Manuscript Oai227-A@1 · Oai227-A.tex · section · 7f45f607

A probability-law formulation of the same certificate

Stage 6: the one explicit finite certificate › A probability-law formulation of the same certificate · lines 1458–1479

Extend the stationary function to the compact by

A_{2, J} (s, t) = A_{2, J} (t) .

For either disorder law, sample the same $J$ for $A_{2, J}$ and $B_{2, J}$ . This is an explicit coupling of their laws on $C (K)$ , with

A_{2, J} - B_{2, J}_{\infty, K} < ε_{cert} almost surely .

Thus the within-disorder function laws admit a coupling with this uniform error.

This restates the finite theorem. It is not a coupling between Gaussian and Rademacher environments.

Displayed formulas in this passage

unlabeled · line 1461
$A_{2, J} (s, t) = A_{2, J} (t) .$
unlabeled · line 1468
$A_{2, J} - B_{2, J}_{\infty, K} < ε_{cert} almost surely .$