Quantum Gravity from the Recognition Ledger: Weighted One-Loop Graviton Responses with Finite Real Parts, 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, 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. Our one-loop statements concern a stated diagram model: a unit-weight gauge term, no ghost line, the Feynman iε. A measure weight built from the ledger's cost of curvature, under two named identifications and a channel count, gaps all four doublers, vanishes to fourth order on long waves, and gives the real parts of the model's one-loop graviton responses finite ordered limits (Theorem 7.4), where the native sum diverges. With it the record-time roots of the Gaussian path sum are stable in the continuum limit at fixed weight strength, given an inference about off-circle modes, though the sum is damped on the lattice, where the bare Regge sum has a growing mode. A Ward identity decides whether the model's graviton stays massless at one loop: the zero-momentum remainder is i/2 times a difference of two Ward defects and vanishes if the 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. 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 κ = 1, and every long-wave vertex of the action is Einstein's, as field equations on sampled jets. A massless graviton, through a derived action with a vanishing Ward defect, is the paper's precise open target, and the exclusions are results; κ, the measure identifications, a ghost or gauge measure, a norm-preserving record-time evolution and the nonlinear limit are open too.