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: its lattice, action and measure must be derived from the ledger rather than chosen; it must be finite, beginning with its ordered one-loop limit; its graviton must be massless, which at one loop requires a vanishing zero-momentum remainder; and at long wavelengths it must be Einstein gravity. On the Kuhn lattice with flat metric I − κ11⊤ , massless corner doublers occur exactly when κ = 1, the value at which the lattice interval reads the ledger's posting order; the foundation as stated does not decide κ, so deriving κ = 1 and deriving the doublers are the same problem. A measure weight built from the ledger's cost of curvature at each hinge, under two named identifications, 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. A Ward identity decides whether the graviton stays massless: invariance of the one-loop determinant under volume-preserving linear moves of the vertices removes the zero-momentum transverse-traceless remainder, in general i/2 times the difference of the second- and first-order Ward defects. No real weight or density restores the symmetry, no phase cS + Σ_h δ_h³ Γ_h has it or changes the remainder, and every candidate computed to convergence leaves a nonzero remainder. For the native path sum it cannot vanish in the ordered limit, given one local bound at the origin of the zone; read through named identifications, the computed values put the one-loop graviton mass scale fifty-one orders of magnitude above the measured bound. In record time the identified Regge path sum has a growing mode, which rules out a positive norm preserved by the evolution. Classically, linearized Einstein gravity, Newton's law with G = 1/(8πs) and the second order with matter that conserves postings are recovered at long wavelengths at the identified value κ = 1. 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. Preprint of the Recognition Physics Institute, 46 pages. Not peer reviewed.