Dark Matter as the n = 0 Kaluza–Klein Condensate of S²: A Positive Geometric Identification of Dark Matter, with the MOND Phenomenology as the Galactic-Scale Limit of the Condensate and the Derived Dark-Matter Fraction α₂₃ = 0.8428

Four decades of direct-detection experiments (XENONnT, LZ, PandaX, DarkSide, and earlier generations) have found no trace of the WIMP. Four decades of axion searches (ADMX, HAYSTAC, CASPEr, IAXO) have found no trace of the axion. Four decades of sterile-neutrino searches have found no unambiguous signal. Yet gravitational observations remain unambiguous: Omega_DM = 0.264 from Planck 2018, with dark matter outweighing visible matter by a factor of 5.4 in every well-measured gravitational system. The observational evidence says dark matter is real; the experimental evidence says it is not any of the candidates conventional particle physics has proposed.

We show that in the Three Time Dimensions (3+3) spacetime framework (de Haan 2026, book manuscript, DOI 10.5281/zenodo.19633127), in which the third time dimension t_3 is compactified as a discrete two-sphere S^2 with 2^152 Planck-area cells, dark matter has a positive identification: it is the n = 0 Kaluza–Klein zero-mode of the compact S^2 — the same sphere that hosts the photon (n = 1), the electron (n = 1/2), and the proton (n_p = 918). The zero-mode is a uniform, featureless condensate with no angular variation and no winding number; by topology it carries zero electromagnetic coupling, zero weak coupling, and zero strong coupling. It interacts only through gravity. The non-detection in four decades of searches is not a fine-tuning fact; it is a topological theorem about a mode with no charge under any gauge group.

Three quantitative predictions follow. First, the condensate acquires an effective mass from t_2 dynamics: m_c = sqrt(E_L  hbarH_0)  Omega_m  alpha = 4.3e-21 eV, placing the framework in the fuzzy dark matter regime with de Broglie wavelength of kpc scale at galactic velocities — 43x above the Lyman-alpha bound and producing solitonic galactic cores (~1.4 kpc for a 10^9 M_sun dwarf, ~0.24 kpc for the Milky Way). Second, the condensate naturally reproduces MOND phenomenology at galactic scales: the Milgrom acceleration a_0 = 1.2e-10 m/s^2 emerges as the condensate oscillation scale a_0 = cH_int, explaining the long-standing ‘a_0 ≈ cH_0’ coincidence as a natural relation rather than an accident. Third, the dark-matter fraction alpha_23 = Omega_DM/Omega_m = 0.8428 is derived from the t_2 equation of motion without free parameters, matching the Planck-inferred value to 0.02%.

The framework makes two firm empirical commitments. First, no WIMP, no axion-as-dark-matter, no sterile-neutrino dark matter will ever be detected: all are forbidden by the n = 0 identification, which has zero electroweak charge by topology. Every null result from XENONnT, LZ, PandaX, DarkSide, ADMX, HAYSTAC, and future searches is consistent with the framework; a single confirmed positive detection falsifies the topological identification. Second, the framework predicts specific positive signatures: solitonic cores in dwarf spheroidals (testable through stellar kinematics in Fornax, Sculptor, Ursa Minor); MOND-like rotation curves; a condensate depletion zone within ~14 pc of Sgr A* (testable by GRAVITY and Keck monitoring of S-stars); CDM-like behaviour on cosmological scales with the cosmic web as the Gross–Pitaevskii ground state.