u^μ ∇_μ u^ν = −c² h^νμ ∇_μ ln A
Automatically orthogonal to four-velocity: u_μ f^μ = 0, so u·u = −c² is preserved. The cohomological piece is −c² α_Q h^νμ ∇_μ 𝒥_Q.
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Particle and fluid from one A
The variational paper's central claim is not a new force law pasted onto GR. Point particles and perfect fluids are two descriptions of S_m[Ψ_m, A²g]. Dots follow −∇ ln A. Arrows are the Euler field. At vanishing pressure they must agree.
Sp = −m ∫ A(φ, ℳ, 𝒥Q) ds
⟨A⟩
1.000
conformal on
Dust residual
0.000
agrees with worldline
α_Q
0.350
β_Q / Λ_Q⁴
p / ρ
0.00
dust at 0
Worldlines · Euler field · Ã = A² g
Couplings
β_Q0.35
Λ_Q1.00
α_φ0.22
α_A0.12
pressure p0.00
Conformal A
Freeze Y (F_Q → 0)
Nested nulls (λ_Q, β_Q)
a^ν = −h^νμ ∇_μ p / (ρ+p) + T_m / (ρ+p) h^νμ ∇_μ ln A
T_m = −ρ + 3p. Dust (p = 0) collapses to the worldline equation. That identity is the paper's consistency test — not an extra postulate.