Preikaphian Framework Model: A Geometric-Computational System for Multi-Domain Transitions

The Preikaphian framework defines a geometric-computational system for representingand evaluating transitions across disjoint domains. Conventional mathematical andphysical formalisms often encounter discontinuities, singularities, or loss of identitywhen moving between incompatible coordinate systems. In contrast, the presentframework embeds multiple three-dimensional domains within a unified four-dimensional structure connected through a hierarchy of zero-valued transportpathways.State evolution within this system is governed by deterministic operators and a cost-based selection mechanism, enabling structured transitions that preserve identityacross domains. From these rules, the framework produces non-trivial emergentbehavior, including constrained transition graphs, asymmetry in state evolution,effective dimensional reduction, and persistent configurations arising from an inverserelationship between transition cost and accessible degrees of freedom.Interpreted computationally, the framework enables continuous evaluation acrossdomains while replacing singularities with structured transport pathways within theframework. Interpreted geometrically, it suggests a basis for structural analogs tophysical phenomena, including alignment-driven attractor-like behavior and domain-dependent cascade dynamics. These interpretations are presented as correspondencesrather than equivalences.The framework yields a set of testable implications, including constrained transitionpathways, asymmetric state distributions, and persistence of high-cost configurations.While the system is internally consistent, it does not yet provide direct mappings toestablished mathematical formalisms or quantitative physical predictions.Accordingly, the Preikaphian framework is best understood as a structured, partiallyformalized system intended to support further mathematical development andexploration of cross-domain relationships. This work builds upon the previously definedJello Math (JM) framework.