This repository contains the complete, reproducible implementation of the theoretical framework presented in the paper
“Time-Crystalline Emergence of Relational Time at a CPT-Symmetric Boundary.”
The work proposes a minimal dynamical mechanism by which physical time emerges from an atemporal, CPT-invariant vacuum. Time is not assumed as a fundamental parameter, but arises as an ordered phase of a relational degree of freedom via spontaneous time-translation symmetry breaking (STTSB), analogous to a time-crystal phase.
Timeless formulations of quantum gravity suggest that the universe may originate from a static, atemporal state. In this work, we show that:
- An atemporal CPT-symmetric vacuum can be dynamically unstable.
- A nonlinear phase equation naturally develops a stable limit cycle.
- This limit cycle defines a robust relational clock, without introducing external time.
- The resulting phase exhibits properties directly analogous to classical time crystals.
Time therefore emerges as a non-equilibrium dynamical phase, not as a background structure.
This project is fully reproducible.
- All figures appearing in the paper are generated directly from the scripts provided in
src/. - No proprietary software or hidden steps are required.
- The numerical integration uses a minimal, transparent implementation of a nonlinear oscillator admitting a stable limit cycle.
- The LaTeX manuscript compiles without external bibliography files.
To reproduce the main phase-portrait figure:
python src/make_phase_portrait_emergent_time.py