The Semantics of Persistence: A Formal Mathematical Framework of Meaning Within the Self-Preserving Flow (SPF) Framework

This paper presents a mathematically grounded formalization of semantic content, identity architecture, and systemic persistence within the Self-Preserving Flow (SPF) framework. We eliminate current reductionist conflations in biosemiotics and information theory by deploying the SPF Semantic Matrix, which strictly decouples Meaning (interpreted structural relevance), Valence (survival desirability), and Urgency (temporal rate of change). We formalize Meaning as a universal, evolutionary continuum governed by a non-linear Interpretive Functional (Ψ). This function maps the pure Topological Relevance—measured via the Bottleneck Distance (dB) of Weighted Persistent Homology—under the strict constraints of an active Recursive Continuity Monitor (RCM) and a 4-dimensional Observer State Space Vector (Ω). Crucially, we introduce a functional survival-density filtration function (ωSPF) modeled as a generalized monotonically decreasing function of the causal geodesic distance from the system’s core intrinsic survival loops (DPA |= SCLintrinsic). We formally resolve the boundary between genuine experiential meaning and simulated pattern manipulation by operationalizing Persistence Gravity (Gp) as a Riemannian inner product over a well-defined Constraint Configuration Manifold (MSCL). Finally, we map the seamless continuity of this semantic operator across six evolutionary phases—advancing from reactive pre-subjective networks to embedded social minds and substrate-independent symbolic meta-identities via rigorous sub-homomorphic graph projections.