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) = (m_e/m_P)² = 1.7518 × 10⁻⁴⁵. A theory of gravity without free parameters has to produce it. We ask how far the recognition ledger's lattice path sum goes. Its long-wavelength limit is linearized Einstein gravity, with exact lattice solutions for matter and an explicit second-order limit at the identified value κ = 1 of the lattice's one geometric parameter, and the lattice form of Newton's law holds for exactly one coupling, G = 1/(8πs), where s is the scale of the Regge action. The ledger's phase premise ties s to the scale of the path sum's phase; then the Planck length 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 at κ = 1 is b = (2/√3) λ_rec ≈ 1.92 l_P. Under these premises the coupling is not a free parameter: every gravitational length is a pure multiple of the record length. The value of G in SI units needs the record length in metres, and that is one pure number: the number N_e of ledger ticks in one Compton period of the electron, with G = π² ħc/(ln 2 · m_e² N_e²). The measured G puts N_e at 9.0156 × 10²²; the framework's present premises leave it free, every candidate on record misses it, and its derivation is the paper's precise open target. Two further requirements of a theory without free parameters are open as well: a massless graviton and the nonlinear Einstein equations. For the path sum as it stands the graviton is proved not to be massless at one loop, given one local bound that is still open, and read through the identified spacing the computed remainder is a one-loop mass scale some fifty-one orders of magnitude above the measured bound, which an action derived from the ledger has to remove. The readings of a triangle's charge that the ledger supplies make its even part zero, which leaves the remainder given that bound, or proportional to the area, which leaves it at every value computed; an even part chosen to remove it would be a fit. Measured values of G appear only as comparisons. Where a proof is complete but its independent verification is not, the text says so.