q4 moment region — covering frozen

Light theory for lifting the aspect-≤4 compact bound $\gamma_4=0.901924$ ( $1/\gamma_4=1.1087$ ) via endpoint algebra and moment inequalities alone. Not a dent.

Setup

Newton kernel $g(r,u)=(r^3+u^3)/(2\max(r,u))$ , $Q=I/D$ , $D=\int r^2\,dm$ . Mass-stationarity at both ends of $\mathrm{supp}\subseteq[1,R]$ gives

M_{-1}=Q+(Q-1)D,\qquad M_3=(Q-1)R^3+QDR.

Positivity $M_{-1}>0$ , $M_3>0$ $\Leftrightarrow$ $D\in\bigl(\frac{1-Q}{Q}R^2,\, \frac{Q}{1-Q}\bigr)$ , nonempty iff $Q>R/(R+1)$ .

Extra linear constraints from $\mathrm{supp}\subseteq[1,R]$ : Markov $M_{-1}\in[1/R,1]$ , $M_3\in[1,R^3]$ , $D\in[1,R^2]$ ; Hölder $\exists M_1\in[1,R]$ with $M_1M_{-1}\ge1$ and $D\ge M_1^2$ ; Chebyshev from $(r-1)^2\ge0$ .

Hausdorff step: $(M_{-1},1,D,M_3)$ must be a moment sequence of a probability on $[1,R]$ (checked by $2/3$ -atom reconstruction in moment_region.py).

Answers

1. $R=4$ , target $\gamma_4=0.901924$ , cut $4/5=0.8$

layer	$Q<\gamma_4$ ?	witness
positivity + endpoint identities only	nonempty	scan finds pairs with $Q\approx0.80$ , $D\approx4$ in the positivity slab
+ Hölder / Chebyshev / Markov	nonempty	best scan: $Q\approx0.894$ , $D\approx2.69$
+ Hausdorff / measure on $[1,4]$	empty (scan)	sample above fails $3$ -atom reconstruction; no endpoint-consistent $2$ -atom has $Q<\gamma_4$

So endpoint identities plus the listed moment inequalities do not alone force $Q\ge\gamma_4$ : the abstract $(Q,D)$ slab still has points below $\gamma_4$ until representability is imposed.

Actual mass-stationary measures with used aspect $\ge4$ sit far above $\gamma_4$ : forced-endpoint mass-opt $Q\approx0.923$ (large_aspect.py); $3$ -point stationarity min $Q\approx0.941$ ; $2$ -atom mass-critical $Q=0.97$ (two_atom_crit.py). None of these is a counterexample to a lift — they confirm the numerical floor is above $\gamma_4$ .

Rigorous $R=4$ lift from moments alone: no. The cut $4/5<\gamma_4$ does not lift; the linear moment package without Hausdorff still leaves a nonempty sub- $\gamma_4$ region; Hausdorff closes that region in scan but is not a short pen-and-paper bound.

2. $R=8$ and $R=9$

$R$	$\gamma_R$	cut $R/(R+1)$	cut $>\gamma$ ?	support-only $Q<\gamma_R$	full moment $Q<\gamma_R$
8	$0.900500$	$8/9\approx0.8889$	no	empty (scan)	empty
9	$0.900257$ (interp.)	$0.9$	no	empty	empty

At $R=8$ the positivity slab below $\gamma_8$ is narrow ( $\approx0.012$ wide); Chebyshev/Hölder already kill every scanned pair. At $R=9$ the gap cut– $\gamma$ is $\approx2.6\times10^{-4}$ ; again empty after linear checks.

Improving the large-aspect cut $Q>R/(R+1)$ cannot beat q3’s $1.1118$ here: at $R=4$ the cut is $0.8\ll\gamma_4$ ; at $R=8,9$ the cut is below $\gamma_R$ and linear moments add nothing below $\gamma_R$ .

3. One extra interior stationarity radius

Mass-stationarity at $\{1,t,R\}$ does not close the abstract $(Q,D)$ hole at $R=4$ (support-feasible pairs below $\gamma_4$ still exist without a representing measure). For actual $3$ -atomic stationary measures the minimum $Q$ rises to $\approx0.941$ at $R=4$ , well above $\gamma_4$ . Interior stationarity is stronger than needed for the lift question and does not produce a simple closed form.

4. Replay

python3 moment_region.py

Writes moment_region.json.