Spacetime from a Prime Sieve: Wave Dynamics, Entanglement Area Law, and a Closed Emergent-Gravity Cosmology

We present a closed framework in which spacetime, gravity, the cosmological background, and the elec tromagnetic sector emerge from a single arithmetic growth rule: a prime sieve whose additive projection provides time and whose multiplicative projection provides space. Wave dynamics on the resulting lattice — a Kronig–Penney ff-chain and its three-dimensional shell embedding — exhibits rational transmission valleys whose strengths follow the Hardy–Littlewood singular series, verified in blind tests. The chain’s entanglement entropy saturates at a plateau governed by a mean-field law 𝐹(Δ) = 1 6 ln(1/Δ) + 0.4621 with the conformal coefficient 6 1 = 𝑐/6 exact and ultraviolet-universal. From the shell’s channel counting we derive an entanglement area law with 𝑘𝐹ℓ0= 𝑑/2 (locking 𝑑 = 3.000 ± 0.002), an Euler–Maclaurin theorem showing that the heat-kernel curvature coefficient 𝑎2= 𝜒/6 is generated by lattice boundary terms, and an explicit constraint algebra that gauges away the 𝑙 = 0 mode. Gravity closes statically: 𝐺 is fixed by the entropy–area coefficient 𝛼 = 0.10604 (0.42% from observation), the horizon temperature 𝑇 = ℏ𝑐/2𝜋𝑅𝐻 follows from the first law, and Jacobson’s argument then yields the Einstein equation; de coupling, a c-theorem argument, and an explicit heat-kernel ff-function — all verified by coarse-graining numerics — show that 𝐺 does not run, consistent with lunar laser ranging. The same growth rule yields a background cosmology with no free parameters: Ω 𝑚= 0.3055, ΩΛ= 0.6945, 𝑞0= −0.5417,𝐻0𝑡0= 0.9592, 𝑤 = −1 and Ω𝑘= 0 exactly, and 𝜌Λ∼ ℓ−2𝑃𝑅−2𝐻 from three convergent mechanisms. New in v2: the electromagnetic sector. The photon is the tangential tension wave of the skeleton coupling medium: masslessness forces a purely derivative (inertial) coupling of the prime shells, the two transverse polarizations are the dimension of the shell tangent plane, and the transverse-elasticityvacuum-Maxwell map is exact. Charge is the occupation imbalance of the filling field: quantization 𝑞 ∈ ℤ follows from sieve exclusion (per-mode Pauli), particles and holes give antimatter for free, and the Coulomb law follows from the Gauss constraint with the shear modulus. The fine-structure constant was subjected to a pre-registered audit — a single locked expression 𝛼 −1 = 4𝜋𝜅 skel /𝑒2 def , three frozen estimator chains, three comparisons — and all three comparisons missed (12.6–13.6, then a conditional closure at Poisson ratio 𝜆/𝜇∗ = 2.88, then the gravity-core fast attack outside its band). The ff account is therefore sealed as a documented falsification, not a fit. The sector leaves sharp observational residue: three strict zero predictions (no smooth linear Lorentz violation, no vacuum birefringence, no photon mass) and a parameter-free rational-comb dispersion signature near the Planck energy. Falsifiable targets include 𝑁 eff = 3.013 (CMB-S4), a rational-comb template in the primordial spectrum (fixed hierarchy, one amplitude), and a +0.2498√ 𝐴 black-hole entropy correction. All discarded branches (including a Ġ prediction excluded by LLR at 15ff, and the ff programme itself) are documented. A companion paper audits the fermion-mass programme that grew out of the electromagnetic sector’s core-energy formula.Thirty-six scripts reproduce every number.