A Finite-Horizon, Zero-Offset Future-Mass Kernel for FMP Gravity: Passing the Cosmology-Light Test in Background and Linear Growth

A Finite-Horizon, Zero-Offset Future-Mass Kernel for FMP Gravity: Passing the Cosmology-Light Test in Background and Linear Growth Farzad Lali∗ October 2, 2025 Abstract We revisit the “Future-Mass Projection” (FMP) idea—a nonlocal, future-inclusive response of the effective gravitational source—and present a physically constrained formulation in which the future kernel has finite horizon and vanishing DC offset. In the small-horizon limit, the kernel induces a well-defined, band-averaged modification of the linear growth source while leaving the homogeneous background expansion effectively unchanged at the sub-percent level. We formulate a transparent “Cosmology-Light” test: (i) background con- sistency H(z) versus ΛCDM (BAO/SN/chronometer range), and (ii) linear growth D(z) and f σ8(z) in the redshift window probed by RSD. With the finite-horizon, zero-offset kernel we show: (a) H(z) deviations remain ≲ O(10−2) out to z ∼ 1, thus BAO-safe; (b) f σ8(z) can be suppressed by ∼ 10–15% around z ∼ 0.3–0.7, in the right direction to ease current growth- tension trends, while keeping the early-time CMB anchor intact. We detail the falsifiability of this tuned FMP: a combined BAO/RSD program with conservative scale cuts can exclude the model if |∆H|/H 3% in 0.3 ≲ z ≲ 1 or if f σ8 suppression fails to fall in the predicted window at z ≃ 0.3–0.7. Local constraints (PPN γ, GW speed cGW = c) remain satisfied because the kernel correction vanishes in the local limit. Our analysis preserves the original FMP motivation (late-time effective growth damping) while repairing the background drift in earlier drafts, thereby passing the Cosmology-Light gate and specifying clear near-term observational kill-tests.