Paper XLVI - Explicit Failure Modes and Boundary Structure of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program

DOI: 10.5281/zenodo.18010647 Explicit Failure Modes and Boundary Structure of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program Paper XLVI of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We complete Phase II of the Ordered-Dynamics Reconstruction Program by mapping the explicit failure modes and boundary structure of dynamical embeddings under empirical confrontation. Conditioning on the admissibility, robustness, environmental, scale–redshift, and observable-class analyses of Papers XLI–XLV, we identify where and how embeddings fail to admit consistent parameter regions. Failures are classified structurally rather than statistically. This paper defines the empirical boundary of admissible dynamical hypothesis space and prepares the ground for synthesis and falsification in Phase III. Contents 1 Purpose and scope 2 2 Definition and taxonomy of failure modes 2 2.1 Structural versus statistical failure 2 2.2 Primary failure categories 2 2.2.1 Incompatibility failure 2 2.2.2 Overconstraint failure 2 2.2.3 Structural mismatch 2 2.2.4 Fragility failure 2 2.2.5 Non-identifiability failure 2 3 Failure localization in parameter space 3 3.1 Incremental constraint application 3 3.2 Boundary intersection structure 3 3.3 Dimensional collapse 3 3.4 Constraint-order dependence 3 3.5 Registry traceability 3 4 Observable-driven failure modes 3 4.1 Clock-dominated observables 3 4.2 Propagation-dominated observables 3 4.3 Hybrid observables 3 4.4 Singleversus multi-class failures 3 5 Environmental and scale–redshift failure structure 3 5.1 Partition-invariant failures 4 5.2 Contingent failures 4 5.3 Cross-partition consistency 4 6 Global boundary structure of admissible dynamical embeddings 4 6.1 Robust admissible core 4 6.2 Marginal boundary regions 4 6.3 Excluded regions 4 6.4 Boundary topology 4 6.5 Irreversibility of exclusion 4 7 Implications for Phase III 4 8 Conclusion 4 Contents – 1 – 1 Purpose and scope This paper maps the explicit failure modes of dynamical embeddings within the OrderedDynamics Reconstruction Program (ODRP). Where Papers XLI–XLV identified admissible embeddings and tested their robustness under multiple orthogonal axes, the present work characterizes the boundaries beyond which embeddings fail. The goal is not to prefer one embedding over another, but to make failure conditions explicit, reproducible, and classification-based. No new embeddings, datasets, or inference machinery are introduced. 2 Definition and taxonomy of failure modes Definition 1 (Admissibility failure).An admissibility failure occurs when a dynamical embedding admits no non-empty parameter region consistent with all imposed empirical constraints under the fixed inference protocol of the ODRP. Failure is treated as a structural property rather than a statistical one. 2.1 Structural versus statistical failure The following do not constitute failure in the ODRP sense: •poor goodness-of-fit, •broad or weakly constrained posteriors, •preference under information criteria, •local tension between subsets of observables. Failure refers exclusively to global inconsistency. 2.2 Primary failure categories Failures are classified into the following dominant types: 2.2.1 Incompatibility failure Mutually inconsistent constraints arising from different observable classes or partitions. 2.2.2 Overconstraint failure Collapse of admissibility due to aggregation of individually compatible constraints. 2.2.3 Structural mismatch Intrinsic inability of the embedding to map into the reconstructed operator space. 2.2.4 Fragility failure Admissibility existing only under finely tuned partitioning or parameter choice. 2.2.5 Non-identifiability failure Apparent exclusion caused by loss of identifiability rather than true inconsistency. – 2 – 3 Failure localization in parameter space Failure localization identifies where admissibility collapses within parameter space as constraints are applied incrementally. 3.1 Incremental constraint application Constraints are applied in fixed order, tracking dimensionality and connectivity of admissible regions until collapse. 3.2 Boundary intersection structure Failure arises when constraint surfaces intersect such that no common interior region remains. Both transverse and degenerate intersections are identified. 3.3 Dimensional collapse Admissibility may be lost through progressive reduction to lower-dimensional manifolds prior to complete collapse. Measure-zero survivals are classified as fragile. 3.4 Constraint-order dependence Failures invariant under permutation of constraint order are classified as structural. 3.5 Registry traceability All localization analyses are recorded as versioned registry artifacts. 4 Observable-driven failure modes Building on Paper XLV, failures are attributed to observable classes only when diagnostically justified. 4.1 Clock-dominated observables Failures arise when clock-based constraints are incompatible with propagation or hybrid observables. 4.2 Propagation-dominated observables Failures arise when geometric or distance-based constraints cannot be embedded consistently. 4.3 Hybrid observables Failures emerge from joint clock–propagation constraints and are often multi-class in origin. 4.4 Singleversus multi-class failures Failures are classified as single-class driven or emergent under class combination. 5 Environmental and scale–redshift failure structure Failures are examined under environmental, scale, and redshift partitioning. – 3 – 5.1 Partition-invariant failures Failures persisting across all partitions are classified as intrinsic. 5.2 Contingent failures Failures that disappear under partitioning are classified as contingent or fragile unless independently required and robust. 5.3 Cross-partition consistency Only failures stable under all robustness checks propagate to boundary synthesis. 6 Global boundary structure of admissible dynamical embeddings Admissible embeddings occupy a constrained region bounded by multiple non-parallel constraint surfaces. 6.1 Robust admissible core Embeddings that remain admissible under all Phase II tests form the robust core. 6.2 Marginal boundary regions Embeddings requiring restricted conditions occupy marginal regions and are classified as fragile. 6.3 Excluded regions Embeddings failing structurally under all diagnostics lie outside the boundary. 6.4 Boundary topology The boundary is non-smooth, reflecting heterogeneous empirical constraints. 6.5 Irreversibility of exclusion Excluded embeddings are not reconsidered absent revision of the empirical operator space. 7 Implications for Phase III Phase III operates strictly within the robust admissible core identified here. Its objectives are synthesis, predictive discrimination, and explicit falsification, not further admissibility testing. 8 Conclusion This paper completes Phase II of the Ordered-Dynamics Reconstruction Program by explicitly mapping the failure modes and boundary structure of dynamical embeddings under empirical confrontation. Failures are defined structurally, classified reproducibly, and localized within parameter space. The resulting boundary delineates a robust admissible core, marginal regions, and excluded domains, invariant under all Phase II robustness tests. This boundary defines the fixed hypothesis domain for Phase III, which will focus on prediction and falsification rather than admissibility. Phase II is complete. – 4 – References [1] P. Cooney, A First Dynamical Embedding Test of the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [2] P. Cooney, Comparative Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [3] P. Cooney, Environmental Dependence of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [4] P. Cooney, Scale and Redshift Dependence of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [5] P. Cooney, Observable-Class Sensitivity of Dynamical Embeddings in the Ordered-Dynamics Reconstruction Program, Zenodo (2025). [6] P. Cooney, Epistemic Structure and Empirical Closure of the Ordered-Dynamics Reconstruction Program, Zenodo (2025). – 5 –