The Ontological Structure of the S³-Manifold

This work presents a philosophical and mathematical research highlight on the ontological structure of space understood as a compact, simply connected, boundaryless three-dimensional manifold: M ≃ S³. Instead of treating space as a local product of physical emergence, the treatise argues that the manifold should be regarded as a persistent global structure, while cosmological evolution describes changing states on it: g(t), Φ(t), ρ(t), C(t). The central distinction is therefore simple but decisive: it is not M that emerges, but states on M. From this perspective, time, causality, observability, motion, polarity, rhythm, and physical reconstruction are interpreted as internal modes of organization within the same compact structure. The S³ framework provides a rigorous way to think about finitude without boundary: ∂S³ = ∅. Space may be globally finite while having no external edge, no outside container, and no privileged external observer. Local knowledge remains necessarily incomplete: Knowlocal(M) ⊊ Struct(M), because every observer satisfies Obs ⊂ M and can access only a finite causal domain. Light is interpreted not as a substance, but as the limiting structure of causal coherence: Dc = πR, tcoh = πR/c, with local propagation still constrained by ds² = 0 and vsignal ≤ c. The model also emphasizes that compactness naturally leads to discrete spectral structure without implying a discrete lattice: λl = l(l + 2)/a², ml = (l + 1)². As a reconstructive cosmological proposal, the treatise places standard ΛCDM within a broader compact-topological interpretation: ΛCDM remains a powerful local effective model, while S³-cosmology asks whether the observable domain Dobs ⊂ M may be only a partial reconstruction of a global compact manifold. The dark sector is treated with formal caution as an Einstein residual, not automatically as a new substance: residual ≠ substance. The result is a disciplined framework that separates established physics, mathematical modelling, speculative extension, and metaphysical interpretation, offering a compact S³ ontology in which global coherence, unitary evolution U†U = I, causal limitation, and local observational reconstruction are held together within one continuous structure.