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 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: the path sum as it stands is proved not to have a massless one-loop graviton, given local bounds at four points of the Brillouin zone, 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. Version 1.1 answers the referee report on version 1.0 at Pith (https://doi.org/10.67347/pith.preprint.2026.000019). It separates the tree-level coupling from the measured long-wave coupling throughout, corrects the drift factor, lists the candidates for the pure number explicitly, and adds the results proved since version 1.0, each with its verification status. Preprint of the Recognition Physics Institute, 20 pages. Not peer reviewed.