Lichen-Universe-Unified-V2
π-Time : Universal Cosmic Clock
Le Standard Temporel Fractal
“Time is not a line. It is a coordinate in the circle.”
π-Time replaces the arbitrary linear second with the π-Cycle. It acts as the heartbeat of the Lichen Universe, providing a self-verifying timestamp format where every moment is a specific digit in the infinite sequence of $\pi$.
🕰️ The Format
\(\pi[CYCLE].[SUB].[POSITION].[DIGIT]\)
Example: π1234.057.890321.4
🌌 Core Features
- Universal Anchoring: Time based on math constants, not political timezones.
- Proof of Time: The final digit acts as a checksum. If the digit doesn’t match the position in $\pi$, the timestamp is fake.
- Lichen Native: Used to synchronize FC-496 cells across the distributed system.
📂 Contents
docs/specification.md: Technical specs and BBP algorithm integration.poc/pi_clock.js: A working JS prototype of the clock and validator.FORMULAS.md: The mathematical definitions of the π-Cycle.
V2.0
Pi-Time : Mathematical Index
Scope: Temporal Indexing & Spiral Mapping.
1. The Spiral Projection (Phyllotaxis)
How we map linear time $t$ into a 2D Holographic Disk.
📐 LaTeX
\[\theta_n = n \times \Psi_{gold} = n \times 2\pi(1 - \frac{1}{\varphi})\] \[r_n = c \sqrt{n}\]Where:
- $n$ is the $\pi$-Index (derived from time).
- $\Psi_{gold}$ is the Golden Angle ($\approx 137.508^\circ$).
- $c$ is a scaling factor.
2. The $\pi$-Index Function
The conversion from linear Unix time to the immutable $\pi$ sequence.
📐 LaTeX
\[I_{\pi}(t) = \lfloor t \cdot \pi \cdot 10^k \rfloor \pmod{L_{max}}\](Simplified for simulation. In production, this maps to the exact offset in the Chudnovsky algorithm stream).
3. Stability Metric
Determining if a moment in time is “structurally sound” for write operations.