Audit 1: stationary edge input

For Gaussian disorder the external route completes $W$ by an independent Gaussian diagonal. The added diagonal changes the spin energy by a configuration-independent constant, because ${{\sigma }_i}^2=1$. This completion can supply Gaussian orthogonal eigenframe structure without changing the finite heat-bath dynamics OAI227Background.

The audit must nevertheless keep three objects separate:

1. the spectral edge of $W$;

2. the projected spin coordinates in (eq:finite-entrance);

3. the spectrum and semigroup of the nonlinear heat-bath generator in (eq:dynamical-trace).

Equation (eq:spin-action) shows why diagonalizing $W$ does not generally diagonalize $L_J$.

The stationary edge law, its conditioned measure, convergence of finite-coordinate integrals, and the limiting energy form are substantive external inputs. Even if all of them are supplied, the extra factor $w_s$ in the quench trace still requires control. Stationary convergence is not the quench theorem.