Allen Orbital Lattice Framework – Pattern Field Theory

Allen, James Johan Sebastian

Allen Orbital Lattice Framework (AOL)

The Allen Orbital Lattice (AOL) is the foundational geometric and algebraic structure of Pattern Field Theory (PFT). It provides the discrete curvature framework from which quantum behaviour, cosmology, resonance coherence, and emergent physical laws arise. The AOL replaces continuous spacetime with a prime-indexed curvature lattice whose geometry permits recursive pattern formation, closure, and stability conditions defined by PFT operators.

1. Formal Definition of the Lattice

The Allen Orbital Lattice is defined as a discrete set:

A = { a_i,j ∈ ℝ² × ℤ | (i,j) ∈ ℤ², i ≥ 0, j ≥ 0 }

where each lattice element a_i,j carries:

a curvature value κ_i,j
a prime index p(n)
a phase value φ_i,j ∈ [0, 2π)
a layer coordinate ℓ ∈ ℕ corresponding to orbital depth

The geometry is hexagonal by construction. Define the primitive basis vectors:

e₁ = (1, 0), e₂ = (1/2, √3/2)

Then each lattice coordinate is:

a_i,j = i e₁ + j e₂

This basis yields a minimal energy tessellation for curvature packing and supports the Crystalline Coherence Equation (CCE).

2. Prime-Seeded Curvature Assignment

Each lattice point inherits curvature from a prime sequence:

κ_i,j = f(p(n), ℓ)

where p(n) is the n-th prime and ℓ is the orbital shell. A commonly used representation is:

κ_i,j = p(n) / (2π ℓ)

This embeds prime discreteness into geometric curvature. Higher primes correspond to higher curvature shells, allowing stacking:

κ_ℓ+1 > κ_ℓ

This rule creates the orbital structure of AOL: curvature increases radially, forming concentric shells analogous to orbital levels, but governed strictly by prime progression instead of arbitrary quantization.

3. The AOL Curvature Operator

The curvature operator acts on lattice regions:

ℒ_AOL : A → ℝ, ℒ_AOL(a_i,j) = κ_i,j

and satisfies:

Discrete curvature conservation:

∑ κ_i,j (within region R) = constant under allowed PAL transformations

Radial curvature monotonicity:

κ(r+1) ≥ κ(r)

Hexagonal symmetry constraint:

κ(i,j) = κ(i',j') if (i',j') is a symmetry transform of (i,j)

These constraints guarantee lattice-level coherence and permit higher-level physics to emerge from discrete fields rather than continuous assumptions.

4. Crystalline Coherence Equation (CCE)

A region R of the lattice is coherent if:

CCE(R) = 0

where:

CCE(R) = ∑ (Δκ)² + ∑ (Δφ)²

with Δκ and Δφ representing curvature and phase mismatches across edges in the region. The region is stable if:

CCE(R) ≤ ε

with ε a small tolerance determined by lattice depth. This equation is central to:

Quantum coherence
Entanglement stability
Cosmic web structural formation
Biological resonance stability

Allen Orbital Lattice – 3D Interactive Visualization

Rotate, zoom and inspect a simplified Allen Orbital Lattice with several orbital layers. Each sphere represents a lattice node; layers correspond to increasing curvature shells.

5. Pattern Closure on the Lattice

PFT uses the closure operator:

𝒞 : A → A

A lattice region closes if:

𝒞(R) = R

This means:

all curvature mismatches can be resolved under PAL
phase alignment constraints are satisfied
CCE(R) is minimized

Closure is a necessary condition for:

quantum identity persistence
stable wave modes
cosmic emergence after null-state

This makes closure the “admissibility rule” for physical configurations.

6. Phase Alignment Lock (PAL) on the AOL

The PAL operator:

PAL : φ → φ'

aligns phase values across curvature-compatible nodes. Formally:

PAL(φ_i,j) = φ_i,j + δ

where δ is chosen such that phase conflicts vanish under CCE. PAL must satisfy two constraints:

Lattice constraint: only edges with κ-matched curvature may align
Coherence constraint: alignment must reduce the global CCE

PAL is responsible for:

entanglement formation
superposition stability
interference patterns
neural resonance coherence

7. Orbital Layering and Depth

Orbital layers ℓ define the depth of the lattice. Each layer is assigned:

κ_ℓ = p(ℓ) / (2π ℓ)

Then the lattice curvature field becomes:

κ(i,j,ℓ) = κ_ℓ · g(i,j)

where g(i,j) encodes hexagonal geometric attenuation. These curvature layers form discrete shells analogous to orbitals, but with no relation to quantum mechanical hydrogenic orbitals. They arise strictly from prime geometry and PAL coherence constraints. This depth structure enables:

multi-scale resonance
nested emergence levels
cosmic structure replication
identity continuity across quantum modes

8. Lattice Dynamics: Differentiat Operator

The Differentiat operator defines local change:

𝔇_pft(a_i,j) = (∂κ/∂t, ∂φ/∂t)

This generalizes differential calculus into discrete pattern evolution. A configuration is stable if:

𝔇_pft(a_i,j) = 0

It is metastable if:

||𝔇_pft|| ≤ δ

and unstable if:

||𝔇_pft|| > δ

These criteria are used in QPF, cosmology, and biological pattern formation.

9. Observable Projection

Physical measurements extract only a subset of lattice information:

Π_obs : (κ, φ, ℓ) → measurable quantities

Examples:

Quantum: probability amplitude shaped by curvature volume
Cosmology: curvature → density → lensing potential
Neural: resonance → amplitude → measurable oscillation bands

10. High-Level Consequences

The AOL enables PFT to provide structural explanations for:

entanglement as lattice-coherent pattern locking
interference as PAL-phase curvature propagation
cosmic anisotropy (Planck data) from lattice-level recursion direction
fractal cosmic web (D≈2) from curvature carrier surfaces
biological coherence from resonance in multi-layer AOL depth
identity persistence from pattern closure and recursive coherence

These consequences tie QPF, cosmology, morphogenesis, and consciousness into one structural architecture.

11. Further Reading

The Allen Orbital Lattice is the mathematical heart of Pattern Field Theory and the generative foundation for coherence, emergence, symmetry, and identity in the physical world.

Pattern Field Theory