On the distinction attractor, the projected linearized theory carries only the two transverse-traceless spin-2 modes. The spin-1 sector is gauge, and the spin-0 sector is constrained by the Jacobi/lapse equation rather than propagating.
Start with a flat background and perturb the metric:
g_{μν} = η_{μν} + h_{μν}
Decompose h_{μν} into transverse-traceless, vector, and scalar pieces:
h_{μν} = h_{μν}^{TT} + ∂{(μ}V{ν)} + (∂μ∂ν − ¼ η{μν}□)σ + ¼ η{μν}χ
with
∂^μ h_{μν}^{TT} = 0, η^{μν}h_{μν}^{TT} = 0, ∂^μ V_μ = 0.
In the sPNP projection, the spacetime perturbation is obtained by smearing configuration-space curvature through the kernel:
h_{μν}(x) = ∫ dμ(X) ρ₀(X) K₀(X,x) δR_{IJ}(X) E^I_μ(x) E^J_ν(x)
The trace/scalar part is
Ψ(x) = ∫ dμ(X) ρ₀(X) K₀(X,x) δR_F(X),
where δR_F = G₀^{IJ} δR_{IJ}. So the projected scalar is literally the pushforward of the scalar curvature perturbation on configuration space.
The vector sector is removed by diffeomorphism equivariance. The DGNSZ-style result says the extraction map is covariant under spacetime symmetry, so the spin-1 part is pure gauge, as in GR. That is a symmetry statement, not an attractor statement.
The scalar sector is removed by the Jacobi constraint. The fundamental relation is
G^{IJ}(X) ∂_I S ∂_J S = 2 [E_eff − V_tot(X)]
At the attractor, the distinction charge is stationary:
δ[∫ R_F dμ(X)] = 0
So the scalar source is fixed on-shell, and the lapse is not an independent propagating field. In the sketch, this appears as the Poisson-like lapse relation
N²(x) = 2 ∫ dμ(X) ρ₀ K₀ [E_eff − V_tot(X)] / ∫ dμ(X) ρ₀ K₀
That is the crucial point: the scalar is determined by a constraint, not by a wave equation. In the long-wavelength regime, that constraint is elliptic, so the scalar is fixed mode by mode rather than propagating.
So in Fourier space, the scalar and vector pieces are removed as follows:
V_i(k) = 0 from gauge/equivariance σ(k) = 0, χ(k) = 0 from the Jacobi/lapse constraint on the attractor
leaving only
h_{ij}^{TT}(k).
Equivalently, after integrating out lapse and shift, the quadratic action reduces to the TT kinetic term:
S^(2) ∝ ∫ d⁴x ∂λ h{ij}^{TT} ∂^λ h_{ij}^{TT}
That is the Fierz–Pauli spin-2 content. The mode count is therefore 2 propagating degrees of freedom on the attractor.
Boundary is still the same: this is exact in the attractor / IR / elliptic-lapse regime.
So: Jacobi constraint + elliptic lapse ⇒ scalar fixed; diffeomorphism equivariance ⇒ vector gauge; therefore only h^{TT}_{μν} propagates.
Proof architecture is: On the distinction attractor and in the IR regime kQ₀ ≪ 1, the scalar sector of the projected linearized theory is non-propagating. This follows from three independent structural facts: (i) the Madelung action is first-order in τ, so K₂ = 0 upstream; (ii) the static projection kernel does not generate new time derivatives; (iii) the Jacobi constraint makes the lapse and longitudinal shift instantaneous, so O_con = C(k) and O_mix = Q(k) are both purely spatial on the attractor. The Schur complement then gives O_red = P_red(k), which is ω-independent, and the scalar propagator has no pole. The vector sector is separately removed by DGNSZ diffeomorphism equivariance. The only propagating sector is the TT spin-2 field, with two degrees of freedom. Eisenhart Lift, Pushforward Geometry, and the Newtonian Limit
In this subsection we connect three ingredients of sPNP:
(i) the Jacobi–Fisher dynamics on relational configuration space, (ii) the Eisenhart/Bargmann lift to a null-geodesic picture, and (iii) the pushed-forward 4D ADM metric whose weak-field limit reproduces the Fisher–Poisson Newtonian potential.
Lemma (Projected distinction source and Newtonian limit). Let K(X, x) be the normalized, causal, RG-stable projection kernel of width Q₀, and let ρ_eff(x) denote the pushed-down density on the effective 3D sector. Define the local Fisher distinction density by
D_eff(x) := ρ_eff(x) |∇ ln ρ_eff(x)|².
Then, to leading order in the long-wavelength approximation, convolution with the Green’s function of the flat 3D Laplacian yields a scalar potential Φ satisfying
∇²Φ(x) = 4π G_eff ρ_eff(x),
where G_eff is the effective coupling fixed by the kernel normalization and the coarse-graining scale Q₀. In this toy Newtonian regime, ρ_eff plays the role of the effective source density, while D_eff is the local distinction content that generates the projected potential.
Proposition (Eisenhart lift and Einstein–Hilbert emergence). Consider the relational Jacobi–Fisher system on configuration space and its Eisenhart/Bargmann lift to a null-geodesic geometry. Under the causal projection induced by K(X, x), the lifted geometry pushes forward to an emergent 4D spacetime metric g_μν whose lapse satisfies
N² ≃ 1 + 2Φ/c²
in the weak-field, slow-motion limit. Since the underlying configuration-space theory is relational and diffeomorphism invariant, the infrared effective action for g_μν must be a local diffeomorphism-invariant functional. At scales large compared with Q₀, the unique leading two-derivative term is the Einstein–Hilbert action,
S_eff[g] ≃ (1 / 16πG_N) ∫ d⁴x √−g R[g] + O(Q₀²R², ∇R, …).
Thus the coefficient 1 / 16πG_N is not inserted by hand; it is the matched infrared coupling produced by the projection map, with its scale set by Q₀ and its numerical value fixed by the weak-field limit.
Corollary (Interpretation of gravitational weakness). The smallness of the Einstein–Hilbert prefactor reflects the coarse-graining scale Q₀: gravity appears weak because the projection from configuration space suppresses fine-grained distinction structure, leaving only a smooth low-energy spacetime shadow.
