Intrinsic matrix variance alone does not control pilot-centered covariance truncation

Fix $\beta =10$, so $k=36$, and choose a fixed integer $m$ arbitrarily large with $m\ge 2C\beta$, $b_{\min }=\lfloor m\mathrm{/}k\rfloor \ge 2$, $k(2\mathrm{/}3{)}^{b_{\min }}<1\mathrm{/}100$ and $18\sqrt{2\beta \mathrm{/}m}<1\mathrm{/}2$. For each $p\ge 2$, set ${\lambda }_1=1$, ${\lambda }_j=p^{-1\mathrm{/}2}$ for $j\ge 2$, $S_p={\sum }_j{\lambda }_j^2=2-1\mathrm{/}p$, and $q_j={\lambda }_j^2\mathrm{/}{S}_p$. Let $X$ take $\pm \sqrt{{\lambda }_j\mathrm{/}{q}_j}{e}_j$, each with probability $q_j\mathrm{/}2$. It is centered and finitely supported. Direct summation gives ${\mathrm{\Sigma }}_p=\mathrm{diag}({\lambda }_j)$, $\mathrm{\parallel }{\mathrm{\Sigma }}_p{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=1$, $E[\mathrm{\parallel }X{\mathrm{\parallel }}^{2}X{X}^{\mathsf{T}}]=S_p{I}_p$ and ${\sigma }_0^2=d=S_p<2$. Thus $m\ge Cd\beta$ for all $p$, but $\mathrm{tr}{\mathrm{\Sigma }}_p\to \mathrm{\infty }$. Tail-axis observations $j\ge 2$ have common squared radius $R_p^2=S_p\sqrt{p}$.

Let $E_p$ be the event that every block contains at least one tail-axis draw and that all sampled tail axes are distinct. A block has no tail draw with probability $(1\mathrm{/}{S}_p{)}^{b_j}$; conditional tail-axis labels are uniform on $p-1$ coordinates. Union bounds give $P(E_p)\ge 1-k(1\mathrm{/}{S}_p{)}^{b_{\min }}-m(m-1)\mathrm{/}(2(p-1))>49\mathrm{/}50$ for large $p$. Every geometric median of finite points lies in their convex hull: if $x$ lies outside that compact hull and $z$ is a closest hull point, differentiating the squared distance along the segment from $z$ to any hull point $q$ gives $⟨x-z,q-z⟩\le 0$. Expanding the square then gives $\mathrm{\parallel }x-q{\mathrm{\parallel }}^2\ge \mathrm{\parallel }x-z{\mathrm{\parallel }}^2+\mathrm{\parallel }z-q{\mathrm{\parallel }}^2>\mathrm{\parallel }z-q{\mathrm{\parallel }}^2$, so replacing $x$ by $z$ strictly reduces every summand, contradicting median minimality. Write therefore $\widehat{\mu }={\sum }_l{w}_l{M}_l$ with $w_l\ge 0$, ${\sum }_l{w}_l=1$. On $E_p$ the tail parts of the block means have disjoint supports and norms at least $R_p\mathrm{/}{b}_{\max }$, where $b_{\max }=\lceil m\mathrm{/}k\rceil$. Thus $\mathrm{\parallel }{\widehat{\mu }}_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}{\mathrm{\parallel }}^2\ge R_p^2{\sum }_l{w}_l^2\mathrm{/}{b}_{\max }^{}\ge R_p^2\mathrm{/}(k{b}_{\max }^{})$, and $\mathrm{\mid }{\widehat{\mu }}_1\mathrm{\mid }\le \sqrt{S_p}$.

For an axis-one sample, $\mathrm{\parallel }{X}_i-\widehat{\mu }{\mathrm{\parallel }}^2\ge \mathrm{\parallel }{\widehat{\mu }}_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}{\mathrm{\parallel }}^2$. For a tail draw in block $l$, its unique tail coordinate in the pilot is exactly $w_l\mathrm{/}{b}_l$ times its sample coordinate, so the difference has magnitude at least $R_p(1-w_l\mathrm{/}{b}_l)\ge R_p\mathrm{/}2$. Hence uniformly in $i$, $\mathrm{\parallel }{X}_i-\widehat{\mu }{\mathrm{\parallel }}^2\ge c_m{R}_p^2$, with $c_m=\min \{1\mathrm{/}(k{b}_{\max }^{}),1\mathrm{/}4\}>0$, whereas $\mathrm{\mid }(X_i-\widehat{\mu }{)}_1\mathrm{\mid }\le 2\sqrt{S_p}$. Choose the exact scale bracket ${\sigma }_{\min }={\sigma }_{\max }={\sigma }_0=\sqrt{S_p}$. Its grid is $\{0\}$, so adaptive and fixed-scale estimates coincide. With $\theta ={\sigma }_0^{-1}\sqrt{\beta \mathrm{/}m}$, all centered samples are clipped for large $p$, and their rank-one formula gives $e_1^{\mathsf{T}}\widehat{\mathrm{\Sigma }}{e}_1\le 4S_p\mathrm{/}(\theta c_m{R}_p^2)=4\mathrm{/}(\theta c_m\sqrt{p})\to 0$. The true entry is one, so operator error exceeds $1\mathrm{/}2$ on $E_p$ eventually. Both claimed thresholds are below $1\mathrm{/}2$ by step 1.1, whereas their claimed failure budget is $5d{e}^{-10}<10e^{-10}<1\mathrm{/}1000$. This contradicts $P(E_p)>49\mathrm{/}50$. Since $m$ was chosen after an arbitrary proposed $C$, no universal such $C$ exists.

To obtain a nonvacuous contradiction even with the corrected ambient prefactor, let an integer $b\to \mathrm{\infty }$ satisfy $b\ge C$, and take ${\beta }_b=2^{b\mathrm{/}2}\mathrm{/}100$, $k_b=\lfloor (7\mathrm{/}2){\beta }_b\rfloor +1$, $m_b=k_{b}b$ and $p_b=\lceil e^{{\beta }_b\mathrm{/}2}\rceil$. These satisfy ${\beta }_b>1$ eventually, $m_b>(7\mathrm{/}2)b{\beta }_b\ge Cd{\beta }_b$ and balanced blocks all of size $b$. The no-tail-block error in step 2.1 is at most $(9\mathrm{/}2){\beta }_b(2\mathrm{/}3{)}^b=(9\mathrm{/}200)(2\sqrt{2}\mathrm{/}3{)}^b\to 0$; the collision error is $O({\beta }_b^2{b}^2{e}^{-{\beta }_b\mathrm{/}2})\to 0$. Here $c_{m_b}=1\mathrm{/}(k_b{b}^2)$ eventually and $1\mathrm{/}\theta =\sqrt{S_{p_b}{k}_{b}b\mathrm{/}{\beta }_b}=O(\sqrt{b})$. Thus the upper bound for the estimated leading entry in step 3.1 is $O({\beta }_b{b}^{5\mathrm{/}2}{e}^{-{\beta }_b\mathrm{/}4})\to 0$, whereas its true entry remains one. The larger advertised error radius is at most $18\sqrt{4\mathrm{/}(7b)}\to 0$, while $5p_b{e}^{-{\beta }_b}\le 10e^{-{\beta }_b\mathrm{/}2}\to 0$. Both failures therefore occur with probability tending to one even though the ambient-prefactor bound would require it to tend to zero. Each asserted vanishing follows since ${\beta }_b$ grows exponentially in $b$ and $e^{-c{\beta }_b}$ dominates every fixed power of ${\beta }_b,b$; taking logarithms proves these limits directly. This also disproves the dimension-uniform source-shaped guarantee under only $m\ge Cd\beta$.