Newton's Constant and the Recognition Ledger: Einstein's Equations, the Graviton, and the One Number That Fixes G

In units where the speed of light and Planck's constant are fixed, Newton's constant is one pure number: for the electron, α_G = G m_e²/(ħc) = 1.7518 × 10⁻⁴⁵. We ask how far the recognition ledger's lattice path sum goes toward deriving it. At the identified value κ = 1 of the lattice's one geometric parameter, its long-wavelength limit is Einstein gravity at linear order and, with matter, at second order, and the lattice form of Newton's law holds for exactly one tree-level coupling, G_tree = 1/(8πs), where s is the scale of the Regge action. Under the ledger's phase premise the Planck length of that coupling is a fixed fraction of the record length, l_P = λ_rec/(2√ln 2) ≈ 0.6006 λ_rec, and if light crosses one record length per record rank the lattice spacing is b ≈ 1.92 l_P. In SI units G_tree = π²ħc/(ln 2 · m_e² N_e²), with N_e the number of ledger ticks in one Compton period of the electron. A laboratory measures the long-wave coupling Z G_tree, with a positive factor Z that the quantum path sum fixes and that is not yet computed, so the measured G fixes N_e/√Z = 9.0156 × 10²². The present premises leave N_e free, every candidate on record misses that value by many orders of magnitude, and deriving N_e is the paper's precise open target. A massless graviton and the nonlinear equations are open too: in a stated one-loop model, the path sum as it stands is proved not to have a massless graviton, and read through two named maps its remainder corresponds to a mass scale fifty-one orders of magnitude above the measured bound, a conditional diagnostic. Measured values enter only as comparisons. Where a proof is complete but its independent verification is not, the text says so.