Introduction

Pattern Field Theory (PFT) and the Allen Orbital Lattice (AOL) treat coherence as a finite, geometric resource rather than an idealised, unbounded capacity. Phase Alignment Lock (PAL) is the basic structural rule that enforces this: configurations are admissible only when their phases remain sufficiently aligned relative to their prime indices on the lattice.

Several later papers in the programme depend on PAL: coherence-constrained computation, PAL-constrained complexity classes, PAL-based inference, and PAL-constrained event cascades. In each case, PAL is assumed as background structure. The present paper extracts PAL, formalises it, and provides an explicit methods document that can be used as the reference point for replication.

The goals are:

• to give a concise but complete definition of the Allen Orbital Lattice and the PAL inequality;

• to derive simple bounds and approximations that link the inequality to angular separation and capacity;

• to provide concrete algorithms for checking and enforcing PAL coherence;

• to specify replication protocols for the key quantitative results that depend on PAL.

Throughout, the focus remains on methods rather than on new conceptual claims. The intent is to reduce the PAL machinery to a set of reproducible procedures.

Allen Orbital Lattice: Minimal Recap

This section recalls only the elements of the Allen Orbital Lattice needed to state and work with PAL. Full context appears in the Pattern Field Theory foundational paper and in the CCCT document.

Geometric substrate

The Allen Orbital Lattice is a hexagonal lattice embedded in the complex plane. Let \(V \subset \mathbb{C}\) be the set of lattice vertices and \(E\) the set of edges connecting nearest neighbours. Each vertex \(v \in V\) corresponds to a point in a regular hexagonal tiling.

For the purposes of PAL, only adjacency and indexing matter. The exact lattice scale can be normalised to \(1\) without loss of generality, since PAL is defined on phases rather than on spatial coordinates.

Prime indexing

A prime-indexing map \[\sigma : V \to \mathbb{P}\] assigns to each vertex \(v \in V\) a distinct prime \(p_v = \sigma(v)\). In finite regions of the lattice, primes are assigned in increasing order along a fixed scan pattern (for example, spiral-out from the origin or row-major ordering). The specific scan pattern does not affect the PAL inequality as long as distinct vertices carry distinct primes.

In numerical experiments, a finite patch \(V_R \subset V\) of radius \(R\) is used and prime indices are truncated to the first \(\lvert V_R \rvert\) primes.

Phase labels and active sets

Time is treated as discrete: \(t \in \mathbb{Z}_{\ge 0}\). At each time \(t\), a finite active set \[S_t \subset V\] is selected together with a phase assignment \[\theta_t : S_t \to [0,2\pi).\] Vertices outside \(S_t\) are considered inactive at time \(t\) and do not contribute to coherence constraints. The dynamics of a particular model (computational, inference, event cascades) specify how \((S_t,\theta_t)\) evolves to \((S_{t+1},\theta_{t+1})\).

PAL is a constraint on each instantaneous configuration \((S_t,\theta_t)\).

Phase Alignment Lock: Definition and Geometry

PAL-coherent configurations

PAL is defined as a pairwise cosine bound on phase differences, weighted by prime indices.

The right-hand side is strictly greater than \(-1\) for all finite primes. PAL therefore forbids antipodal differences (\(\pi\) phase flips) and enforces tight clustering for small primes.

Angular radius approximation

For small angular differences \(\Delta\theta\), the cosine admits the usual quadratic approximation: \[\cos(\Delta\theta) \approx 1 - \frac{\Delta\theta^2}{2}.\] Substituting this into ([eq:PAL]) yields \[1 - \frac{\Delta\theta^2}{2} \;\gtrsim\; 1 - \frac{1}{p_u p_v},\] and therefore \[\begin{equation} \Delta\theta^2 \;\lesssim\; \frac{2}{p_u p_v}. \label{eq:PAL-radius-approx} \end{equation}\]

This motivates the PAL angular radius \[\begin{equation} r_{uv} \;=\; \sqrt{\frac{2}{p_u p_v}}, \label{eq:PAL-radius} \end{equation}\] which acts as an approximate upper bound on the allowed phase separation between vertices \(u\) and \(v\) inside a PAL-coherent set.

In many numerical applications, inequality [eq:PAL-radius-approx] is used as a fast filter before the exact cosine computation.

PAL balls and coherence capacity

For a fixed configuration, choose a reference vertex \(u_0 \in S_t\) and define the centred phase differences \[\Delta\theta_t(v) \;=\; \theta_t(v) - \theta_t(u_0) \mod 2\pi.\] PAL coherence implies that all \(\Delta\theta_t(v)\) lie inside the intersection of pairwise angular windows determined by ([eq:PAL-radius]). This intersection can be visualised as a PAL ball in the circle.

The coherence capacity of a region of the lattice is then the maximal cardinality of a set \(S_t\) for which such a PAL ball exists. Later papers refine this concept into explicit bounds for computational and inferential models. This methods paper focuses on how to compute such capacities numerically.

Algorithmic PAL Checking

This section specifies reference algorithms for verifying PAL coherence in a finite active set. They are written in a way that can be translated directly into code in any imperative language.

Naive pairwise check

Let \(S\) be a finite active set with \(n = \lvert S \rvert\).

Input

• An array of vertices \([v_1,\dots,v_n]\).

• Phase angles \(\theta[i] \in [0,2\pi)\) for \(i=1,\dots,n\).

• Prime indices \(p[i] \in \mathbb{P}\) associated to each vertex.

Output

• Boolean flag is_PAL_coherent.

Procedure

1. For \(i\) from \(1\) to \(n\):

   1. For \(j\) from \(i+1\) to \(n\):

      1. Compute \(\Delta\theta = \theta[i]-\theta[j]\).

      2. Wrap \(\Delta\theta\) into \((-\pi,\pi]\) to avoid large jumps.

      3. Compute \(c = \cos(\Delta\theta)\).

      4. Compute the PAL bound \(b = 1 - \frac{1}{p[i]\cdot p[j]}\).

      5. If \(c < b\), return false.

2. Return true.

This algorithm runs in \(O(n^2)\) time. For the small and medium-sized active sets used in reference experiments, this cost is acceptable and keeps the definition transparent.

Radius-based prefilter

To accelerate the check for larger \(n\), a radius-based prefilter can be added using the approximation ([eq:PAL-radius-approx]).

1. Precompute \(r[i,j] = \sqrt{2/(p[i]\,p[j])}\) for all pairs.

2. For each pair \((i,j)\), if \(|\Delta\theta| > r[i,j]\), immediately flag a PAL violation without computing the cosine.

This approach replaces some trigonometric calls with cheaper comparisons at the expense of precomputation. It remains exactly faithful to the PAL inequality if the cosine step is still used as the final arbiter; the radius check is then a sufficient condition for violation.

Incremental PAL maintenance

Many dynamical models update a configuration by small modifications: adding or removing vertices, or nudging phases by small increments. In these settings, PAL can be maintained incrementally.

Phase update

Suppose a single vertex \(v_k\) changes its phase from \(\theta[k]\) to \(\theta'[k] = \theta[k] + \delta\). PAL can be checked by:

1. For all \(j \neq k\):

   1. Compute new \(\Delta\theta' = \theta'[k] - \theta[j]\).

   2. Wrap into \((-\pi,\pi]\).

   3. If \(\cos(\Delta\theta') < 1 - 1/(p[k]p[j])\), PAL is broken.

Vertex insertion

To insert a new vertex \(v_{\text{new}}\) with phase \(\theta_{\text{new}}\):

1. For each existing vertex \(v_j\):

   1. Compute \(\Delta\theta = \theta_{\text{new}} - \theta[j]\).

   2. Check PAL inequality.

2. If all pass, add \(v_{\text{new}}\) to \(S\).

These incremental rules are used in the replication protocols for event cascades and coherence-limited computations.

PAL-Constrained Dynamics

PAL does not prescribe a particular dynamics; it restricts the allowable configurations at each time step. This section describes a simple reference dynamics used for replication experiments.

Local update rule

Consider a lattice region \(V_R\) with active set \(S_t\) and phases \(\theta_t\). A generic local update rule has the form \[\begin{equation} \theta_{t+1}(v) = \theta_t(v) + F\bigl(v, S_t, \theta_t\bigr) \mod 2\pi \quad\text{for } v \in S_t, \label{eq:generic-update} \end{equation}\] where \(F\) is a function of local neighbourhood data (for example, neighbour phases along edges in \(E\)).

A reference choice is: \[\begin{equation} F(v,S_t,\theta_t) = \lambda \sum_{u \sim v} w_{uv} \sin\bigl(\theta_t(u)-\theta_t(v)\bigr), \label{eq:Kuramoto-like} \end{equation}\] where the sum is over neighbours \(u\) of \(v\), \(\lambda\) is a coupling constant, and \(w_{uv}\) are weights (often \(1\)). This is a Kuramoto-type alignment rule adapted to the lattice.

PAL-enforced update

PAL is enforced by rejecting updates that would exit the PAL-coherent region.

PAL-enforced step

1. Given \((S_t,\theta_t)\) PAL-coherent, compute tentative \(\theta_{t+1}\) via ([eq:generic-update]).

2. Check PAL coherence of \((S_t,\theta_{t+1})\) using the pairwise or incremental algorithm.

3. If PAL holds, accept the update.

4. If PAL fails, rescale the increment: \[\theta_{t+1}(v) = \theta_t(v) + \alpha\, F(v,S_t,\theta_t)\] with \(0 < \alpha < 1\) chosen (for example via binary search) to be the largest factor that preserves PAL coherence.

This procedure defines a PAL-constrained flow that never leaves the admissible region of phase space.

Coherence time and PAL-lock convergence

Given an initial configuration \((S_0,\theta_0)\) that may not be PAL-coherent, one can apply the PAL-enforced update until a PAL-coherent configuration is reached (or until convergence fails).

PAL-lock times are a key observable in numerical experiments. They characterise how quickly generic configurations converge into coherence funnels on the lattice.

Replication Protocols

This section specifies concrete replication tasks. Each task is defined so that independent implementations can compare outputs and verify consistency.

Replication 1: PAL inequality validation

Objective. Verify that the PAL inequality behaves as expected across a range of prime indices and random phase assignments.

Steps.

1. Choose a finite set of primes \(P = \{p_1,\dots,p_m\}\).

2. Generate random pairs \((p_i,p_j)\) and random phase differences \(\Delta\theta \in (-\pi,\pi]\).

3. For each triple \((p_i,p_j,\Delta\theta)\), record whether \[\cos(\Delta\theta) \ge 1 - \frac{1}{p_i p_j}\] holds.

4. Estimate the empirical boundary in \((p_i p_j,\Delta\theta)\) space where PAL transitions from satisfied to violated.

5. Confirm that the empirical transition tracks the theoretical radius \(r_{ij} = \sqrt{2/(p_ip_j)}\) within the expected approximation error.

Replication 2: Coherence capacity estimates

Objective. Estimate the maximal size of PAL-coherent sets in finite lattice patches.

Steps.

1. Fix a lattice patch \(V_R\) and assign primes to its vertices.

2. For each target cardinality \(k\):

   1. Sample random subsets \(S \subset V_R\) of size \(k\).

   2. For each \(S\), assign random phases and project them into a small angular interval around a randomly chosen centre.

   3. Check PAL coherence.

3. For each \(k\), estimate the fraction of random configurations that are PAL-coherent.

4. Identify the threshold \(k^\ast\) beyond which PAL coherence becomes rare.

This experiment provides empirical capacity estimates that can be compared to analytic bounds in later papers.

Replication 3: PAL-constrained computation toy model

Objective. Reproduce a simplified version of the coherence-limited computation results used in CCCT.

Model sketch.

• Encode binary strings as phase patterns on a fixed subset \(S\) of the lattice.

• Define a small set of local update rules that implement logical gates while preserving PAL coherence when possible.

• Measure the maximal input length for which the computation remains PAL-coherent along the entire trajectory.

Steps.

1. Choose a set of \(n\) vertices and assign primes.

2. Define two base phases to represent logical \(0\) and \(1\), both inside a PAL ball determined by the smallest primes in the set.

3. Construct simple circuits (for example, parity or majority) and simulate their execution as PAL-constrained dynamics.

4. Record when PAL coherence breaks as input size or circuit depth grows.

These toy models provide intuition for the full coherence-constrained hierarchy used in the complexity results.

Replication 4: PAL in Bayesian Coherence Collapse

Objective. Reproduce the PAL-based constraints on hypothesis sets in Bayesian Coherence Collapse.

Steps.

1. Implement PAL-coherent hypothesis sets \(S\) using the same pairwise inequality as for active sets.

2. For small \(n\), enumerate all possible subsets \(S\) within a fixed lattice patch and test PAL coherence.

3. Record the maximal cardinality of PAL-coherent hypothesis sets as a function of the rank or prime index range.

4. Compare these empirical capacities to the analytic bounds stated in the BCC paper.

Even small-scale experiments already show the constraint that PAL imposes on hypothesis multiplicity.

Physical and Analogue Approximations

While PAL originates as a formal constraint on a mathematical lattice, several physical systems approximate its behaviour. This section sketches such systems to guide experimental exploration.

Coupled oscillator arrays

Arrays of coupled oscillators (mechanical, electrical, or optical) with controllable coupling strengths can be tuned so that their phase differences remain within PAL-like bounds for a range of parameters. By mapping oscillators to lattice vertices and coupling strengths to effective prime weights, one can design experimental platforms that approximate PAL-constrained dynamics.

Phase-locked loop architectures

Phase-locked loops (PLLs) and related phase alignment circuits in electronics already implement practical versions of phase locking. By arranging PLLs in hexagonal networks and modulating their lock ranges according to prime-like weights, engineers can construct circuits that mimic PAL-enforced coherence regions.

Neural phase alignment

In neural systems, low-frequency phase alignment relative to external rhythms and internal patterns can be studied as a biological analogue of PAL. While the underlying substrate is not the AOL, the concept of a finite coherence capacity and phase alignment constraints may still apply at an effective level.

These analogues are not definitions; they are routes to approximate PAL in laboratory settings.

Discussion

Phase Alignment Lock is the structural rule that turns the Allen Orbital Lattice from a static geometric background into a coherence-limited medium. Once PAL is imposed, several consequences follow:

• not every configuration of phases is allowed;

• not every computational trajectory remains admissible;

• not every hypothesis family can coexist in a single coherent state.

Later papers in the PFT series translate these consequences into concrete results in complexity theory, inference, and event cascades. This methods paper exists to make replication straightforward. Any researcher can work from the definitions and algorithms here, implement independent code, and verify that the same constraints appear.

Document Timestamp and Provenance

This document is part of Pattern Field Theory (PFT) and the Allen Orbital Lattice (AOL). It defines the Phase Alignment Lock (PAL) constraint and specifies methods and replication procedures used by subsequent papers in the series.

© 2025 James Johan Sebastian Allen — Pattern Field Theory — patternfieldtheory.com. All rights reserved.

Pattern Field Theory (PFT) and related marks are claimed trademarks. This work is licensed under the Pattern Field Theory Licensing framework (PFTL). Any research, derivative work, or commercial use requires an explicit license from the author.