Dark Matter as the Wedderburn dim-24 Adjoint Sector: Why It Couples Gravitationally but Not to Gauge Bosons
Authors/Creators
Description
Description
Dark matter is identified with the dim-24 adjoint sector of the finite algebra A_F = ℂ ⊕ M₂(ℂ) ⊕ M₃(ℂ), the internal algebra of the Standard Model fixed by the integer pair (N_w, N_c) = (2, 3). The Wedderburn decomposition of A_F has four primary sectors with degeneracies (1, 3, 8, 24): the singlet, the SU(2) weak adjoint (d₂ = 3), the SU(3) colour adjoint (d₃ = 8), and the mixed sector d₄ = (N_w² − 1)(N_c² − 1) = 24, the joint adjoint of SU(2) ⊗ SU(3) and the largest primary of the algebra.
This mixed sector carries energy-momentum, so it gravitates, but it does not transform under any of the smaller Wedderburn sectors that contain the gauge fields. The gauge bosons live in the individual adjoints (the weak boson in d₂ = 3, the gluon in d₃ = 8); a single gauge boson acts within one individual factor and has no exchange channel into the joint product sector. The dim-24 sector therefore has no gauge coupling and is invisible to electromagnetic, weak, and strong probes — dark not by a small coupling but by the absence of a coupling channel, with direct-detection cross-section identically zero. It is stable (no product-conserving decay into visible sectors) and is a Planck-scale composite, behaving as cold, collisionless matter that clusters gravitationally and passes through baryonic gas and through itself without scattering.
The dark-matter-to-baryon ratio is set by the non-dark block dimension weighted by the thermal-conformal structure at freeze-out: Ω_DM/Ω_b = (d₁ + d₂ + d₃) · π / β₀ = 12π/7 = 5.39, against the observed 5.41 (the naive block ratio 24/12 = 2 is explicitly not the correct reading). With the matter partition Ω_m = 6/19 this gives Ω_DM ≈ 0.266 and Ω_b ≈ 0.049, and with the 13/19 dark-energy partition of the companion black-hole paper the cosmic budget closes. The central prediction is null: because the sector has no gauge coupling, no dark-matter particle will be produced in a collider or recoil in a direct-detection target. The only external inputs are the Planck mass and the Z mass; there are no free parameters.
This paper is part of the Crystal Topos program, which derives the Standard Model and cosmological observables from the finite spectral geometry of A_F. It builds on the four foundational papers (the discrete core, the MERA scaling spectrum, the Type II∞ core, and the triangular duality) and the black-hole lifecycle paper.
Keywords
dark matter, noncommutative geometry, spectral triple, finite spectral triple, Wedderburn decomposition, adjoint representation, joint adjoint, SU(2), SU(3), gauge coupling, gravitational coupling, collisionless dark matter, Planck-scale relic, dark matter abundance, Omega_DM, cosmological parameters, dark sector, direct detection, Connes-Chamseddine, spectral action, Standard Model, finite algebra, Crystal Topos, 2x3 substrate, zero free parameters
Built on the Crystal Topos foundation: load the foundational papers and this paper's prerequisites before this one, one at a time in separate uploads, to give a large language model (Claude, ChatGPT, Gemini, or similar) full context — a single bundled upload compresses each paper into a thin summary and loses the algebra. Between uploads, prompt the model to confirm it has read the paper and can summarize its main theorem and Wedderburn projectors before sending the next.
- (1) Discrete Core — https://doi.org/10.5281/zenodo.20232800
- (2) MERA Scaling Spectrum — https://doi.org/10.5281/zenodo.20236511
- (3) Type II Core — https://doi.org/10.5281/zenodo.20257603
- (4) Triangular Duality — https://doi.org/10.5281/zenodo.20257953
- (4b) Spacetime Constants — https://doi.org/10.5281/zenodo.20485003
- (9) Black Hole Lifecycle — https://doi.org/10.5281/zenodo.20421277
- (10) Higgs Singlet Projector — https://doi.org/10.5281/zenodo.20422007
After all seven are ingested, upload this paper as the eighth turn. The model will then read this paper against the foundation rather than in isolation, and answer questions about why a given sector or Wedderburn projector takes its form from the actual algebraic construction rather than by pattern-matching.