Quantum Gravity from the Recognition Ledger: A Finite One-Loop Path Sum, Einstein Gravity at Long Wavelengths, and the Massless-Graviton Problem

A quantum theory of gravity built from the recognition ledger must meet four demands: a lattice, action and measure derived from the ledger, not chosen; finiteness, beginning at one loop; a massless graviton; and Einstein gravity at long wavelengths. On the Kuhn lattice with flat metric I − κ11ᵀ, massless corner doublers occur exactly at κ = 1, where the lattice interval reads the ledger's posting order; the foundation does not decide κ, so κ = 1 is an identification, and deriving it means deriving the doublers. A measure weight built from the ledger's cost of curvature at each hinge, under two named identifications and a channel count, gaps all four doublers, vanishes to fourth order on long waves, and gives the real part of the ordered one-loop anisotropy of the graviton self-energy a finite limit, where the native path sum diverges logarithmically; with it the Gaussian path sum is unitary in record time in the continuum limit at fixed weight strength, though damped on the lattice, while the bare Regge path sum has a growing mode. A Ward identity decides whether the graviton stays massless at one loop: the zero-momentum transverse-traceless remainder is i/2 times the difference of two Ward defects and vanishes if the one-loop determinant is invariant under volume-preserving linear vertex moves. No real weight or density restores the symmetry, no phase cS + Σ_h δ_h³ Γ_h has it or changes the remainder, every candidate computed to convergence leaves a nonzero remainder, and the native remainder cannot vanish in the ordered limit, given local bounds at four points of the zone. Under two named maps the unrepaired sum's remainders correspond to a mass scale fifty-one orders above the measured bound, a conditional diagnostic. Classically, linearized Einstein gravity, Newton's law with G = 1/(8πs) and, under named identifications, the second order with matter that conserves postings are recovered at long wavelengths at κ = 1; on sampled metric jets the metric part of the lattice field equations is the full Einstein density. A massless graviton, through an action derived from the ledger with a vanishing Ward defect, is the paper's precise open target, and the exclusions are its results; a derivation of κ, the measure identifications, exact unitarity and the nonlinear limit are open too.