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φ/ψ Constants Reference
Golden ratio-derived constants used throughout ARIA, including values and primary usage contexts.
ARIA φ/ψ Constants Reference
This document provides a reference table of all golden ratio-derived constants used throughout the ARIA system.
Base Constants
| Symbol | Name | Value | Definition |
|---|---|---|---|
| φ (PHI) | Golden Ratio | 1.618033988749895 | (1 + √5) / 2 |
| ψ (PSI) | Consciousness Constant | 1.378605249903986 | φ^(1/φ) |
| π (PI) | Pi | 3.141592653589793 | Circle constant |
| e (E) | Euler's Number | 2.718281828459045 | Natural exponential base |
| τ (TAU) | Tau | 6.283185307179586 | 2π |
Derived Constants
Powers of φ
| Expression | Value | Common Usage |
|---|---|---|
| φ² | ~2.618 | Slow timescale constant |
| φ³ | ~4.236 | Very slow timescale constant |
| 1/φ | ~0.618 | Fast timescale constant, basin centers |
| 1/φ² | ~0.382 | Very fast timescale constant, basin radius |
| 1/φ³ | ~0.236 | Ultra-fast adaptation rates |
Timescale Hierarchy (τ values)
| Name | Expression | Value | Purpose |
|---|---|---|---|
| τ_very_slow | φ³ | ~4.24 | Proto-semantic plasticity (v4) |
| τ_slow | φ² | ~2.62 | Relation matrix updates (v3) |
| τ_medium | φ | ~1.62 | SSV aggregation (v2) |
| τ_fast | 1/φ | ~0.62 | Symbol adaptation (v1) |
| τ_very_fast | 1/φ² | ~0.38 | Latent channel updates (v0) |
Usage in ARIA Layers
CFM v0–v2: Coupled Field Model
| Parameter | Expression | Value | Description |
|---|---|---|---|
| omega_global | 1/φ | ~0.618 | Global oscillator frequency |
| omega_local | φ/φ² = 1/φ | ~0.618 | Local oscillator frequency |
| basin_center_c | 1/φ | ~0.618 | Coherence basin center |
| basin_center_e | 1/φ | ~0.618 | Energy basin center |
| basin_center_s | 1 - 1/φ² | ~0.618 | Stability basin center |
| basin_radius | 1/φ² | ~0.382 | Attractor basin radius |
| basin_strength_inner | 1/φ² | ~0.382 | Inner basin strength |
| basin_strength_outer | 1/φ³ | ~0.236 | Outer basin strength |
| coherence_target | 1/φ + 0.1 | ~0.718 | Target coherence level |
| energy_target | 1/φ | ~0.618 | Target energy level |
| stability_target | 1 - 1/φ² | ~0.618 | Target stability level |
| alignment_target | 1/φ | ~0.618 | Target alignment level |
ARIA v0: Latent Concept Channels
| Parameter | Expression | Description |
|---|---|---|
| Projection weights | φ-derived | CFM → Latent projection |
| Cluster softness | 1/φ² | Attractor membership sharpness |
ARIA v1: Proto-Symbolic Layer
| Parameter | Expression | Description |
|---|---|---|
| Symbol count | 8 | φ-inspired codebook size |
| Similarity threshold | 1/φ | Activation threshold |
| Entropy smoothing | 1/φ³ | Entropy calculation smoothing |
ARIA v2: System State Vector
| Parameter | Expression | Description |
|---|---|---|
| SSV dimension | 12 | State summary vector size |
| Aggregation weights | φ-derived | Layer contribution weights |
ARIA v3: Relational Memory
| Parameter | Expression | Description |
|---|---|---|
| Relation matrix | 8×8 | Symbol pair relationships |
| Plasticity gate | 1/φ | Learning rate modulation |
| RSV dimension | 12 | Relation summary vector |
ARIA v4: Proto-Semantic Layer
| Parameter | Expression | Description |
|---|---|---|
| Code count | 16 | Semantic codebook size |
| Plasticity | 1/φ³ | Very slow learning rate |
| Pattern threshold | 1/φ² | Activation threshold |
Cross-Channel Resonance (CFM v2)
The resonance calculation uses φ-weighted contributions:
target_resonance = (
coherence_correlation * (1/φ) +
energy_correlation * (1/φ²) +
stability_correlation * (1/φ²) +
phase_coherence * (1/φ³)
) / (1/φ + 2/φ² + 1/φ³)
Attractor Basin Geometry
The 3D attractor basin uses φ-derived radii:
- Inner radius:
r_inner = basin_radius / φ - Outer radius:
r_outer = basin_radius * φ
This creates nested shells with φ-proportional spacing, ensuring smooth transitions between attraction regimes.
Mathematical Properties
The golden ratio φ has special properties that make it ideal for dynamical systems:
- Self-similarity: φ = 1 + 1/φ (recursive definition)
- Fibonacci convergence: Fₙ₊₁/Fₙ → φ as n → ∞
- Minimal resonance: Powers of φ avoid integer ratios, preventing mode-locking
- Aesthetic proportion: Appears in natural growth patterns
The consciousness constant ψ = φ^(1/φ) provides:
- Bounded self-reference: A fixed point of the operation x → φ^(1/x)
- Moderate value: ~1.38, providing balanced scaling
- Mathematical elegance: Combines φ with its own recursive structure
All constants are defined in math_consts.py as the single source of truth.