Scale Covariance and the Metacontinuum
Date: 2025-10-08
Abstract. We propose that cosmic “expansion” is not a ballooning of an external volume but the stepwise opening of interior degrees of freedom across fractal levels of a hexagonal field. A one-dimensional hex harmonic seeds a 2D Allen Orbital Lattice (AOL) plane; symmetry breaking ignites a 3D hexagonal supercell (the Big-Bang moment), after which dynamics are stored and related through an eight-vector meta-frame (a tesseract-like registry of frames). A boundaryless metacontinuum mediates recursion between stacked, scale-disjoint universes. “Flatness” emerges naturally as planar coherence within each sheet.
1. From 1D Hex Harmonic → 2D Lattice → 3D Supercell
The seed is a one-dimensional hexagonal harmonic (a six-phase ratio code). Rotation into six 60° projections closes a 2D AOL plane. Persistent planarity cannot differentiate; symmetry breaks by opening a new axis, releasing stored potential as a 3D hexagonal supercell. This dimensional transition—not an explosion into space—constitutes the cosmological ignition.
2. Perspective Inversion and the Angle of Fractal Depth
Viewed end-on, recursive scale reads as “distance”; viewed side-on, it reveals stacked windows (rings). The angle of fractal depth captures how small per-level realignments accumulate into apparent curvature over long ranges. In the AOL basis, the 60° hex edge and its complementary 30° projection form the reciprocal pair that maps scale to space.
3. Stacked Universes and the Metacontinuum
Each lattice level is a self-contained domain (“universe”) with its own metric. Between levels lies a metacontinuum—a dimensionless boundary where units collapse and only relations persist. A hexagonal π-particle (a coherence seed) transits this boundary and unfolds a new AOL sheet: a new fractal window. Levels are scale-disjoint and mutually invisible to local instrumentation.
4. Tesseract Frames: Eight Projections as Stored States
Once volumetric, the field supports eight projection vectors: six lateral (in-plane) and two vertical (up/down). These define a tesseract-like registry of stored frames—orientations of the same universe rather than new universes. The eight-vector closure is the minimal supercell capable of holding a full cycle of field states.
5. Flatness as a Built-in Consequence
Planar hex propagation makes the global mean curvature of each universe’s sheet approximately zero. Curvature belonging to adjacent tiers does not project into the local sheet, explaining observed near-flatness without fine-tuning.
6. Testable Implications
- Fractal field convergence: Recursive refinement of boundary geometry should converge physical fields (potential, charge, stress) in a manner analogous to mesh subdivision—predicting edge/corner intensification profiles consistent with AOL sub-lattices.
- Scale-covariant drifts: Small anomalies that track with logarithmic scale (rather than linear distance) in lensing, background correlations, or structure growth indicate the angle-of-depth mapping.
- Eight-vector anisotropy: Weak, octadic symmetries in large-scale correlations could reveal the supercell registry.
7. Definitions (Reader Primer)
- Self-similar sub-lattice: Each boundary hex resolves into a miniature AOL, preserving ratios at new resolution levels.
- Metacontinuum: A boundary of relation, not of distance; a dimensionless junction where frames hand off.
- π-particle (hex): Minimal hexagonal coherence loop that seeds a new lattice window.
8. Conclusion
“Expansion” is the in-folding of new degrees of freedom—growth by enlarging through minimization. The Big Bang marks the unique symmetry break from 2D coherence to 3D activity; thereafter, an eight-vector registry organizes the universe’s frames. Flatness, invisibility between tiers, and fractal convergence of fields follow naturally.
References
- H. Chen et al., “Fractal charge distribution on closed surfaces generated by triangular mesh subdivision,” Scientific Reports (2025).
- Fractal, Wikipedia, accessed 2025-10-08.