The Mathematics of ONE — Minimal Thickness and the Geometric Program

The Mathematics of ONE — Minimal Thickness and the Geometric Program (V1.6) This preprint is a supplement to The Mathematics of ONE (DOI 10.5281/zenodo.21263119). In that framework, all of reality is a single Absolute Unity (ONE), which pulsates between two limits that are never reached, Alpha and Omega. Part I – The Minimal Thickness Principle (MTP). No real physical object has zero thickness in any direction. Pure dimensions 1 and 2, the ideal line and the ideal surface, are therefore mathematical limits rather than physical entities. Every physical extent satisfies L ≥ κ·ℓP, with κ of order one. Here the Planck length ℓP is read as a minimal uncertainty of distance, not as a lattice spacing. Dimension is redefined in wave terms: it is the number of directions a wave can explore freely at a given energy. This links the MTP to two known results of theoretical physics: the reduction of the spectral dimension towards 2 at the Planck scale, and the holographic principle. Part II – The Geometric Program and Dark Coupling (working hypothesis). A bound wave carries a geometric signature Σ = (R, Q), made of its internal representation and its topological invariants. Interactions require terms that are neutral under the symmetries of both sectors. This condition yields the “ports” between the visible and the dark sector: Higgs, vector, neutrino and axion, while gravity is universal. A program distance, Δ = d_min − 4, grades their strength. Without a “bridge wave”, the kinetic mixing of a dark photon would arise only gravitationally, at a reference order of ε ~ 10⁻¹³. Current experimental limits (ATLAS, LZ, Parker Solar Probe, TASEH, HAYSTAC, CAST, galaxy-cluster collisions) are compiled and interpreted, and seven falsifiable predictions are listed. Every claim is labelled with one of four epistemic levels, from established physics (A) to speculative hypothesis (D). Null experimental results are treated as compatible with the hypothesis, never as validation. Developed with AI research partners: Gemini (Google), Perplexity and Claude (Anthropic). Thanks for the effort and continuous work. Marius Iacob Lumineanu.