Mathematical Implications of Non- Homogenious Para-Topological Convergences Between Orthogonal Unbridged n-Spaces I

Investigated is a hypothetical mathematical framework describing a dynamical crossing betweentwo otherwise orthogonal and causally unbridged (n)-dimensional spaces or spacetimes, interpretedhere as distinct universes. The crossing is represented by a localized, time-dependent kernel thattemporarily modifies the effective mode spectrum of the coupled system.The central object is a non-homogeneous para-topological crossing kernel which induces a transientmixing between the two spaces without requiring a permanent geometric bridge. In an effectivefield description, the crossing may generate a temporarily negative effective squared frequency,[ ω 2 k ( t )<0 ] , thereby producing a transient tachyonic regime. Shown is, that this does not necessarilyimply a persistent tachyonic degree of freedom after the crossing. Instead, the process can berepresented by a Bogoliubov transformation characterized by coefficients ( α k ) and ( β k ) .Subsequently introduced ergo is an effective entropy/information sector and investigated is itsfeedback on the crossing dynamics. The resulting system admits stabilizing, destabilizing andpotentially symmetry-breaking regimes. In particular, entropy transport between the two universescan be distinguished mathematically from entropy production, allowing the possibility ofasymmetric post-crossing states even when the initial configuration is symmetric. May be, that thephenomenon of matter transportation or of radiation energy could occur between the two spacemanifolds over the crossing zone (“skip effect“) but this special situation is discussed in a laterpaper. Two similiar crossing zones between the same universes then will be defined as a„congruence“.The present work however concentrates on and constitutes a phenomenological mathematicalframework rather than a claim of an established physical theory. Its principal purpose is to identifythe necessary dynamical structures, stability criteria and falsifiable mathematical consequences ofsuch a crossing scenario. Some more treatises to this theme discussing intesively these sort ofphenomena, will follow.