Topological Phase Averaging and Vorticity Regularity on a π-Length Toroid: A Geometric Collapse Framework

This deposit contains an exploratory paper, a companion note, and an interactive visualization model investigating how domain topology can enforce vorticity regularity through phase averaging.

The main paper shows that on a toroid of circumference L = π, the twelve-phase curvature rosette from the 9-form/12-phase collapse framework tiles the domain commensurately (exactly six cycles per revolution). This forces the modulation integral to vanish identically over every toroidal circuit, preventing the sustained phase locking required for finite-time blow-up. Numerical demonstrations using the continuous Möbius collapse system and spectral CLM/De Gregorio simulations on the π-domain confirm the regularity mechanism.

The companion note documents the interactive React/Three.js/D3 visualization (rtm_exploration.jsx), mapping each 3D layer and analysis tab to its source — distinguishing elements grounded in the main paper from exploratory structures originating in the broader Structured Recursive Geometry (SRG) research programme. The SRG toy model, which informed several of the visualization's bridge layers, is available separately (https://zenodo.org/records/18215874).

Files included:

• toroidal_ns_paper.pdf — Main paper (9 pages)
• companion_note.pdf — Companion note documenting the visualization model (4 pages)
• rtm_exploration.jsx — Interactive visualization source (~2400 lines, React/Three.js/D3)