On the Dual Origin of Ontological Deltas: Spectral Necessity vs. Axiomatic Minimality in the TNA–ISSP Framework

We clarify the conceptual and structural distinction between the Theory of Axiomatic Necessity (TNA) and the Induced Spectral Stability Principle (ISSP), establishing their precise domains of validity and their hierarchical relationship. TNA is introduced as a general ontological framework defining minimal axiomatic structures required for the existence of coherent systems, independently of physical realization. ISSP is then formulated as the physical manifestation of this necessity within differentiable systems, where structural stability emerges through the spectral decomposition of induced metrics. We demonstrate that ISSP does not replace TNA, but provides its concrete operational mechanism in physical systems: while TNA defines ontological deltas as irreducible necessity anchors, ISSP identifies their realizable counterparts as spectral modes of stability. This separation resolves long-standing ambiguities between mathematical structure, physical realizability, and ontological necessity, and establishes a unified framework in which linear systems, decoherence, and quantum measurement can be interpreted as boundary-induced spectral selection under axiomatic constraints. The resulting hierarchy—TNA ⊃ ISSP ⊃ BIC—offers a coherent foundation for interpreting collapse, stability, and structural emergence without invoking ontological randomness or observer-dependent postulates. Diagram https://drive.google.com/file/d/1RLVINPhsgjcNvlLv4W3ee8avLM5wzTya/view?usp=sharing