β₁ = 3 as a Topological Invariant of Minimal Relational Closures: Numerical Evidence from the PDL and OFN Frameworks

Two independent pre-geometric frameworks — the Projective Dynamic Logo (PDL) and the Ontological Fundamental Network (OFN) — independently arrive at the first Betti number β₁ = 3 as a key topological invariant of their respective foundational structures. In PDL, β₁(K₄) = 3 is forced by four combinatorial axioms and is a necessary condition for the cosmological leakage formula to be non-degenerate, matching the Planck 2020 value of Λ to 0.41 ppm with no free parameter. In OFN, β₁(G_H) = 3 is the cyclomatic number of the Hamming subgraph induced by the vacuum manifold Ω₂₁ ⊂ Q₆, corresponding to the three generations of fermions in the Standard Model.

We present a structured numerical investigation — four Python scripts, fully reproducible in Google Colab — establishing that this convergence is not due to a structural identity between the two frameworks, but reflects a deeper topological property: n = 6 is the minimal dimension of a binary space {0,1}ⁿ admitting the construction of three topologically independent cycles. Both frameworks have independently selected this minimal dimension.

Key results: (1) K₄ is the unique connected graph on 4 vertices with β₁ = 3 (exhaustive over 38 graphs); (2) β₁ = 3 is a necessary condition for the PDL cosmological formula — deviations reach 8×10³⁰ ppm with β₁ = 1; (3) b₁(Ω₂₁) = 3 independently verified; (4) no natural bijection exists between K₄ and Q₆; (5) n = 6 is the dimension maximising the probability of β₁ = 3 for structures of intermediate scale (~21/64). All scripts are available at the linked GitHub repository.