Emergent Spin-2 Gravity from Gauge-Invariant Network Transport: Continuum Limit and Cosmological Dynamics

We construct a discrete geometric–transport framework whose continuum limit reproduces classical General Relativity without modifying its constraint structure. Starting from a Regge- type discretization supplemented by a purely gradient transport functional, we derive the effec- tive Einstein–Hilbert action in the long-wavelength limit and identify the emergent Planck scale in terms of geometric stiffness parameters. We demonstrate that the Hamiltonian and momentum constraints close according to the standard Dirac algebra, ensuring preservation of full diffeomorphism invariance. The propa- gating sector contains exactly two tensor degrees of freedom. The quadratic tensor action is free of ghost and gradient instabilities, and linear scalar perturbations introduce no additional propagating degree of freedom, preserving the spin-2 character of gravitational interactions. In homogeneous configurations, the transport sector does not generate a local vacuum energy contribution. Consequently, the cosmological constant appears as a constant of integration associated with the Hamiltonian constraint rather than as a radiatively unstable vacuum term. Upon reduction to FLRW symmetry, the model reproduces the standard Friedmann equations, with the homogeneous transport mode behaving as a stiff component (w = 1) that dilutes as a^{-6} . At the level of linear cosmological perturbations, the theory yields a modified Poisson equa- tion characterized by a scale-independent effective gravitational coupling Geff(a), while preserv- ing Φ = Ψ and introducing no gravitational slip. The background expansion coincides with ΛCDM in the absence of early-time modifications, and a reproducible geometric diagnostic is provided to test background consistency with the CMB sound-horizon angle when high-H0 sce- narios are considered. A representative early-time modification (Variant A) was implemented in CLASS v3.3.4 via a background-only replacement G → Geff(a). We verify that CMB tempera- ture spectra satisfy |Rℓ −1| ≲ 0.5% for ℓ ≳ 50, while linear growth-rate deviations remain at the few-percent level (≲ 3% in fσ8), with no additional propagating degrees of freedom introduced. The framework thus provides a consistent discrete origin for classical spacetime geometry and gravitational dynamics, together with controlled background- and linear-level cosmological tests, while leaving the matter sector unspecified. Its scope is restricted to the classical low-energy regime, where it remains structurally equivalent to General Relativity.