratigraphic levels (hence the generation count), while the projective-resolution growth $\Lambda_{\mathrm{proj}}(n)$ determines the inter-level separations (hence the mass ratios).
- Selection of $2I$: imposing the neutral (order-four) generator convention of the admissibility programme singles out the binary icosahedral group at the group level, though it does not yet fix the generator set within $2I$.
- Central open problem: deriving $\Lambda_{\mathrm{proj}}(n)$ from first principles.
Interpretation
The analysis reveals a natural separation between the structural origin of particle generations and the physical mechanisms controlling mass scales.
- Group topology determines the number of spectral strata and therefore the number of particle generations.
- Projective dynamics determines the separation between stabilisation depths and therefore the hierarchy of masses.
- Admissibility constraints determine which spectral sectors stabilise first.
Within this perspective, the existence of three particle generations can be interpreted as a consequence of discrete spectral organisation rather than an arbitrary property of the Standard Model.
Relation to the Cosmochrony program
Spectral stratigraphy complements other components of the Cosmochrony research program. Spectral admissibility characterises which modes can exist under bounded relational flux, while spectral capacity measures the aggregate admissible content of a relational system.
Spectral stratigraphy adds a dynamical dimension by analysing when different sectors stabilise along the relaxation cascade. Together these elements form a coherent framework linking relational spectra to particle phenomenology.
References
Jérôme Beau. Three Particle Generations and Mass Hierarchy from Spectral Stratigraphy. Preprint, Zenodo.