Preliminaries and Notation

Let \(\mathcal{G}=(V,E)\) denote the Allen Orbital Lattice graph (or cell complex), with vertex set \(V\) and adjacency relation \(E \subseteq V\times V\).

Each vertex \(v\in V\) carries a prime index \(p(v)\in\mathbb{P}\), where \(\mathbb{P}\) is the set of primes. Composite indices are inadmissible as identity anchors in this formalism.

Define the prime-index curvature load: \[\begin{equation} \kappa(v) \;:=\; \kappa(p(v)) \;=\; \log p(v), \end{equation}\] where \(\log\) is the natural logarithm (any fixed log base yields an equivalent ordering and scaling).

Let \(t\) denote a progression parameter (time-like ordering of transport steps). Let \(S(v,t)\ge 0\) denote the curvature surplus present at vertex \(v\) at step \(t\).

PAL Coherence Predicate

Phase Alignment Locking (PAL) is encoded as a local coherence predicate on adjacent vertices.

For each oriented edge \((u,v)\in E\), define a coherence function \[\begin{equation} \Phi(u,v,t) \in [0,1], \end{equation}\] interpreted as the degree of phase alignment between \(u\) and \(v\) at step \(t\).

Fix a coherence threshold \(\Phi_\star\in (0,1]\). The PAL condition at step \(t\) is: \[\begin{equation} \mathrm{PAL}(u,v,t) \;:\iff\; \Phi(u,v,t)\ge \Phi_\star . \end{equation}\]

A subset \(W\subseteq V\) is PAL-coherent at step \(t\) if \[\begin{equation} \mathrm{PAL}(u,v,t)\quad \forall (u,v)\in E \text{ with } u,v\in W. \end{equation}\]

Admissible Basin Neighborhoods

Fix a prime-index anchor vertex \(v_p\in V\) with \(p(v_p)=p\).

Let \(\mathcal{N}_r(v_p)\) be the radius-\(r\) neighborhood of \(v_p\) in graph distance, and let \[\begin{equation} \mathcal{B}_p(t)\;\subseteq\;\mathcal{N}_r(v_p) \end{equation}\] be the maximal PAL-coherent admissible basin containing \(v_p\) at step \(t\), i.e. the largest subset containing \(v_p\) such that:

1. \(\mathcal{B}_p(t)\) is PAL-coherent at step \(t\),

2. all vertices in \(\mathcal{B}_p(t)\) are prime-indexed identity support vertices,

3. \(\mathcal{B}_p(t)\) is maximal under inclusion within \(\mathcal{N}_r(v_p)\).

The radius \(r\) is an order parameter for locality; it may be tier-dependent (Section 7).

Capacity Functional

Define the basin capacity \(C_p(t)\) as the maximum integrated surplus admissible within \(\mathcal{B}_p(t)\) before PAL failure occurs.

Introduce a basin weight functional \(w:V\to \mathbb{R}_{>0}\), representing local geometric participation (degree, cell-volume, or other AOL-specific measure). The effective load budget of the basin is \[\begin{equation} L_p(t) \;:=\; \sum_{v\in \mathcal{B}_p(t)} w(v)\,\kappa(v). \end{equation}\]

Let \(\alpha\in(0,1]\) be an admissibility factor encoding that only a fraction of the geometric budget can be occupied by surplus without destabilizing PAL. Then define: \[\begin{equation} C_p(t) \;:=\; \alpha\,L_p(t). \end{equation}\]

In applications, \(w\) and \(\alpha\) are lattice-internal quantities; they are not fit to measurement space but computed from AOL neighborhood structure and PAL constraints.

Basin Capacity Inequality

Define the integrated basin surplus: \[\begin{equation} Q_p(t)\;:=\;\sum_{v\in \mathcal{B}_p(t)} S(v,t). \end{equation}\]

Theorem 1 (Basin Capacity Bound)

If the basin \(\mathcal{B}_p(t)\) is PAL-coherent at step \(t\) and \[\begin{equation} Q_p(t) \;\le\; C_p(t), \label{eq:capacitybound} \end{equation}\] then there exists at least one admissible transport update to step \(t+1\) such that:

1. the updated basin remains PAL-coherent: \(\mathcal{B}_p(t+1)\) exists and contains \(v_p\),

2. surplus remains confined to the basin neighborhood: \(Q_p(t+1)\) is finite and supported on \(\mathcal{N}_r(v_p)\),

3. no forced redistribution (spill) occurs outside \(\mathcal{B}_p(t)\).

If instead \[\begin{equation} Q_p(t) \;>\; C_p(t), \label{eq:saturation} \end{equation}\] then PAL coherence cannot be maintained for all edges internal to \(\mathcal{B}_p(t)\) under any local update that preserves admissibility, and a redistribution event must occur: \[\begin{equation} \exists\, (u,v)\in E \text{ with } u,v\in \mathcal{B}_p(t) \text{ such that } \Phi(u,v,t+1)<\Phi_\star. \end{equation}\]

Interpretation.

Inequality [eq:capacitybound] characterizes stable excitation persistence and transport within the basin. Violation [eq:saturation] defines saturation-triggered instability: curvature surplus must spill into adjacent admissible basins or decay into lower-order echoes.

Redistribution as Deterministic Spill

When [eq:saturation] holds, define the spill set at step \(t\): \[\begin{equation} \Sigma_p(t)\;:=\;\{v\in \mathcal{B}_p(t): \exists u\in \mathcal{B}_p(t)\ \text{s.t.}\ (u,v)\in E \ \wedge\ \Phi(u,v,t+1)<\Phi_\star\}. \end{equation}\]

Define the excess surplus: \[\begin{equation} X_p(t)\;:=\;Q_p(t)-C_p(t)\;>\;0. \end{equation}\]

The deterministic redistribution postulate is: \[\begin{equation} X_p(t)\;=\;\Delta Q_{\mathrm{out}}(t)\;+\;\Delta Q_{\mathrm{echo}}(t), \label{eq:redistribution} \end{equation}\] where:

• \(\Delta Q_{\mathrm{out}}(t)\) is the portion transferred to adjacent admissible basins (structural spill),

• \(\Delta Q_{\mathrm{echo}}(t)\) is the portion dissipated into lower-order curvature echoes (emission/decay signatures).

No annihilation term exists. Redistribution is conservation under admissibility.

Tier Structure and Basin Radius

Excitation tiers correspond to basin radius orders \(r\) and the resulting capacity scaling.

Let \(r_1<r_2<r_3<\dots\) denote neighborhood orders. Define tier-\(k\) basins by: \[\begin{equation} \mathcal{B}^{(k)}_p(t)\subseteq \mathcal{N}_{r_k}(v_p),\qquad C^{(k)}_p(t)=\alpha\,\sum_{v\in \mathcal{B}^{(k)}_p(t)} w(v)\kappa(v). \end{equation}\]

Deep basins (higher tiers) have larger coherent neighborhoods and therefore larger capacity. This yields a mechanical persistence ladder without imposing boundaries: \[\begin{equation} C^{(1)}_p(t)\;\le\;C^{(2)}_p(t)\;\le\;C^{(3)}_p(t)\;\le\;\cdots \end{equation}\]

Intrinsic Cutoff and Divergence Prevention

Because each basin is a finite coherent neighborhood, \(L_p(t)\) is finite and therefore \(C_p(t)\) is finite. Thus any curvature surplus accumulation is bounded by [eq:capacitybound] or forced into redistribution [eq:redistribution].

This supplies an intrinsic cutoff mechanism: \[\begin{equation} Q_p(t) \not\rightarrow \infty \quad \text{under admissible evolution.} \end{equation}\]

Hence divergence is structurally impossible inside the theory, and renormalization is replaced by basin capacity enforcement.

Conclusion

We formalized basin capacity as a prime-indexed, PAL-coherent admissibility bound on integrated curvature surplus within the Allen Orbital Lattice. The basin capacity inequality separates stable excitation persistence from saturation-triggered redistribution, providing a deterministic mechanism for decay, emission, interaction, and an intrinsic cutoff preventing divergence.

Notes for implementation. To compute \(C_p(t)\) concretely, the remaining task is to specify:

• the AOL adjacency structure and neighborhood measure,

• the coherence function \(\Phi(u,v,t)\) and threshold \(\Phi_\star\),

• the geometric weighting \(w(v)\) derived from lattice participation,

• the admissibility factor \(\alpha\) from PAL margin conditions.