Shell-Alignment from Projection Locking: A Discrete Admissibility Theorem under Born–Infeld Saturation O22 — Spectral Admissibility Sub-programme

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 $A_n \leq {c_{\mathrm{BI}}}/\sqrt{\lambda_n}$ 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 ${c_{\mathrm{BI}}} \to A_n^{\max} \to \mathcal{L}_{\mathrm{BI}} \cap \mathcal{N}_q \to {n_{\mathrm{sat}}} \in \mathbb{N}$ introduces no parameter external to the Cosmochrony substrate. The problem of determining which shell is selected — equivalent to deriving $\Sigma_c(n_3) = 3$ — remains open and is the subject of O23.