A Joint Bound on the Bounce Amplitude from the Full Distance Ladder: Fisher Combination of Sixteen Rungs

This paper combines the rung-by-rung frame sensitivities computed across the distance ladder into a single bound on the bounce amplitude, using a Fisher combination of the independent tolerances. Treating each probe's tolerance as a Gaussian constraint on the asymptotic amplitude epsilon, the acoustic transverse and radial modes, the sound horizon, the supernova diagram, the chronometers, and the time-delay distances dominate the combination, while the local stellar rungs contribute negligibly, exactly as their individual sensitivities require. The combined standard deviation is sigma_epsilon = 0.030, giving a 95 percent upper limit of epsilon = 0.058 when the fiducial amplitude is zero. Removing the acoustic probes entirely loosens the limit to 0.228, quantifying how much of the ladder's verdict is carried by the plasma scale. A bounce amplitude of 0.10 already lies outside the combined bound, while amplitudes below 0.06 remain open by the ladder as a whole even though several individual rungs would permit them. This study is computed from stated assumptions, not measured.