Alysis as a Structural Diagnostic of Reconstructibility

Scope and Intent of this Preprint This work introduces Alysis as a diagnostic concept for identifying when reconstruction from relational data ceases to be structurally meaningful under boundary-induced aggregation. The Alysis Coefficient (AC) is treated as a normalized scalar/order diagnostic for the collapse of hierarchical reconstructions toward degenerate, brush-like or star-like forms under deterministic, non-invertible boundary operations. The contribution of this paper is diagnostic rather than dynamical. It does not propose new physical laws, new reconstruction algorithms, new entropy measures, or new holographic principles. Instead, it introduces a representation-bound criterion for distinguishing regimes in which declared relational data still support meaningful hierarchical reconstruction from regimes in which reconstruction becomes structurally underdetermined by the available data. Relation to Existing Concepts Alysis is distinct from entropy and probability-based measures. Entropy characterizes statistical uncertainty, information loss, or disorder under coarse-graining, whereas Alysis diagnoses the loss of reconstructible relational hierarchy. High entropy and high Alysis need not coincide. The Alysis Coefficient therefore captures a structural aspect not contained in entropy alone. It focuses on whether relational data, after boundary aggregation, still support differentiated hierarchical reconstruction within the declared representation, aggregation protocol, and admissible reconstruction procedure. Epistemic Status The framework is explicitly epistemic. It diagnoses limits imposed by boundary access, coarse-graining, restricted observables, or related interface operations between system and description. No ontic claim is made that structure is necessarily “truly lost” in the underlying physical system. The diagnostic concerns what can and cannot be reconstructed from the available relational data under admissible deterministic and stable reconstruction procedures. Applications Discussed The paper applies this diagnostic perspective to selected domains in modern physics where structure is inferred rather than directly observed: - emergent geometry in entanglement renormalization and holography;- quantum channels and open quantum systems;- thermodynamic coarse-graining and macroscopic irreversibility;- schematic critical-phenomena contexts. In each case, the Alysis Coefficient is used to clarify when boundary-induced flattening of relational distinctions can lead to structural degeneracy of hierarchical reconstructions. No new empirical or domain-specific physical result is claimed. These examples serve as paradigmatic contexts for illustrating the diagnostic approach. Appendices and Supplementary Material All essential definitions, formal statements, and interpretations are contained within the manuscript and appendices. Appendix A provides an explicit finite construction illustrating boundary-induced structural collapse with intermediate AC values. Appendix B shows how selected quantum-channel families can induce boundary aggregation at the level of accessible relational data. Appendix C gives a schematic route for applying the framework to critical-phenomena settings through relational matrices derived from connected correlations. It is not presented as a numerical Ising simulation. The supplementary file ac_boundary_aggregation.zip contains a deterministic Python demonstration of the Appendix A boundary-aggregation calculation. The accompanying outputs.zip contains the generated report and figures. Outlook The present work is restricted to physics and mathematically declared relational representations. Possible extensions to non-physical relational systems may be considered in future work, but would require separate specification of the relational matrix, aggregation protocol, and admissible reconstruction procedures.