Graviton Dynamics from Modular Non-Commutativity: John's Equations and the D(3,2) Doubleton on Kinematic Space (IGS V)

This paper explores a pathway where the massless graviton wave equation (hμν=0), complete with exactly two physical polarisations, emerges from the quantum-information structure of a holographic boundary CFT. In this framework, the derivation does not rely on postulating the Einstein-Hilbert action or a gravitational variational principle. The proposed mechanism rests on two primary inputs: Physical Input: The non-commutativity of modular Hamiltonians of overlapping causal diamonds ([K̂A, K̂B] ≠ 0). Geometric Input: The Ouroboros global consistency condition on kinematic data. Operating within the 6-dimensional kinematic space M_ = SO(3,2)/[SO(2,1)(1,1)] of AdS₄/CFT₃ , we observe that these conditions constrain the kinematic data to satisfy John's integrability equations. The resulting solution naturally aligns with the $D(3,2)$ doubleton representation of SO(3,2) (saturating the unitarity bound with E₀=3, C₂=6) and takes its canonical normalization from the KMS boundary condition via the First Law of Entanglement. Key Findings: Explicit Poisson Bracket Formula: We provide a closed-form expression for , kB\ that applies to general causal diamonds (both spherical and tilted), verified computationally via SymPy. Polarisation Lemma: An algebraic reduction of the spin-2 John's condition to the well-established scalar case, offering a self-contained proof that helps address existing caveats in the kinematic-space literature. Spin-1 Constraint: We demonstrate a structural limitation indicating that photon dynamics (C₂(D(2,1))=0) do not emerge from this specific 6D kinematic space mechanism. Testable Implication (TT̄ Deformation): The framework suggests a bulk graviton mass of m² = ε(3+ε) ≈ 3ε under a TT̄ deformation that shifts the lowest weight E₀ → 3+ε. Series Context: As Paper V in the Information-Geometric Spacetime (IGS) series, this work attempts to bridge a conceptual gap by offering an explicit mechanism that ties kinematically reconstructed fields to the bulk graviton wave equation.