The Sharon Instability Theorem: Generic Instability of the M-Invariant

This paper resolves an open question regarding the robustness of the M-invariant introduced by Guidolin, Divason, Romero, and Vaccarino. The author proves the Sharon Instability Theorem, demonstrating that the invariant is generically unstable under perturbations of the birth and death locus. Specifically, for any epsilon greater than zero, the map fails to be an isomorphism whenever a simplex appears exactly at the death corner. Overview Multiparameter persistent homology has emerged as a central tool in topological data analysis, extending the classical theory of persistence to filtrations indexed over partially ordered sets beyond the integers. While single-parameter persistence enjoys a complete discrete invariant (the barcode or persistence diagram) with strong stability properties, the multiparameter case is fundamentally more complex. Research has shown that no complete discrete invariant exists for multiparameter persistence modules, as the classification problem reduces to wild representation type. The M-Invariant In their influential work, Guidolin, Divason, Romero, and Vaccarino introduced the M-invariant, which captures information about homology classes born at a birth downset and dying at a death downset. This invariant, defined via spectral systems, provides finer information than the rank invariant and can distinguish filtrations that the rank invariant cannot. The original authors demonstrated its computability using the Kenzo system and effective homology techniques. Resolution of Robustness A fundamental question regarding the robustness of this invariant was left open, with the original authors noting the need to investigate its behavior with respect to small changes in the multiparameter filtration. This paper resolves that open question definitively: the M-invariant is generically unstable. Small perturbations of the birth or death locus can cause the invariant to collapse entirely. Key Contributions The Sharon Instability Theorem: Establishes that small perturbations of the birth or death locus can cause the invariant to collapse entirely. The Sharon Bound: Provides a quantitative upper bound on the rank drop, showing it is determined by the number of simplices appearing exactly at the corners of the death downset. Tightness of Bound: Demonstrates that the rank drop can exactly match the number of simplices at the corner through specific bifiltration examples. Stabilisation via e-Pruning: Proposes a modified e-pruned invariant that protects simplices near the death boundary from being counted as dying early. Stability Theorem: Provides a stability proof for the triangle case when the perturbation delta is less than or equal to one-third of the pruning parameter epsilon. Computational Verification: Validates findings via a Python implementation, showing a 100 percent instability rate across tested random bifiltrations where a simplex exists at the death corner.