Coherence as the Structural Condition of Stability and Identity

This paper clarifies the structural relations between coherence, tolerance, stability, and identity within the Bellori Framework. While previous formulations established that systems persist when coherence between successive configurations is preserved within tolerance, the structural role of coherence and the dependency of tolerance remained implicit. The present work makes this relation explicit by positioning coherence as the primary structural condition for sequence formation and persistence. Coherence is defined as the preservation of structural relations across successive configurations, enabling structural dependence within configuration chains. From this, tolerance is redefined not as an independent domain, but as a domain derived from coherence, specifying the range of variation within which such relations can be preserved. A key result is that tolerance is not freely adaptive. Its variation is itself constrained by the structure of coherence. This introduces a non-trivial limitation on system persistence: not all transformations can be accommodated, and continuity is bounded by the capacity of coherence to sustain variation. Stability is consequently reformulated as the bounded preservation of coherence within a dynamically evolving but structurally constrained tolerance domain. Identity is treated as the observational recognition of such preserved coherence, rather than as a defining condition of persistence. The framework further introduces the concept of coherence regimes to describe domains of structurally consistent persistence, and shows that transitions between regimes correspond to changes in relational structure rather than the loss of structure itself. The central claim is that systems persist not because stability or identity is imposed upon them, but because coherence constrains both transformation and the tolerance of transformation. This establishes a domain-independent structural condition for persistence that applies across physical, biological, technological, and social systems.