VREL — Vertical Resonance Extraction Lens
A SARG lens for extracting structural invariants from any substrate.
VREL is the primary lens in SARG. It reads a substrate's structure and extracts the features that persist under mirror‑axis transformations — vertical, horizontal, and dual. These invariants become the substrate's resonance fingerprint, which SARG then maps to universal anchors.
VREL is substrate‑agnostic: it operates identically whether the input is a glyph, a crystal lattice, a harmonic series, or a planetary envelope.
1. What VREL Extracts#
VREL applies mirror‑axis analysis to a substrate and returns three invariant classes:
1.1 Vertical Invariants#
Features that remain unchanged when the substrate is reflected across a vertical axis.
- In glyphs: letters like A, H, M, T, U, V, W, X, Y
- In geometry: shapes with bilateral vertical symmetry
- In crystals: planes of vertical mirror symmetry
- In symbolic substrates: operators or tokens with left–right equivalence
1.2 Horizontal Invariants#
Features that remain unchanged when the substrate is reflected across a horizontal axis.
- In glyphs: letters like B, C, D, E, H, K, O, X
- In geometry: shapes with bilateral horizontal symmetry
- In acoustic substrates: waveforms with top–bottom amplitude symmetry
- In biological substrates: dorsal–ventral mirror planes
1.3 Dual Invariants#
Features that persist under both vertical and horizontal reflection. These are the most structurally stable elements in any substrate.
- In glyphs: H, I, O, X
- In geometry: circles, squares, regular polygons with even‑order symmetry
- In crystals: highly symmetric unit cells
- In cosmological substrates: isotropic structures (spherical shells, uniform fields)
Dual invariants are the strongest candidates for resonance anchor mapping.
2. How VREL Behaves Across Substrates#
VREL adapts its extraction logic to the substrate's native structure. The lens itself does not change — the interpretation of "axis" changes.
| Substrate Type | Vertical Axis | Horizontal Axis | What VREL Reveals |
|---|---|---|---|
| Linguistic | Left–right midline of a glyph | Top–bottom midline of a glyph | Stroke families and symmetry classes |
| Geometric | Primary symmetry axis | Secondary symmetry axis | Axes, planes, and symmetry orders |
| Acoustic | Time‑axis symmetry of a waveform | Amplitude‑axis symmetry | Harmonic stability and phase coherence |
| Symbolic | Relational left–right equivalence | Relational top–bottom equivalence | Structural roles and operator symmetry |
| Biological | Sagittal plane | Transverse plane | Body‑plan invariants and mirror organs |
| Cosmological | Polar axis | Equatorial plane | Shell symmetries and radial invariants |
| Lostational | Visible‑side axis | Inversion‑side axis | Dimensional drift and resonance shells |
The same lens produces the same invariant types (vertical, horizontal, dual) regardless of domain. Only the substrate determines what those invariants are.
3. Invariants Produced by VREL#
Every VREL extraction produces a structured invariant set:
"invariants": {
"vertical": ["A", "H", "I", "M", "O", "T", "U", "V", "W", "X", "Y"],
"horizontal": ["B", "C", "D", "E", "H", "I", "K", "O", "X"],
"dual": ["H", "I", "O", "X"]
}