q1 bounds — HPS / Nam replay

Compute notes for compute/q1/. dent / residue as in AGENTS.md.

How to run

python3 replay_hps.py
python3 tighten_hps.py
python3 replay_nam_beta.py

or ./run_all.sh from this folder (invokes the three scripts if present). Needs numpy, scipy, mpmath. Writes certs/hps_replay.json, certs/hps_tight.json, certs/nam_beta.json.

Record replayed

Opened 2026-08-27: HPS 2504.18487v1 HTML, Nam 1009.2367v3 HTML, FHJN 1808.09017 HTML.

HPS §7 chain with 1.456 from FHJN Theorem 1 (not replaced). A later LT claim 1.44655 (arXiv:2403.04347) was not replayed and is not used.

Closed forms for $b(2)$, $b(3)$ match a numerical max of $(1+t^{s-1})/(1+t^s)$ on $[0,1]$. $C_1^{-1}=2.34131154\ldots$ and $C_2^{-1/2}=2.21537683\ldots$ match Lemma A.6 / (A.18), the displayed closed forms, and the three Lieb integrals for $f_p$.

HPS said the supremum of $a_1(x)$ on $[\beta_3^{-1},5/2]$ is at the left endpoint. Independently it is a minimum in the middle and the max is at $x=5/2$. So $3.893$ is only an enclosure of the left-endpoint formula, not of their stated supremum. $3.90$ still covers $x=5/2$.

Nam $g(\lambda)$ at the maximiser $\lambda\approx 0.8434764$ equals $0.82180662586\ldots$. The AM-GM equality geometry sits in the triangle, so $\inf W_\lambda/(|x|+|y|)=g(\lambda)$ there. A $(r,s,$ angle$)$ grid agrees to a few $10^{-7}$ and is not a lower bound. $1/0.8218=1.21684108\ldots$, which Nam rounds to $1.22$.

What tightened (dent)

Same HPS theorem, tighter arithmetic, interval enclosures.

1. Exact $b(3)$ in $a_1$, not $1.1185$.
2. Lieb $N/Z<2+1/Z$ for the Prop. 2.5 range $Z\ge 4$, so $N/Z<9/4$ not $5/2$. Then $a_1(x)$ on $[b(3),9/4]$ is maximised at the left, and $a_1<3.892$.
3. Evaluating the actual $a_1(x)$ is what shows (2). At $x=b(3)$ the crude $Z<N\beta_3$ is equality.
4. Scanning $\lambda$ does not move $a_1$: HPS already minimises the left-endpoint coefficient.
5. Intervals on every claimed decimal.

Certified inequalities (see certs/hps_tight.json):

• $N_c<b(2)Z+2.953\,Z^{1/3}$ for $Z\ge 2$ (printed $2.96$).
• $N<b(3)Z+3.892\,Z^{1/3}+0.0134+0.184\,Z^{-1/3}+0.0196\,Z^{-2/3}$ for $Z\ge 4$ (printed $3.90$; extras unchanged).
• $N_c<b(3)Z+3.9781\,Z^{1/3}$ and $N_c<1.1185Z+3.9781\,Z^{1/3}$ for $Z\ge 4$ (printed $4$).

$r=\lambda(N\beta_3)^{-1/3}<0.5$ still holds on the contradiction set $N\ge\beta_3^{-1}Z$, $Z\ge 4$.

What did not

No new argument for $\beta_3$, so no dent of the leading $1.1185$. A second Nam trial (64 atoms on $[1,10]$) gives $\beta\le 0.87021$ and sits next to Nam's own $0.8702$ remark; that tightens only the upper end of $\beta$, which does not improve an ionization upper bound. Optimising a single power $r^{-\alpha}1_{[1,L]}$ recovers Nam's $(\alpha,L)=(3/2,9)$ and does not beat $115/81-(1/2)\ln 3$.