Abstract
This paper introduces Invariant-Guided Dimensional Dynamics (IGDD), a rule-based framework for bodies whose dimensional and graded expressions evolve under admissible operations. It presumes an infinite-dimensional, non-degenerate, spatially unbounded universe in which every physical body has nonzero expression along all constitutive dimensions. Accessible bodies may nonetheless have finite or infinite representations produced by projection, section, embedding, filtering, thresholding, measurement, transformation, or related operations. Numerical dimension and problem-specific grade are treated separately, so a body may change dimensional stratum, grade stratum, or both. State descriptors combine invariants with variable attributes and guide sequential or simultaneous schedules subject to compatibility and conflict rules. IGDD supports fixed or adaptive, finite, condition-terminated, and infinite trajectories, including unity, recurrence, periodicity, divergence, branching, and domain-specific outcomes. Worked constructions illustrate same- and cross-stratum operations, recurrent and divergent simplex dynamics, and a geometric-biophysical trajectory linking polytope geometry with biological, physical, and chemical conditions. Meta-bodies provide higher-dimensional envelopes for families of structures or models, while dimensional revealing occurs when a suppressed expression crosses an observational or operational threshold. IGDD is intended to organize relations among bodies, dimensions, grades, operations, representations, and trajectories across multiple scientific and engineering domains.