A Spectral and Geometric Perspective on Immune Repertoire Selection

<jats:p>The immune system delivers a highly diverse receptor repertoire while preserving self-tolerance, functional robustness and effective pathogen recognition. Although the mechanisms governing clonal selection are characterized, the mathematical principles potentially underlying repertoire-wide organization are less understood. We propose a geometric, probabilistic and spectral perspective in which immune repertoire selection could be interpreted as constrained sampling from a high-dimensional space of receptor configurations. Within this perspective, we connect developments in modern probability and combinatorics with distinct, testable aspects of repertoire organization: polynomial stability could define a global landscape of admissible repertoires; negative dependence could constrain simultaneous selection of similar or redundant clonotypes; entropy maximization could preserve the broadest receptor distribution compatible with self-tolerance and antigen recognition; log-concavity could organize compatible configurations into broad basins supporting gradual adaptation. Still, Kadison–Singer subset preservation could explain how comparatively small lymphocyte samples retain statistical and spectral properties of larger repertoires, whereas random-forest connectivity could quantify collective integration generated by local receptor relationships. Finally, spectral independence could constrain propagation of clonal perturbations, allowing local responses without uncontrolled repertoire-wide cascades. These mathematical issues could provide measurable descriptors of diversity, redundancy, representativeness, connectivity and perturbation propagation, generating falsifiable predictions accessible to repertoire sequencing and network analysis. Rather than replacing established immunological mechanisms, we introduce a complementary ensemble-level description shifting the mathematical object of interest from the individual clonotype to the organization of the repertoire as a whole.</jats:p>