Overview
This article continues the spectral admissibility programme after O21. While O21 removed external thresholds and fit parameters from the persistence criterion, it still relied on a structural shell-alignment condition.
O22 resolves this remaining gap. Its central objective is: to prove that shell-alignment follows necessarily from the Born–Infeld admissibility structure and the discrete observable support induced by the non-injective projection \(\Pi\).
The paper introduces the notion of projectively admissible shell support \[ \mathbb{N}_q, \] and shows that the continuous Born–Infeld saturation locus \[ L_{\mathrm{BI}} \] can be physically realised only where it meets this discrete support.
The key structural statement is that saturation is not merely a continuous threshold crossing: it is an admissible realisation event, and admissible realisation is discrete.
Core contributions
- Discrete admissibility support: definition of the projectively admissible shell set \[ \mathbb{N}_q, \] with observable space decomposed as \[ O = \bigsqcup_{n \in \mathbb{N}_q} O_n. \]
- Continuous saturation locus: the observable is treated as a discrete sampling of an effective decay law \[ \tilde{\sigma}(t) \sim C\, t^{-\delta_{\mathrm{pair}}}, \] so the Born–Infeld condition defines a continuous candidate saturation depth.
- Projection locking theorem: admissible saturation can occur only when the continuous locus \[ L_{\mathrm{BI}} \] meets the discrete admissible shell support \[ \mathbb{N}_q. \]
- Derived shell-alignment: the shell condition of O21 is no longer postulated, but recovered as a corollary of projection locking.
- Internal derivation chain: the result is obtained entirely within the framework, \[ c_\chi \to A_n^{\max} = \frac{c_\chi}{\sqrt{\lambda_n}} \to L_{\mathrm{BI}} \to L_{\mathrm{BI}} \cap \mathbb{N}_q \to n_{\mathrm{sat}} \in \mathbb{N}. \]
- Logical clarification: O22 proves that saturation must occur on a shell, but does not yet derive which shell is selected.
- Foundational closure: this is the first theorem-level closure of the shell-alignment problem in the O-series.
Interpretation
O22 changes the logical status of shell-alignment.
- O21: shell-alignment is required
- O22: shell-alignment is explained
The apparent resonance between continuous decay and discrete BFS geometry is therefore no longer a primitive principle. It is the observable signature of projection locking.
The central insight is that physical saturation is not just a value crossing. It is an admissible realisation event constrained by the discrete support of \(\Pi\).
Relation to the Cosmochrony program
O22 occupies a decisive position in the O-series. It follows the intrinsic crossing construction of O21 and removes the remaining structural hypothesis on shell-alignment.
The programme now reads: O16 (pair observable), O17 (pair dynamics), O18 (fibre structure), O19 (canonical normalisation), O20 (persistence criterion), O21 (intrinsic saturation rank), O22 (projection locking and shell-alignment).
After O22, the shell condition is no longer conjectural, but derived from the internal interaction between continuous saturation and discrete admissible support.
Current outcome and open directions
O22 establishes that admissible saturation must occur on a shell. The shell-level condition is now structurally internal and theorem-level established.
Remaining directions include:
- Shell selection: derive which shell is selected, not merely that saturation must occur on one.
- Threshold derivation: explain the emergence of \[ \Sigma_c(n_3)=3 \] and the privileged role of the three-dimensional stable sector.
- Large-\(q\) asymptotics: determine how the locking residual behaves as \(q\) increases.
- Universality: test projection locking beyond Heisenberg graphs.
- Full transfer closure: complete the unconditional derivation of \[ \delta_{\mathrm{pair}} \to \beta^* \] once shell selection itself is derived.
References
Jérôme Beau. Shell-Alignment from Projection Locking: A Discrete Admissibility Theorem under Born–Infeld Saturation.