Small-step martingale actions cannot hold everpresent Lambda constant: an exact bound beyond Gaussian histories

In everpresent Lambda the cosmological constant is an accumulated action divided by a growing four-volume, Lambda = S/V. A recent width-law theorem shows that, for jointly Gaussian (or linear-homoscedastic) histories, a present value in the typicality window is compatible with low-redshift constancy in conditional mean square only if the width barely grows into the past, and notes that a non-Gaussian maximal coupling of the same marginals escapes it. We consider every construction of the form Lambda = S/V in which the action is a martingale in the volume clock of nested past regions, that is, the action of new elements has zero conditional mean given the past, whatever its distribution or its dependence on the past. For continuous martingale actions we obtain the exact supremum of the probability that Lambda stays within a fraction epsilon of its value while the volume grows by a factor K >= rho: (1/2) rho^((1+epsilon)/(2 epsilon)) K^(-(1+epsilon)/(2 epsilon)), with rho = (1+epsilon)/(1-epsilon). For general martingale actions an optional-stopping argument gives a bound of the same form plus an overshoot term that vanishes when downward jumps of the action are small; rare downward jumps of order the total action are the one way around it. On the flat LambdaCDM past cone (K about 27 since z = 1) the supremum is 2.1e-8 for 10% and 8.7e-5 for 20% constancy. The maximal coupling cannot be realised by a martingale action. A drift that returns a fraction lambda of the current Lambda to the action relaxes the exponent by 1 - lambda; for lambda > 1/2 the fluctuation scales as V^-(1-lambda) and depends on the earliest volume as a power, so its amplitude must be tuned. Even at lambda = 1/2 the supremum for 10% constancy since z = 1 is 1.8e-4. The deposit contains the scripts that compute the past-cone volume ratios, the exact supremum, the general bound, a dynamic-programming check and a Monte Carlo check.