The leading-soft cubic graviton self-interaction on the black-hole horizon
We expand the Einstein--Hilbert action to cubic order about the Schwarzschild horizon, in the even Regge--Wheeler gauge of the Gaddam--Groenenboom--'t~Hooft near-horizon framework, and derive the cubic graviton self-interaction. Our central result is a vanishing theorem: at leading soft order the self-coupling of the purely traceless longitudinal polarizations is identically zero. This is a framework-specific statement: even RW gauge, GGV sector, leading soft order. The surviving interaction lives in the trace sector; its on-shell equal-$\ell$ weight $W(λ,λ,λ)=-3λ(2λ^2+λ+3)/(λ+1)^2$, follows from two independent derivations agreeing to machine precision. We simulate the resulting Hamiltonian at two complementary scales. On IBM Qiskit/Aer, with exact cross-checks, the hardware-format circuits confirm quantitatively what two structural facts already predict, a conserved charge that only the cubic vertex violates (opening $ϕϕ\to hh\to4ϕ$) and a resonance-free boost spectrum (gap $\to1/2$): the longitudinal channel is perturbatively rigid, and the simulation measures the residual dressing ($d_{\rm eff}=1.06$; multiplicity far from thermal, Poisson, and Haar). A symmetry-graded matrix-product-state evolution then climbs the multipole ladder to sixty modes, the $120$-qubit register, and settles the one question the multiplet cannot: on a late-time measure common to both engines the inelasticity converges to $\barη_\infty\simeq0.16$ (understated by the single multiplet), the dressing stays area-law with peak entanglement $\simeq0.45$ nats far below the Page value, and the result is cutoff-converged and robust to $96$ qubits at larger occupation. The rigidity therefore persists rather than scrambling as the register grows.