diff --git a/sk_model_critical_slowing_v2.txt b/sk_model_critical_slowing_v2.txt
index 7dc92b2..f2d448f 100644
--- a/sk_model_critical_slowing_v2.txt
+++ b/sk_model_critical_slowing_v2.txt
@@ -3745,3 +3745,118 @@ including when $k_n=0$.
 
 Intersect these two lower-bound events with the event in Proposition 9.4 to obtain all four asserted bounds simultaneously. $\square$
 
+
+---
+
+## 10. Appendix: why the equilibrium model alone has no mixing time
+
+The main theorems specify both the update rule and the clock. Two elementary constructions show why these are mathematical assumptions, even if the equilibrium law and reversibility are already fixed.
+
+A third calculation distinguishes high-probability asymptotic bounds from essential bounds over every disorder realization.
+
+### Proposition 10.1 (Arbitrary refresh times)
+
+Fix any realization of the SK measure on $\Omega_n=\{-1,+1\}^n$, $n\ge2$. For every positive integer $r$, there is an irreducible, aperiodic, reversible Markov kernel with stationary law $\mu$ whose worst-state threshold-$1/4$ mixing time equals $r$.
+
+#### Proof
+
+At each step, refresh the whole configuration from $\mu$ with probability $p$ and otherwise leave it fixed. In operator notation,
+
+$$
+\Pi(x,y)=\mu(y),
+\qquad
+K_p=(1-p)\mathrm{Id}+p\Pi,
+\qquad
+0<p\le1.
+$$
+
+All entries of $K_p$ are positive, and
+
+$$
+\mu(x)K_p(x,y)=\mu(y)K_p(y,x).
+$$
+
+The identity $\Pi^2=\Pi$ gives, for $0<p<1$,
+
+$$
+K_p^k=\Pi+(1-p)^k(\mathrm{Id}-\Pi),
+$$
+
+and
+
+$$
+\max_x\|K_p^k(x,\cdot)-\mu\|_{\mathrm{TV}}
+=(1-p)^k(1-\mu_{\min}).
+$$
+
+Here
+
+$$
+\mu_{\min}\le2^{-n}\le1/4.
+$$
+
+Choosing
+
+$$
+1-p=\left[\frac{1}{4(1-\mu_{\min})}\right]^{1/r}
+$$
+
+gives distance exactly $1/4$ at $r$ and strictly larger at every earlier integer. The case $p=1$ also has mixing time one. $\square$
+
+The same freedom remains if transitions are required to change at most one spin. Given the uniform-site heat-bath kernel $P$, the kernel
+
+$$
+(1-p)\mathrm{Id}+pP
+$$
+
+has the same stationary law, detailed balance, irreducibility, and transition restriction. Starting from a minimum-mass state, its distance after $k$ steps is at least
+
+$$
+(1-p)^k-\mu_{\min},
+$$
+
+by the event that no update was attempted. Its mixing time consequently diverges as $p\downarrow0$. Allowing $p$ to depend on $n$ changes possible growth classes as well.
+
+We next fix the dynamics completely. The remaining distinction is between typical disorder as $n$ grows and all disorder at a fixed dimension. Unbounded Gaussian couplings already separate those notions for two spins.
+
+### Proposition 10.2 (A two-spin example)
+
+For $n=2$ and the discrete uniform-site heat-bath kernel, the mixing time has no finite deterministic essential upper bound over the Gaussian coupling, although it is finite for every realization.
+
+#### Proof
+
+A large positive coupling keeps the two aligned states persistent.
+
+This can be quantified using the linear observable
+
+$$
+g(x)=x_1+x_2.
+$$
+
+Put $w=W_{12}$. Spin-flip symmetry gives $\mu[g]=0$, and conditional expectation computes the eigenvalue exactly:
+
+$$
+Pg=\theta g,
+\qquad
+\theta=\frac{1+\tanh w}{2}.
+$$
+
+Starting at $(1,1)$, the expectation of $g$ after $k$ steps is $2\theta^k$. Since $\|g\|_\infty=2$, the dual total-variation bound yields
+
+$$
+\|P^k((1,1),\cdot)-\mu\|_{\mathrm{TV}}
+\ge\tfrac12\theta^k.
+$$
+
+For each fixed $k$, this exceeds $1/4$ when $w$ is sufficiently large and positive. Every such Gaussian tail has positive probability.
+
+The chain for any finite $w$ remains finite, irreducible, and aperiodic, so its own mixing time is finite. $\square$
+
+Proposition 10.2 concerns essential bounds over all disorder realizations at fixed $n$. It does not contradict the typical-disorder asymptotic statements in Theorem C.
+
+---
+
+## 11. End of proof
+
+The static spectral estimates of Section 6, the spherical estimates of Section 7, and the Haar transfer of Section 8 prove the small-projection theorem and the squared-overlap bound. Section 9 uses those inputs to prove Theorem A, Theorem B, and Theorem C. This completes the proof.
+