Mathematics of Hyper Movement on An Energentic Lemniscate: Formalizing the Upper Dimensionality of the Trawin Zero Point

Description This paper formalizes the higher-dimensional geometry, temporal-strain memory, and light–dark exterior transfer dynamics of an energetic lemniscate centered at the Trawin Zero Point (TZP). By modeling the lemniscate as a two-lobed projected flow on $S^2$, the apparent central crossing is resolved via a hidden Hopf phase motion on $S^3$. The trajectory is then recursively lifted through successive one-dimensional suspension coordinates from $S^4$ up to $S^8$. The complete internal-plus-exterior space is closed using a signed normal coordinate $\Xi_{LD}^1$, where the equatorial hypersphere functions as the transfer seam between pure-light ($L$) and pure-dark ($D$) oriented sectors. Additionally, this work incorporates a non-trivial temporal-strain ladder ($TS_{1-8}$) governed by hyperspherical volume-breathing dynamics and $32/27$ factorial weighting. The strain history acts directly on the seam-localized memory, breaking exact Hopf antipodality and converting the symmetric $TZP$ crossing into a hysteretic transfer seam with hysteresis-driven sector occupancy.