Shell-Alignment from Projection Locking: A Discrete Admissibility Theorem under Born-Infeld Saturation
Year: 2026
Type: preprint
Abstract: Within the spectral admissibility sub-programme of Cosmochrony, the canonical fibre-level observable $\sigma^{\mathrm{can}}_{\mathrm{pair}}(n)$ on the Cayley graph $\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}), S_q)$ governs the charged-lepton mass hierarchy through a capacity decay exponent $\delta_{\mathrm{pair}}$. O21 conjectured that physical admissibility requires the observable saturation rank $n^{\mathrm{obs}}_3$ to align with a BFS shell of this graph — a condition left open as an independent geometric hypothesis.We prove that this conjecture is not independent: it is a theorem. The proof identifies a projection locking mechanism intrinsic to the non-injective projection $\Pi$ of the Cosmochrony framework. The argument rests on a distinction between two types of discreteness. The shell index $n$ is discrete as a matter of graph geometry. The observable $\sigma^{\mathrm{can}}_{\mathrm{pair}}(n)$ is a discrete sample of a continuous effective decay law $\tilde{\sigma}(t) \sim C\,t^{-\delta_{\mathrm{pair}}}$ inherited from the underlying $\chi$-relaxation dynamics; the Born--Infeld admissibility constraint $A_n \leq c_{\chi}/\sqrt{\lambda_n}$ is therefore a condition on a continuously varying, generically irrational amplitude. Nothing in the Born--Infeld dynamics alone forces saturation to occur at an integer depth.Shell-alignment follows from a third ingredient: the observable space of $\Pi$ decomposes as $\mathcal{O} = \bigsqcup_{n \in \mathcal{N}_q} \mathcal{O}_n$, indexed by BFS shells, so that $\Pi$ cannot encode a saturation event at a non-admissible depth. The physically realisable saturation depth is therefore the first element of $\mathcal{L}_{\mathrm{BI}} \cap \mathcal{N}_q$, which lies in $\mathbb{N}$ by construction. At its core, the proof reduces to the incompatibility between a continuous saturation locus and a discrete admissibility support: the discreteness of $n_{\mathrm{sat}}$ is not assumed but derived as a constraint on admissible realisations under~$\Pi$. As a corollary, the logarithmic commensurability condition of O21 is recovered as the observable signature of projection locking, not as an independent resonance principle. The entire derivation is foundational: the chain $c_{\chi} \to A_n^{\max} \to \mathcal{L}_{\mathrm{BI}} \cap \mathcal{N}_q \to n_{\mathrm{sat}} \in \mathbb{N}$ introduces no parameter external to the Cosmochrony substrate. The problem of determining \emph{which} shell is selected --- equivalent to deriving $\Sigma_c(n_3) = 3$ — remains open and is the subject of O23.
Author: Beau, Jérôme
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7147224159
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