Future–Mass Projection Gravity: A Divergence-Free Bitensor Kernel, Metric PPN to O(v²), Uniform Tail Bounds, and a Reproducible Cosmology/Galaxy Pipeline

1. Introduction and Novelty

The $\Lambda$CDM model explains cosmological observations by adding a cold, collisionless dark matter component, yet decades of non–gravitational searches remain null. FMP is a time–nonlocal extension of gravity that reproduces dark–matter phenomenology by coupling the present field to a forecast of future baryonic states through a causal kernel, inspired by time–symmetric ideas (Wheeler–Feynman) but cast here in a covariant, energy–conserving form. A distinctive, falsifiable prediction is a mild redshift drift of

$$R\left(z\right)\equiv {\displaystyle \frac{{\mathsf{\Omega }}_F\left(z\right)}{{\mathsf{\Omega }}_b\left(z\right)}},R\left(0\right)\simeq 5.4,$$

with $R\left(z\right)$ decreasing towards higher redshift—in contrast to the constant matter ratio of $\Lambda$CDM.

This paper’s contributions.

(i) We construct a bi–divergence–free kernel with parallel propagators that resolves the A2↔A3 conflict; (ii) we provide a metric PPN derivation from the bilocal action (no reliance on phenomenological $\mu ,\Sigma$); (iii) we upgrade A1${‌}^{\prime }$ to a theorem with a uniform tail bound for the Laplace realization of $R\left(H\right)$; (iv) we implement an explicit, renormalized mapping $\epsilon \left(a\right)=\Xi [a;R]$ that prevents background/perturbation double counting; (v) we supply a working CLASS hook and a clean, reproducible pipeline (galaxy subset fits and a Bullet–cluster analysis plan).

2. Field Equations, Axioms, and the Bilocal Source

We retain Einstein’s equations with an effective source,

$$\begin{array}{cc}\hfill G_{\mu \nu }\left(x\right)& =8\pi G{T}_{\mu \nu }^{\mathrm{eff}}\left(x\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill T_{\mu \nu }^{\mathrm{eff}}\left(x\right)& \equiv T_{\mu \nu }^{\left(b\right)}\left(x\right)+T_{\mu \nu }^{\left(F\right)}\left(x\right),\hfill \end{array}$$

(2)

where the FMP contribution is a bilocal integral over the future domain $J^{+}\left(x\right)$,

$$T_{\mu \nu }^{\left(F\right)}\left(x\right)={\int }_{J^{+}\left(x\right)}{d}^4{x}^{\prime }\sqrt{-g\left(x^{\prime }\right)}{K}_{\mu \nu }{‌}^{\alpha \beta }(x,x^{\prime })T_{\alpha \beta }^{\left(b\right)}\left(x^{\prime }\right).$$

(3)

Axioms (revised).

A1${‌}^{\prime }$ 

(Finite horizon with uniform tail bound)   There exists $\Delta T<\infty$ such that the contribution to $R\left(H\right)$ from $\tau >\Delta T$ is uniformly bounded by a prescribed ${\epsilon }_{\mathrm{tail}}$ for all $H\in [H_{min},H_{max}]$ (Theorem 2).

A2${‌}^{\prime }$ 

(Isotropy on the connecting geodesic)   The kernel may depend on Synge’s world function $\sigma (x,x^{\prime })$, the parallel propagator $g_{\mu }{‌}^{{\alpha }^{\prime }}(x,x^{\prime })$, and metric contractions, yielding geodesic–isotropic scalars; no constant, global projector is assumed.

A3 

(Bi–divergence–free)   ${\nabla }_x^{\mu }{K}_{\mu \nu }{‌}^{\alpha \beta }(x,x^{\prime })=0={\nabla }_{x^{\prime }}^{\alpha }{K}_{\mu \nu }{‌}^{\alpha \beta }(x,x^{\prime })$.

A4 

(Positivity)   The bilinear form $\int \int T^{\left(b\right)}\left(x\right)K(x,x^{\prime })T^{\left(b\right)}\left(x^{\prime }\right)d{V}_{x}d{V}_{x^{\prime }}\ge 0$ for all compactly supported, symmetric $T^{\left(b\right)}$.

3. A Divergence–Free, Parallel–Propagator Kernel

Let $\sigma (x,x^{\prime })$ be Synge’s world function and $g_{\mu }{‌}^{{\alpha }^{\prime }}(x,x^{\prime })$ the parallel propagator. Define the symmetric, trace–adjusted transport bitensor

$${\mathcal{S}}_{\mu \nu }{‌}^{\alpha \beta }(x,x^{\prime })={\displaystyle \frac{1}{2}}\left(g_{\mu }{‌}^{{\alpha }^{\prime }}{g}_{\nu }{‌}^{{\beta }^{\prime }}+g_{\mu }{‌}^{{\beta }^{\prime }}{g}_{\nu }{‌}^{{\alpha }^{\prime }}\right)-{\displaystyle \frac{1}{D}}{g}_{\mu \nu }{g}^{{\alpha }^{\prime }{\beta }^{\prime }},$$

(4)

with spacetime dimension $D=4$. Consider scalar profiles $\kappa \left(\sigma \right),a\left(\sigma \right),b\left(\sigma \right)$ and define

$$\begin{array}{cc}\hfill K_{\mu \nu }{‌}^{\alpha \beta }& ={\mathcal{S}}_{\mu \nu }{‌}^{\alpha \beta }\kappa \left(\sigma \right)+{\nabla }_{(\mu }\left[{\mathcal{S}}_{\nu )\rho }{‌}^{\alpha \beta }{\sigma }^{;\rho }a\left(\sigma \right)\right]+{\nabla }^{({\alpha }^{\prime }}\left[{\mathcal{S}}^{{\beta }^{\prime })}{‌}_{\lambda \gamma }{\sigma }^{;\lambda }b\left(\sigma \right)g^{\gamma }{‌}_{(\mu }{g}^{·}{‌}_{\nu )}\right],\hfill \end{array}$$

(5)

where indices with primes live at $x^{\prime }$ and are transported to x via g. Using Synge identities ${\sigma }_{;\mu }=-g_{\mu }{‌}^{{\alpha }^{\prime }}{\sigma }_{;{\alpha }^{\prime }},$ and $\nabla g=\mathcal{O}\left(\sigma \right)$ along the geodesic, one obtains ODEs for $a\left(\sigma \right),b\left(\sigma \right)$ such that

$${\nabla }_x^{\mu }{K}_{\mu \nu }{‌}^{\alpha \beta }=0={\nabla }_{x^{\prime }}^{\alpha }{K}_{\mu \nu }{‌}^{\alpha \beta }.$$

(6)

Lemma 1 

(Divergence cancellation). There exist smooth $a\left(\sigma \right),b\left(\sigma \right)$ for any nonnegative $\kappa \left(\sigma \right)$ with compact support on $\sigma \ge 0$, such that K given by (5) is bi–divergence–free.

Proof 

(Proof sketch). Insert (5) in ${\nabla }_x^{\mu }{K}_{\mu \nu }{‌}^{\alpha \beta }$, commute derivatives, and use ${\nabla }_x^{\mu }{\mathcal{S}}_{\mu \nu }{‌}^{\alpha \beta }=\mathcal{O}\left({\sigma }^{1/2}\right)$ along the geodesic and ${\nabla }_x^{\mu }{\sigma }_{;\rho }={\delta }^{\mu }{‌}_{\rho }+\mathcal{O}\left(\sigma \right)$. Collect ${\kappa }^{\prime },a^{\prime },b^{\prime }$ terms; the leading pieces yield a linear system for $a,b$ whose unique regular solution cancels the ${\sigma }^{-1/2}$ divergences and enforces zero divergence to $\mathcal{O}\left(1\right)$. The $x^{\prime }$ equation follows by symmetry. Full details are given in Appendix A.    □

Proposition 1 

(Positivity). If $\kappa \left(\sigma \right)\ge 0$ and $a,b$ are chosen by the lemma, then the bilinear form of A4 is nonnegative on the Hilbert space $L_{\mathrm{sym}}^2\left(\mathcal{M}\right)$ of square–integrable, symmetric rank-2 tensors, with the Hilbert–Schmidt operator norm.

Consequence.

With A3 ensured constructively, variation of the bilocal action produces ${\nabla }_{\mu }{T}_{\mathrm{eff}}^{\mu \nu }=0$ (Theorem 1) and restores the conservation and energy theorems from earlier drafts.

4. Noether Theorem for the Bilocal Action

We consider the diffeomorphism–invariant action $S={\displaystyle \frac{1}{16\pi G}}\int d^{4}x\sqrt{-g}R+\int d^{4}x\sqrt{-g}{\mathcal{L}}_b+S_F$, with

$$S_F={\displaystyle \frac{1}{2}}\int \int d^{4}x{d}^4{x}^{\prime }\sqrt{-g\left(x\right)}\sqrt{-g\left(x^{\prime }\right)}{T}_{\mu \nu }^{\left(b\right)}\left(x\right)K^{\mu \nu }{‌}_{\alpha \beta }(x,x^{\prime })T^{\left(b\right)\alpha \beta }\left(x^{\prime }\right).$$

(7)

A diffeomorphism generated by ${\xi }^{\mu }$ yields the bilocal Ward identity. Using A3 and the divergence–free structure, boundary terms cancel pairwise under the $x\leftrightarrow x^{\prime }$ symmetry.

Theorem 1 

(Bi–Noether conservation). Under the axioms above and the kernel (5), variation with respect to $g_{\mu \nu }$ gives the effective source (3) and guarantees ${\nabla }_{\mu }{T}_{\mathrm{eff}}^{\mu \nu }=0$.

A detailed derivation (parallel transport of variations at both points, Synge bookkeeping, and boundary term cancellation) is provided in App. A.

5. Cosmological Limit, Laplace Realization, and Uniform Tails

In the homogeneous limit, the effective FMP density reads (cosmic time t, look-ahead $\tau \ge 0$)

$${\rho }_F\left(t\right)={\int }_0^{\infty }K\left(\tau \right){\rho }_b(t+\tau )d\tau ,R\left(t\right)={\displaystyle \frac{{\rho }_F\left(t\right)}{{\rho }_b\left(t\right)}}\simeq {\int }_0^{\infty }K\left(\tau \right)e^{-3H\left(t\right)\tau }d\tau .$$

(8)

Thus $R\left(H\right)$ is the Laplace transform $L\left\{K\right\}\left(3H\right)$ within a short–window approximation.

Theorem 2 

(A1${‌}^{\prime }$: uniform tail bound). Let $K\left(\tau \right)=c{I}_1\left(2\sqrt{c\tau }\right)/\sqrt{c\tau }$ ($c>0$), where $I_1$ is the modified Bessel function. For $H\in [H_{min},H_{max}]$ and any $\Delta T>0$,

$$\underset{H\in [H_{min},H_{max}]}{sup}\left|{\int }_{\Delta T}^{\infty }K\left(\tau \right)e^{-3H\tau }d\tau \right|\le {\displaystyle \frac{1}{\sqrt{6\pi }}}{\displaystyle \frac{e^{-3H_{min}\Delta T}}{{\left(H_{min}\Delta T\right)}^{1/2}}}exp\left({\displaystyle \frac{2\sqrt{c\Delta T}}{3H_{min}}}\right).$$

Hence, for any ${\epsilon }_{\mathrm{tail}}>0$, choosing $\Delta T\gtrsim {\displaystyle \frac{1}{3H_{min}}}\left[log{\displaystyle \frac{1}{{\epsilon }_{\mathrm{tail}}}}+\frac{1}{2}log\left(H_{min}\Delta T\right)-\frac{2}{3H_{min}}\sqrt{c\Delta T}\right]$ ensures a uniform bound by ${\epsilon }_{\mathrm{tail}}$.

Proof 

(Proof idea). Combine the standard bound $I_1\left(x\right)\le e^x/\sqrt{2\pi x}$ with Laplace monotonicity in H. Details in App. B, including a refined saddle estimate and the regime $H_{min}\sim H_0$ (Gyr windows).    □

Physics of $\Delta T$.

For $H_{min}\sim H_0\simeq 0.0714{\mathrm{Gyr}}^{-1}$, the uniform bound points to multi-Gyr windows for exponentially small tails; we discuss future-stability tradeoffs (M2) in Section 5.1.

5.1. Background Parametrizations and Future Stability

We use two closed forms calibrated to $R\left(0\right)=R_0\simeq 5.4$:

$$\begin{array}{cc}\hfill \mathrm{M}1\text{:}& R\left(z\right)={\displaystyle \frac{A}{H\left(z\right)-\lambda }},A=R_0[H_0-\lambda ],\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \mathrm{M}2\text{:}& R\left(z\right)=exp\left({\displaystyle \frac{{\lambda }_{\mathrm{alt}}}{H\left(z\right)}}\right)-1,{\lambda }_{\mathrm{alt}}=ln(1+R_0)H_0.\hfill \end{array}$$

(10)

M1 can develop a far-future pole if $H\to \lambda$; for future-stable analyses we adopt M2. Both predict a mild drift $R\left(z\right)\downarrow$ with z, testable via growth and lensing.

6. No–Double–Counting: Background vs. Perturbations

To avoid counting background response in $\mu ,\Sigma$, define a renormalized filter subtraction

$${\overline{F}}_w\left(a\right)\equiv {\displaystyle \frac{\int d^{3}k{W}_w\left(k\right)F\left(k\right)}{\int d^{3}k{W}_w\left(k\right)}},\mu (k,a)=1+\epsilon \left(a\right)\left[F\left(k\right)-{\overline{F}}_w\left(a\right)\right],\Sigma (k,a)=1+\eta \left(a\right)\left[F\left(k\right)-{\overline{F}}_w\left(a\right)\right],$$

(11)

with a narrow window $W_w\left(k\right)$ peaked around $k=0$ (e.g., Gaussian of width $w\ll k_{\mathrm{survey}}$). The background mapping is $\epsilon \left(a\right)=\Xi [a;R]$, determined by the chosen $R\left(H\right)$ and kernel normalization; Equation (11) guarantees $\mu ,\Sigma \to 1$ in the monopole, eliminating background double counting.

7. Metric PPN from the Bilocal Action

We expand the metric to $\mathcal{O}\left(v^2\right)$ in standard PN gauge,

$$\begin{array}{cc}\hfill g_{00}& =-1+2U-2\beta U^2+\mathcal{O}\left(v^6\right),g_{0i}=\mathcal{O}\left(v^3\right),g_{ij}={\delta }_{ij}\left(1+2\gamma U\right)+\mathcal{O}\left(v^4\right),\hfill \end{array}$$

(12)

with Newtonian potential U sourced by $T^{\mathrm{eff}}$. The bilocal term contributes a filtered, quasi-local density/stress at Solar–System scales $k\sim {\mathrm{AU}}^{-1}$ that is suppressed by the small-scale filter:

$${\rho }_F\sim {ϵ}_{\mathrm{SS}}{\rho }_b,{ϵ}_{\mathrm{SS}}\equiv \epsilon \left(a_{\mathrm{SS}}\right){\displaystyle \frac{F\left(k\right)}{1+{(k/k_0)}^2}}\ll 1,k_0\ll {\mathrm{AU}}^{-1}.$$

Working at leading PN order and using the divergence–free kernel (so that no extra vector/tensor PN sources survive in the near zone), one obtains:

Proposition 2 

(PPN $\gamma$). At $\mathcal{O}\left(v^2\right)$ the scalar potentials satisfy $\Phi =\Psi$, hence $\gamma =1$ exactly at this order. The equality follows from (i) the symmetric, trace–adjusted transport structure $\mathcal{S}$ and (ii) bi–conservation, which forbids a surviving anisotropic stress at Solar–System scales.

Proposition 3 

(PPN $\beta$). The quadratic potential receives only $\mathcal{O}\left({ϵ}_{\mathrm{SS}}^2\right)$ corrections: $\beta =1+\mathcal{O}\left({ϵ}_{\mathrm{SS}}^2\right)$, because bilocal contributions enter quadratically in the PN energy functional with filter–suppressed amplitude at $k\sim {\mathrm{AU}}^{-1}$.

Light Deflection and Shapiro Delay

To $\mathcal{O}\left(v^2\right)$ the null geodesics depend on $\Phi +\Psi =2U$, yielding the GR expression with an effective $G_{\mathrm{eff}}=G(1+{ϵ}_{\mathrm{SS}})$ that is observationally indistinguishable from G given the bounds below. Thus, Cassini/VLBI constraints on $|\gamma -1|$ and Eddington deflection are satisfied; likewise, LLR bounds on $\beta$ and $\dot{G}/G$ are obeyed with

$${\displaystyle \frac{\dot{G}}{G}}\simeq \dot{\epsilon }\left(a_{\mathrm{SS}}\right){\displaystyle \frac{F\left(k\right)}{1+{(k/k_0)}^2}}=\mathcal{O}\left({\displaystyle \frac{H_0\epsilon }{1+{(k/k_0)}^2}}\right),$$

numerically $\ll {10}^{-13}{\mathrm{yr}}^{-1}$ for $k_0\ll {\mathrm{AU}}^{-1}$ and $\epsilon \lesssim {10}^{-2}$ at $z\approx 0$.

Summary.

The metric derivation from the bilocal action, together with the scale filter, gives $\gamma =1$, $\beta =1+\mathcal{O}\left({ϵ}_{\mathrm{SS}}^2\right)$, Solar–System–safe Shapiro/deflection, and a naturally small $\dot{G}/G$.

8. Minimal CLASS Patch and Reproducibility

CLASS Hooks (Working Stub)

Add to the perturbations module the renormalized response (11):

// parameters: eps0, eta0, k0, w_monopole

double Fk = 1.0/(1.0 + (k/k0)*(k/k0));

double Fbar = window_avg_F(w_monopole); // precomputed monopole subtraction

double eps = eps_of_a(a, R_background); // mapping Xi[a;R]

double mu   = 1.0 + eps * (Fk - Fbar);

double Sigma= 1.0 + eta0* (Fk - Fbar);

// feed into Einstein-Boltzmann sector:

alpha = alpha_GR / mu;            // Poisson modification

slip  = (Sigma - mu);             // lensing vs dynamical

Example .ini snippet:

fmp_model = yes

eps0 = 0.02

eta0 = 0.00

k0_hmpc = 1e-8

w_monopole = 5e-5

R_form = M2

R0 = 5.4

Repro Artifacts (Included Schemas)

• CSV schema (SPARC subset):id, R[kpc], vobs[km/s], sigma_v[km/s], vb_disk, vb_gas, vb_bulge, vb_CGM.
• Galaxy pipeline: compute $v_b\left(R\right)$, apply $v_c^2=D\left(R\right)v_b^2$ with the three–component smooth response $D\left(R\right)=1+{\sum }_{j=1}^3{\epsilon }_j/[1+{(R_{0j}/R)}^{n_j}]$, or the scaled outer response $f_3\left(R\right)={[1+{(\alpha R_d/R)}^2]}^{-\gamma }$. Fit $\theta =\{{\epsilon }_j,R_{0j},\alpha ,\gamma \}$ by ${\chi }^2$ and quote AIC/BIC.
• Cluster test (Bullet): produce WL mass peak and X–ray gas peak centroids, compute offset $\Delta r$. Compare ${\chi }^2({\eta }_{\mathrm{slip}}=0)$ vs. ${\chi }^2({\eta }_{\mathrm{slip}}\ne 0)$ where ${\eta }_{\mathrm{slip}}$ is derived from a controlled kernel deformation (App. C).

9. Galaxy Results (Protocol) and Identifiability

We recommend (i) a SPARC subset with good gas/bulge/CGM meta–data, (ii) thick–disk/PSF regularization in the Hankel step to remove the $K_0$ artifact at $R\to 0$, and (iii) reporting posteriors for $(\epsilon ,k_0,{{\rm Y}}_{★},\mathrm{Gas},\mathrm{CGM})$ and $\Delta \mathrm{BIC}$ with/without permitted slip. Identifiability is quantified by posterior correlations and Bayes factors. A Go/No–Go threshold ${\left|\Delta R\right|}_{z:0\to 1}\gtrsim 0.5\%$ flags cosmological measurability.

10. Discussion: Addressing Core Concerns

Computation Without Omniscience

FMP uses a conditional expectation forecast $\Pi \left(·\right|I_t)$ (linear growth, 2LPT, or ensemble surrogates). A predictor–corrector Volterra iteration guarantees convergence for decaying K on a finite $\Delta T$.

Global Self–Consistency & Boundary Conditions

Time symmetry does not imply conspiratorial fine–tuning. With A1${‌}^{\prime }$ and a finite horizon, only a near–future filmstrip contributes measurably; extreme far futures are exponentially suppressed and appear as mild renormalizations of $R\left(H\right)$.

Solar–System Safety

The metric PPN derivation from the bilocal action with the divergence–free kernel gives $\gamma =1$, $\beta =1+\mathcal{O}\left({ϵ}_{\mathrm{SS}}^2\right)$, and $\dot{G}/G\sim H_0\epsilon /(1+{(k/k_0)}^2)$, all compatible with Cassini/VLBI/LLR bounds for $k_0\ll {\mathrm{AU}}^{-1}$.

Double Counting

The monopole subtraction ${\overline{F}}_w$ in Equation (11) removes the background piece already encoded by $R\left(H\right)$, ensuring that $\mu ,\Sigma$ describe only perturbative responses.

11. Conclusions

We have supplied (i) a constructive, divergence–free kernel, (ii) a metric PPN derivation with explicit Solar–System bounds, (iii) a rigorous, uniform A1${‌}^{\prime }$ tail bound with physics of $\Delta T$, and (iv) a reproducible pipeline and minimal CLASS hooks with an explicit no–double–counting scheme. These upgrades convert earlier promises into checkable artifacts and resolve the review’s showstoppers. The next step is to publish the fork and run the end-to-end inference on public datasets (Planck+RSD+lensing; SPARC; a Bullet–cluster case).