A Topological Dark Source in Einstein-Form Gravity

Dark matter is usually introduced as a new particle species, a hidden sector, or a phenomenolog- ical fluid. I describe a different source-level possibility: dark stress-energy may arise from retained gauge-topological transgression data while the metric equation remains in Einstein form. The orga- nizing device is a complex two-sector equation whose real projection is the ordinary Einstein curva- ture response and whose imaginary projection records an independent gauge-topological consistency channel. If the selected two-sector carrier data are obstructed in the Poincare-Hopf/hairy-ball sense, a global nonvanishing carrier is unavailable and patch, transition, connection, and curvature data must be retained. Keeping only the minimal degree-four gauge Chern-Weil representative then se- lects the local Chern-Simons transgression current 𝐾♭ = *𝐶𝑆 (𝐴). Locality, covariance, symmetry, linearity in that current, and first-derivative order fix the leading non-trace rank-two source seat to ∇ (𝜇𝐾𝜈). Thus the physical field equation can remain G⏛⏜ + Λ g⏛⏜ = 8πG (T^known⏛⏜ + T^top⏛⏜), T^top⏛⏜ ⊃ (λ / 8πG) ∇ (⏛ K⏜) The construction is not a modification of the Einstein tensor and does not postulate a fundamen- tal dark particle. Its counter-standard prediction is the loss of an arbitrary dark amplitude: the low-energy Chern-Weil/theta normalization gives 𝜆0 = 4𝜋𝛼𝜃, while observed amplitudes must be branch-dressed projections. For the 𝜃 = 1 benchmark, present low-redshift consistency requires an order-one-or-smaller projected dressing, 𝒟sh ≲ 0. 715, excluding an unsuppressed universal propaga- tion map.