Pattern Field Theory

1. 1D Hex Harmonic to 2D Lattice to 3D Supercell

The seed is a one-dimensional hex harmonic: a six-phase ratio code that can be rotated into six 60-degree projections. Closure of those projections yields a 2D AOL plane. A purely planar coherence regime supports stable propagation but limits differentiation. The symmetry break is modeled as opening a new axis that releases stored potential as a 3D hexagonal supercell. The transition is a structural phase change in admissible degrees of freedom, not an explosion into pre-existing space.

2. Perspective Inversion and Fractal Depth Mapping

When viewed end-on, recursive scale can appear as distance. When viewed side-on, it resolves as stacked windows (rings) that are scale-disjoint. The angle of fractal depth describes how small per-level realignments accumulate into apparent curvature over long baselines. In an AOL basis, the 60-degree edge and its complementary 30-degree projection form a reciprocal mapping between scale progression and spatial projection.

3. Stacked Universes and the Metacontinuum

Each lattice level is treated as a self-contained domain with its own local metric regime. Between levels is the metacontinuum: a boundary of relation rather than distance, where unit-meaning collapses and only invariant relations persist. A pi-particle (hex coherence seed) is defined as the minimal closed loop capable of transiting the boundary and unfolding a new AOL sheet as a new fractal window. Levels are scale-disjoint and mutually invisible to local instrumentation because measurement primitives are anchored to the active sheet.

4. Eight-Vector Registry of Stored Frames

Once volumetric, the field supports an eight-vector projection set: six lateral projections (planar basis) and a vertical coupling pair (up and down). This registry is treated as a minimal closure capable of storing a complete cycle of field states as frame orientations of the same universe. The registry organizes how states are related without requiring an external embedding volume.

5. Flatness as a Built-In Consequence

Planar hex propagation produces a near-zero global mean curvature within each active sheet. Curvature belonging to adjacent tiers does not project into the local sheet, so observed near-flatness follows from sheet coherence rather than fine tuning.

6. Testable Implications

• Fractal boundary convergence: Recursive refinement of boundary geometry should produce convergent intensification profiles at edges and corners consistent with AOL sub-lattice structure.
• Scale-covariant drift signatures: Anomalies that track with logarithmic scale progression rather than linear distance in lensing residuals, background correlations, or structure growth support the depth-angle mapping.
• Eight-vector anisotropy: Weak octadic symmetries in large-scale correlation structure may indicate an underlying supercell registry rather than isotropic continuous degrees of freedom.

7. Definitions

• Allen Orbital Lattice (AOL): The 2D hex-based substrate plane used as the primary coherence and address basis for local structure.
• Self-similar sub-lattice: A refinement where each boundary hex resolves into a smaller AOL preserving ratio relations at a new resolution level.
• Metacontinuum: A boundary of relation where unit-meaning collapses and only invariant relations persist during cross-tier recursion.
• Pi-particle (hex coherence seed): Minimal closed coherence loop defined to transit the metacontinuum and unfold a new lattice window.
• Eight-vector registry: Six lateral projections plus an up-down coupling pair, forming the minimal volumetric closure for stored frame cycles.

8. Conclusion

Expansion is modeled as growth by opening interior degrees of freedom across scale-disjoint sheets. The ignition event is the symmetry break from 2D planarity to 3D supercell activity, after which an eight-vector registry organizes stored frame cycles. Flatness follows from planar coherence, and cross-tier invisibility follows from scale-disjoint measurement anchoring.