A Spiral Coherence Field Coupled to General Relativity

A complex coherence field Ψ = A e^(iψ) is coupled non-minimally to general relativity through a curvature term −½ξA²R. The amplitude A encodes Pattern, the phase ψ encodes Rhythm, and their dynamics constitute Process; varying the action yields modified Einstein equations in standard variational form. In the weak-field static limit the field contributes an effective gravitating density ρ_SCC. For an axially regularized azimuthal phase winding ψ = mφ with amplitude A = A₀ sin θ, this density scales as 1/r², giving an enclosed mass M(r) ∝ r and a flat rotation curve without dark matter; because ρ_SCC is genuine stress-energy, gravitational lensing follows automatically, with an effective G_eff = G/(1 − κξA₀²). The winding is shown to be an exact stationary solution of the phase equation, a stable energy minimum, and a topologically protected configuration formed by the galaxy's own rotation through the free-energy competition that nucleates vortices in a rotating superfluid. Evaluating that competition for standard density profiles gives a critical rotation Ω_c = K/J that is independent of the field amplitude and set purely by galaxy size (Ω_c ∝ 1/r_s²), a falsifiable prediction: large galaxies wind easily and flatten, while compact slowly rotating systems may remain irrotational and appear free of dark matter. The same field equations arise from three independent derivations. Beyond the curvature route, an entropic derivation varies the logarithmic coherence entropy S = −ln F to yield the Einstein equations with a k-essence source and an emergent cosmological constant Λ = ln(A₀²X₀) fixed by the coherence vacuum, with no auxiliary field introduced; and a third derivation, in the coherence-gated Spiral calculus, recovers general relativity where coherence is high and predicts controlled deviations where coherence varies in space. The construction is a classical coherence-field realization in the same family as superfluid and ultralight/Bose–Einstein-condensate dark-matter models, sharing the quantized vortex circulation ∮∇ψ·dℓ = 2πm.