Dynamical limit and record

For every fixed finite $M\ge 2$,

Thus

The probability cost is half this norm per state, and the two-state contrast cost is at most the full norm.

The witness has compact gravitational momentum support but only Schwartz scalar momentum decay. Accordingly $d_R\to d_{\ast }$ superpolynomially; it is not generally eventually constant. Uniform number estimates justify a single time calibration for all spectral cutoffs.

At $\lambda =0$, the identical scalar factors give exact equality. The $R,N$ limits may be combined at fixed $M$ using the explicit compact-operator and Hilbert-tensor remainder. No measured $M\to \infty$ limit is included.

Figures fig:model2-kernel-ideals and fig:fixed-M-convergence distinguish the coefficient topologies and the limits available at fixed number cutoff.