Appendix A: Mathematical Constants
This appendix provides the complete mathematical constants that govern the Motion Calendar. These values are structural necessities arising from the geometry of motion itself.
The Fundamental Identity
The thermal quantum K is defined by the exact identity:
K × φ² = 4
Where φ = (1 + √5) / 2 is the golden ratio.
Precise Values
Golden Ratio (φ): 1.618033988749895
Thermal Quantum (K): 1.527864045000421
The Six Motion Thresholds
Each motion function operates at threshold 1/φⁿ:
| Function | Symbol | Threshold | Value |
|---|---|---|---|
| Heat | Σ | 1/φ¹ | 0.6180339887 |
| Polarity | +/− | 1/φ² | 0.3819660113 |
| Existence | δ | 1/φ³ | 0.2360679775 |
| Righteousness | R | 1/φ⁴ | 0.1458980338 |
| Order | Q | 1/φ⁵ | 0.0901699437 |
| Movement | Lin | 1/φ⁶ | 0.0557280900 |
| Sum (= K) | Σ(1/φⁿ) | 1.5278640450 |
The Julia Set Connection
|c| < 1/4 → connected Julia set (spine exists)
|c| ≥ 1/4 → Julia dust (disconnected)
The Existence threshold (≈ 0.236) falls below 1/4 = 0.25, ensuring motion capable of persistence forms connected structures.
The Entropic Constants
12 — movement constant (6 directions × 2 frames)
6 — divisor count; number of motion functions
−1/12 — entropic bound; Ramanujan's ζ(−1)
i — imaginary unit; rotational capacity
φ — golden ratio; self-similar scaling