On the Distinction Between Computational and Physical Ontology and Its Necessity for Operator Realizability

Claudio Bresciano

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 An accompanying short video provides an intuitive overview of the core ontological distinction developed in this paper. Through a brief conversational explanation, it illustrates why theories formulated exclusively within the observable domain cannot be ontologically complete, and why the explicit acknowledgment of a latent structural domain is necessary for conceptual clarity. https://vimeo.com/1157925725?share=copy&fl=sv&fe=ci

Décio Krause

Alyssa Ney

Claudio Bresciano

The observable domain (N}_0) of foundational theories is structurally incomplete and necessitates a latent domain ({N}_1) to account for stability and selection without circularity. This paper formalizes the {N}_1 -> {N}_0 projection, demonstrating that purportedly self-sustaining frameworks implicitly rely on hidden structural remainders. Modern foundational discourse often pursues ontological closure by restricting reality to operational outcomes, yet fails to explain why specific possibilities are realized over alternatives. By analyzing standard quantum interpretations—including Copenhagen, Many-Worlds, and Decoherence—we show that latent structures such as measurement contexts or environment-induced selection are consistently present but unacknowledged. The necessity of this latent domain is presented not as a metaphysical addition, but as a structural requirement analogous to Gödelian incompleteness, where a system cannot justify its own conditions of validity from within. Ultimately, the transition from possibility to realization is reframed as an ontological projection under constraint, providing a path toward structural honesty and clarity in theoretical physics Video Abstract — Intuitive overview of ontological incompleteness in closed theories. Two physicists discuss why any theory confined to observable structures (N₀) necessarily presupposes a latent structural domain (N₁) to account for selection, stability, and necessity. https://vimeo.com/1157867642?share=copy&fl=sv&fe=ci

Jeanette Leue

This monograph establishes a unified ontological and mathematical framework for understanding discrete state transformations across mathematics, physics, and complex systems. The work identifies a fundamental gap in binary logic regarding developmental dynamics and proposes the Primarcode domain \mathcal{P} = \{0, 1, +1\} as the minimal necessary extension to accommodate state change. The central claim is that every realized transition requires a minimal act of realization, formalized as a Surplus Operator (+1), which acts upon a structured space of potentials to compel a unique outcome. By developing this distinction axiomatically, the work demonstrates that the universal increment x = y + 1 is not a domain-specific assumption but a structural necessity of any discrete system capable of change. Key Applications & Results: Quantum Mechanics: Reinterprets Schrödinger dynamics as the evolution of potential structures and provides a formal resolution to the measurement problem via the Observer-Modulator Framework, aligning with experimental decoherence data. Prime Number Mechanics: Introduces an arithmetic resonance framework where prime gaps act as frequencies. It presents a deterministic prime generation algorithm and a geometric embedding (the Prime Helix) that reveals the underlying mod-6 structure of the prime sequence. Riemann Hypothesis: Proposes the Leue Map and the Residual Whiteness Condition, establishing a computationally testable equivalence between the Riemann Hypothesis and the spectral completeness of prime-gap residuals. Topology & Cosmology: Provides a constructive proof for the 3-sphere as the necessary resonance arena for physical and arithmetic processes, offering new insights into the nature of space and Dark Energy. This work renders long-standing conceptual tensions—particularly in quantum measurement and number theory—transparent as necessary consequences of ontological structure.

Shan Gao