HOLE? NO_HOLE_FOUND

Adversarial check of the proposed q3 aspect lift. No dent. No Borel probability with $Q<0.8995$ , and no mass-stationary $k$ -atomic with $Q\le 0.921$ and aspect $\ge 12$ .

Opened this pass: HPS HTML (4.1), Thm 4.2, Prop. 4.5. Class is $D_3=P\cap H^{-1}\cap L_2$ , so $D<\infty$ . Radial shells (1D $k$ -atomic) sit in that class ( $\widehat\mu\sim j_0$ , $\int|\widehat\mu|^2/(1+|k|^2)<\infty$ ).

Strongest objection

Existence of a $k$ -atomic minimizer is a one-line sketch, not a compactness proof. The lift applies compact $\gamma$ only to a minimizer (critical $\Rightarrow$ aspect $<Q/(1-Q)$ ). If for large $k$ the inf were approached only by non-critical measures with aspect $>12$ and $Q<\gamma$ , the lift would fail. I could not build that sequence: stretching the 6-atomic local min out to $R=12\ldots50$ and reoptimising masses sends $Q\to 0.9253$ and breaks criticality (rel err $0.003\to 0.080$ ); forced-aspect $Q$ -mins degenerate to a $(k-1)$ -atomic. So this is an unwritten step, not a numerical break.

Step 6 as written is false: at large $r$ , $V-(Q/2)(r^2+D)\sim (1-Q)r^2/2>0$ , not negative. $F(r)=2V/(r^2+D)\to 1$ , so equality on an unbounded support forces $Q=1$ . Repairable.

The $R=12$ row’s 5 “singular” faces are residual skips ( $10^{-8}$ – $6\cdot10^{-8}$ ), mixed sign, $\mathrm{slsqp}\, \min m^\top Mm\ge 0.009$ . Not a hole in compact $\gamma$ . $D$ -aspect $=$ $m$ -aspect on $(0,\infty)$ . Mass at 0: $V(0)=D/2$ , equality $\Rightarrow Q=1$ . Lagrange is one multiplier: integrating $2V-Qr^2=\lambda$ against $m$ forces $\lambda=QD$ .

Formulas (hold)

On $\mathrm{supp}\subseteq[1,R]$ : $V(1)=D/2+M_{-1}/2$ , $V(R)=R^2/2+M_3/(2R)$ . Max abs err $1.1\cdot10^{-13}$ on 80 random atomics. Criticality $\Rightarrow M_{-1}=Q+(Q-1)D$ , $M_3=(Q-1)R^3+QDR$ . On the q2 6-atomic local min ( $Q=0.923231$ , aspect $2.91$ , eq rel $2\cdot10^{-7}$ ): moment errors $5\cdot10^{-8}$ , $2\cdot10^{-8}$ . Then $M_{-1}>0\Rightarrow D<Q/(1-Q)$ and $M_3>0\Rightarrow D>((1-Q)/Q)R^2$ , hence $R<Q/(1-Q)$ . Algebra: any mass-stationary measure with aspect $12$ has $Q>12/13\approx 0.92308>0.921$ .

Numbers (no killer)

object	$Q$	aspect	notes
compact $\gamma$ $R=12$	$\ge 0.899526$	$\le 12$	D-aspect class
$f_{\min}$	$0.894107$	—	HPS Prop. 4.5
power $\alpha=-2$ , $n=3.5$	$0.920655$	$3.50$	HPS Fig. 2 trial
64-atom of that trial	$0.920678$	$3.43$	cap $Q/(1-Q)=11.607<12$
6-atomic local min	$0.923231$	$2.91$	cap $12.026>12$ ; not a $Q_{\mathrm{hi}}$
2-atomic critical at $R=12$	$0.994576$	$12$	eq rel $10^{-16}$
best critical found at $R=12$ (k=5)	$0.978410$	$12$	still $\gg 0.921$
forced-aspect-12 $Q$ -min (k=8)	$0.922580$	$12$	not critical (rel $0.065$ )
random 500 atomics	$\ge 0.9241$	$\le 200$	none $<0.8995$
two-clump / near-0 / heavy tail	$\ge 0.9224$	up to $10^3$	none $<0.8995$

A trial with $Q<12/13$ is required for the $11.7<12$ slot. The 6-atomic does not give it; the 64-atom / power-law does.

Scripts

probe_hole_aspect.py → probe_hole_aspect.json
probe_crit_largeR.py → probe_crit_largeR.json
dump_R12_singular.c → R12_singular_masks.txt

Replay: python3 probe_hole_aspect.py && python3 probe_crit_largeR.py and gcc -O3 -o dump_R12_singular dump_R12_singular.c -lm && ./dump_R12_singular ../../q2/certs/beta3_mid_R12_n22.txt.