Invariant Temporal Ordering Framework (ITOF) V13: Structure-Dependent Measurable Realization and Residual Physical Response

Invariant Temporal Ordering Framework (ITOF): Structure-Dependent Measurable Realization and Residual Physical Response presents a foundational operational framework distinguishing invariant temporal ordering from measurable physical evolution. Within the framework, temporal ordering is represented as an invariant ontological succession structure rather than as a measurable dynamical substance, energetic flow, or directly deformable physical entity. The framework introduces a comparative residual formalism through which experimentally observable asymmetries are interpreted operationally as structure-dependent physical realization emerging from interaction-dependent system response under shared invariant ordered succession. Measurable evolution is therefore treated as belonging to physical systems and their coupled physical realization rather than to measurable variation of temporal ontology itself. The manuscript develops: • an invariant ordered succession ontology,• a comparative measurable residual framework,• structure-dependent measurable realization,• nonlinear measurable response extensions,• interaction-dependent operational interpretation,• effective Hamiltonian response pathways,• and experimentally oriented pressure-response realizations. The framework further develops nonlinear residual structure through higher-order measurable response behavior, suggesting that measurable realization need not remain universally proportional across structurally distinct physical systems under increasing environmental interaction influence. Operationally, measurable asymmetry is represented through comparative residual structure: \[\delta_{A|B}\] together with nonlinear pressure-response realization: \[\delta_{A|B}(P)\approx(\beta_{AP}-\beta_{BP})\Delta P+\frac12(\gamma_{AP}-\gamma_{BP})(\Delta P)^2\] where measurable response coefficients are interpreted as effective structure-dependent descriptors associated with interaction-dependent physical response. The current Version 13 formulation further develops effective interaction-dependent response interpretation through controlled Hamiltonian realization structure while extending the framework toward experimentally interpretable comparative realization pathways. This Version 13 preprint represents a refined and substantially expanded operational formulation of ITOF integrating invariant temporal ontology, measurable residual formalism, nonlinear comparative realization structure, experimental operational interpretation, and controlled future derivational direction into a unified theoretical framework.