Shell-Alignment from Projection Locking: A Discrete Admissibility Theorem under Born-Infeld Saturation

Within the spectral admissibility sub-programme of Cosmochrony, the canonical fibre-level observable σᶜᵃⁿₚₐᵢᵣ(n) on the Cayley graph Cay(Heis₃(Z/qZ), Sq) governs the charged-lepton mass hierarchy through a capacity decay exponent δₚₐᵢᵣ. O21 conjectured that physical admissibility requires the observable saturation rank nᵒᵇˢ₃ 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 Π 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 σᶜᵃⁿₚₐᵢᵣ(n) is a discrete sample of a continuous effective decay law σ̃(t) ~ C\,t^-δₚₐᵢᵣ inherited from the underlying χ-relaxation dynamics; the Born--Infeld admissibility constraint Aₙ ≤ cχ/√λₙ 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 Π decomposes as O = n ∈ Nq Oₙ, indexed by BFS shells, so that Π cannot encode a saturation event at a non-admissible depth. The physically realisable saturation depth is therefore the first element of LBI ∩ Nq, which lies in 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 nₛₐₜ is not assumed but derived as a constraint on admissible realisations under~Π. 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 cχ → Aₙᵐᵃˣ → LBI ∩ Nq → nₛₐₜ ∈ N introduces no parameter external to the Cosmochrony substrate. The problem of determining which shell is selected --- equivalent to deriving Σc(n₃) = 3 — remains open and is the subject of O23.