Planck-Bandlimited Spin-2 Gravity and a Nonsingular FLRW Geometry

Two foundational problems afflict our description of gravity: ultraviolet divergences in its quantum formulation and a curvature singularity at the Big Bang in its classical one. Both are addressed here by imposing a single kinematic restriction — a Planck-scale mode cutoff on the temporal frequency in conformal time, |k| ≤ ℓₚ⁻¹, on a massless spin-2 field propagating in a structureless void — and deriving its consequences. For the radiation-dominated FLRW case this is equivalent to restricting the spatial momentum magnitude to sub-Planckian values in the comoving frame. The band-limited projection of the singular scale factor a(η) = |η|/ℓₚ yields the closed-form entire function aᴿᵉᶢ(η) = (2/π)[(η/ℓₚ) Si(η/ℓₚ) + cos(η/ℓₚ)], with strictly positive global minimum aᴿᵉᶢ(0) = 2/π and Kretschmann scalar Kᴿᵉᶢ(0) = (3/4)π⁴ℓₚ⁻⁴ ≈ 73.06 ℓₚ⁻⁴; both are parameter-free. The classical Big Bang singularity is replaced by a smooth bounce whose properties are fixed by ℓₚ alone. Two formal results follow from the same restriction: the Deser bootstrap produces the Einstein–Hilbert action within the restricted field space, and all graviton loop integrals are finite by power counting. A companion paper [34] addresses the gauge-consistency question on which both rest, proving an exact one-loop transverse-traceless Ward identity at all sub-Planckian momenta, bounding the deviation from general relativity by a measured function f(k) ≈ k/Mₚ that is of order 10⁻⁶⁰ at cosmological scales, and quantifying the residual Planck-scale breaking of nonlinear diffeomorphism invariance as a prediction rather than an inconsistency. The regulated metric satisfies the projected Einstein equations with a geometry-derived effective stress-energy that recovers radiation asymptotically; the consistency check is given in Section 5.5.