Research log — simon-ionization-excess

Papers, OEIS, failed lookups. Cite every URL opened this session. Forum numbers are leads, not citations.

2026-08-27 — record

• Wikipedia, Simon problems and the mobile page. Map only. 2000 #9: prove $N_0(Z)-Z$ bounded as $Z\to\infty$. 1984 10(a): $\Delta E(N-1,Z)\ge\Delta E(N,Z)$. Definitions of $H(N,Z)$, $E(N,Z)$, $N_0(Z)$.
• MathWorld, Simon's Problems. Map. Points to Simon 2000, mp_arc 00-78.
• HandWiki, Simon problems. Same map.
• Simon, Fifteen problems in mathematical physics (1984 PDF). Fetch timed out this session.
• Nam, New bounds on the maximum ionization of atoms, arXiv:1009.2367v3 (26 Nov 2011). $N_c<1.22Z+3Z^{1/3}$ for fermions; $\beta\in[0.8218,0.8705)$; Lemma 1 and Proposition 1. Beats Lieb for $Z\ge6$. Replay in compute/q1/replay_nam_beta.py.
• Nam, On the number of electrons that a nucleus can bind, arXiv:1209.3642v2 (7 Dec 2012). Survey. Ionization conjecture $N_c\le Z+1$ or $Z+2$. Zhislin: binding for $N<Z+1$. Bosons: $N_c/Z\to t_c\approx1.21$.
• Nam, The ionization problem in quantum mechanics, arXiv:2206.15393v1 (30 Jun 2022). Lieb settles hydrogen. Convexity of $E_N$ and “binding of $N$ implies binding of $N-1$” still open. Lenzmann–Lewin: no eigenvalue if $N\ge4Z+1$.
• Hundertmark–Pattakos–Schulz, On the Excess Charge Problem of Atoms, arXiv:2504.18487v1 (25 Apr 2025). Opened abs and HTML. Theorem 2.2: $N_c<b(s)Z+c(s)Z^{1/3}$ for $s\in(1,3]$, $b(s)=\max_{t\in[0,1]}(1+t^{s-1})/(1+t^s)$. Prop. 2.4 ($Z\ge2$): $N_c<\frac12(\sqrt2+1)Z+2.96Z^{1/3}$. Prop. 2.5 ($Z\ge4$): $N<b(3)Z+3.90Z^{1/3}+0.0134+0.184Z^{-1/3}+0.0196Z^{-2/3}$ with $1.1184<b(3)<1.1185$. Simplified $N_c<1.1185Z+4Z^{1/3}$ for $Z\ge4$. This is the published non-asymptotic record to beat. Independently replayed in compute/q1/.
• Solovej, The Ionization Conjecture in Hartree-Fock Theory, arXiv:math-ph/0012026v3 (22 Apr 2004). HF excess and ionization energy stay bounded as $Z\to\infty$. Does not apply to the many-body Schrödinger operator.
• Lewin, Some open mathematical problems concerning charged quantum particles, PDF. Open Problem 1: $N_{\max}\le Z+CM$. Notes that even a huge $C$ is unknown for the Schrödinger operator. Cites Lieb $N_{\max}<2Z+M$ as the then-best explicit bound (pre-HPS).
• Nam, The Ionization Problem, EMS Newsletter. Same conjectures; LSST compactness; Fefferman–Seco / SSS $Z+O(Z^{5/7})$.
• Frank–Hundertmark–Jex–Nam, The Lieb–Thirring inequality revisited, arXiv:1808.09017v1 (27 Aug 2018). $L_{1,d}/L_{1,d}^{\mathrm{cl}}\le1.456$. This is the factor HPS uses in $\kappa$. Not replaced.
• Frank, The Lieb–Thirring inequalities: Recent results and open problems, arXiv:2007.09326v1 (18 Jul 2020). Survey; 1.456 still the cited bound for $1\le\gamma<3/2$.
• Benguria–González-Brantes–Tubino, New bounds on the excess charge for atomic systems, arXiv:2207.08328v2 (updated 3 Nov 2025). Comment on the abs page: “This version is not definite and has errors.” Not used as a record. HTML 404 this session; PDF 404.
• Lieb, Phys. Rev. A 29, 3018 (1984). Abstract: $N_c<2Z+1$; hydrogen $N_c=2$. APS HTML blocked by Cloudflare this session.

2026-08-27 — small-Z replay sources

• Høgaasen–Richard–Sorba, arXiv:0907.2614 (HTML). Chandrasekhar closed forms. Comparison for the H$^-$ open-shell trial.
• Nakashima–Nakatsuji, J. Chem. Phys. 127, 224104 (2007). Published NR benchmarks for He and H$^-$. Comparison only; not replayed. No arXiv id found (0707.2101 is unrelated).
• Failed: Simon 2000 Imperial College chapter was not fetched (book, not arXiv). Wikipedia is the map to its statement.

What the record does not say

No published paper opened this session proves $N_c\le Z+C$ for a $Z$-independent $C$ in the many-body Schrödinger theory. HPS 1.1185 is a leading coefficient, not a bounded excess. Nam’s $\beta\ge0.8218$ is a lower bound on a classical variational constant, not a new ionization coefficient beyond 1.22.

2026-08-27 — q2, later papers (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2, Remark 2.3 ($s\ge1$ conjectured), Prop. 4.5, Lemma 4.3 ($s\le3$), Figure 2 (min $f$ not sharp). Does NOT prove a coefficient below $1.1185$.
• Nam abs, 1009.2367. Latest v3. STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$.
• Nam, 2206.15393v1. Lieb write-up; conjecture 2 still open.
• Benguria–González-Brantes abs, 2511.07582 (10 Nov 2025, v1 only). STATES a statistics-independent bound that improves Lieb for $Z\ge12$. Does NOT mention $1.1185$.
• 2511.07582v1 HTML. STATES Theorem 1.1 / display (2): $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge12$ (bosonic atom; argument written as statistics-independent). Remark 1.2: improves Lieb $N(Z)<2Z+1$ for $Z\ge12$ in the bosonic case. Does NOT beat Nam $1.22$ or HPS $1.1185$ for fermions; does NOT prove $N_c\le Z+C$; does NOT unique $N_0(Z)$ for $Z>1$.
• 2207.08328 abs. Comment: “This version is not definite and has errors.” Official PDF/HTML of v2 404 earlier; not a record.
• Semantic Scholar ARXIV:2504.18487. STATES citationCount: 0, citations: [].
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• Lewin, CR Physique PDF. Open Problem 1: even $C=10^{100}$ for $N_{\max}\le Z+CM$ unknown (accepted 31 Mar 2025, before HPS).

Failed / unchanged: APS PRA 29 full text still not obtained this pass; 2207.08328v2 body still not a record.

2026-08-27 — q3, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2.
• HPS HTML, 2504.18487v1. Re-read (4.1), Theorem 4.2, Proposition 4.5, Lemma 4.3, Figure 2, Section 7. STATES $\beta_s$ is an inf over $D_s=P\cap H^{-1}\cap L_{s-1}$, and that radial measures achieve it for $s\le 3$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. Latest v3. STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$.
• Benguria–González-Brantes abs, 2511.07582. Still v1 (10 Nov 2025). STATES a statistics-independent bound for $Z\ge 12$. Does NOT beat $1.1185$ for fermions.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. No later fermionic leading coefficient. Hits are older Hartree–Fock / relativistic statistical papers.
• Semantic Scholar ARXIV:2504.18487 returned HTTP 429 this pass. Not used.

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record.

2026-08-27 — q4, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2 / Prop. 2.5 / (4.1). STATES the printed leading $1.1185$. Does NOT prove a coefficient below $1.1185$.
• Nam abs, 1009.2367. Latest v3. STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$.
• Benguria–González-Brantes abs, 2511.07582. Still v1 (10 Nov 2025). STATES a statistics-independent bound for $Z\ge 12$. Does NOT beat $1.1185$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient. Oldest Hartree–Fock / relativistic / DFT papers only.

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-27 — q5, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2, Remark 2.3, Proposition 2.5, Theorem 4.2, Section 7. STATES the printed leading $1.1185$ and $b(3)<1.1185$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. Latest v3 (26 Nov 2011). STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1057$.
• Benguria–González-Brantes abs, 2511.07582. Still v1 (10 Nov 2025). STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1057$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-27 — q6, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2, Remark 2.3, Proposition 2.5, Theorem 4.2, Section 7. STATES the printed leading $1.1185$ and $b(3)<1.1185$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. Latest v3 (26 Nov 2011). STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1035$.
• Benguria–González-Brantes abs, 2511.07582. Still v1 (10 Nov 2025). STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1035$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q5/verify_lift.py (recon $\gamma$ matches, $1/\gamma<1.1057$, cut$>\gamma$); compute/q5/verify_rebuild.py (stdlib $A$ to $10^{-15}$); compute/q1/hylleraas.py ($E=-815/1602$, $N_0(1)=2$).

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-27 — q7, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2.
• HPS v1 abs, arXiv:2504.18487v1. Submission history: [v1] Fri 25 Apr 2025 16:54:13 UTC only.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2, Remark 2.3, Proposition 2.5, Theorem 4.2, Section 7. STATES the printed leading $1.1185$ and $b(3)<1.1185$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. Latest v3 (26 Nov 2011). STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1026$.
• Benguria–González-Brantes abs, 2511.07582. Still v1 (10 Nov 2025). STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1026$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q6/verify_lift.py (recon $\gamma$ matches, $1/\gamma<1.1035$, cut$>\gamma$); compute/q6/verify_rebuild.py (stdlib $A$ to $10^{-15}$); compute/q1/hylleraas.py ($E=-815/1602$, $N_0(1)=2$).

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-27 — q8, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2.
• HPS v1 abs, arXiv:2504.18487v1. Submission history: [v1] only.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2, Remark 2.3, Proposition 2.5, Theorem 4.2, Section 7. STATES the printed leading $1.1185$ and $b(3)<1.1185$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. Latest v3 (26 Nov 2011). STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1021$.
• Benguria–González-Brantes abs, 2511.07582. Still v1 (10 Nov 2025). STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1021$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q7/verify_lift.py (recon $\gamma$ matches, $1/\gamma<1.1026$, cut$>\gamma$); compute/q7/verify_rebuild.py (stdlib $A$ to $10^{-15}$); compute/q1/hylleraas.py ($E=-815/1602$, $N_0(1)=2$).

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-27 — q9, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2. Submission history: [v1] Fri 25 Apr 2025 16:54:13 UTC only.
• HPS v1 abs, arXiv:2504.18487v1. Same page; og:url is .../2504.18487v1.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2 / printed $N_c<1.1185Z+4Z^{1/3}$ for $Z\ge 4$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1020$.
• Benguria–González-Brantes abs, 2511.07582. Still v1. STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1020$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q8/verify_lift.py (recon $\gamma=0.9075071558364936$, $1/\gamma=1.1019196857772997$ prints as $1.1020$, cut $10/11>\gamma$); compute/q8/verify_rebuild.py (stdlib $A$ to $10^{-15}$); compute/q1/hylleraas.py ($E=-815/1602$, $N_0(1)=2$).

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-27 — q10, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2. Submission history: [v1] Fri 25 Apr 2025 16:54:13 UTC only.
• HPS v1 abs, arXiv:2504.18487v1. Same page; og:url is .../2504.18487v1.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2 / printed $N_c<1.1185Z+4Z^{1/3}$ for $Z\ge 4$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1017$.
• Benguria–González-Brantes abs, 2511.07582. Still v1. STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1017$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q9/verify_lift.py (recon $\gamma=0.9077156846635223$, $1/\gamma=1.1016665426142618$ prints as $1.1017$, cut $10/11>\gamma$); compute/q9/verify_rebuild.py (stdlib $A$ to $10^{-15}$); compute/q1/hylleraas.py ($E=-815/1602$, $N_0(1)=2$). Do not re-enumerate the stored $17{,}179{,}869{,}183$ faces.

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-28 — q11, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2. Submission history: [v1] Fri 25 Apr 2025 16:54:13 UTC only.
• HPS v1 abs, arXiv:2504.18487v1. Same page; og:url is .../2504.18487v1.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2 / printed $N_c<1.1185Z+4Z^{1/3}$ for $Z\ge 4$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1013$.
• Benguria–González-Brantes abs, 2511.07582. Still v1. STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1013$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q10/verify_lift.py (recon $\gamma=0.908018018752533$, $1/\gamma=1.1012997312254167$ prints as $1.1013$, cut $10/11>\gamma$); compute/q10/verify_rebuild.py (stdlib $A$ to $10^{-15}$). Do not re-enumerate the stored $34{,}359{,}738{,}367$ faces.

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-29 — q11 wrap, same-day record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2. Submission history still [v1] Fri 25 Apr 2025 only.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2 / printed $N_c<1.1185Z+4Z^{1/3}$ for $Z\ge 4$. Does NOT prove a coefficient below $1.1185$.
• Nam abs, 1009.2367. Latest v3. STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1013$ or the certified $1.1010$.
• Benguria–González-Brantes abs, 2511.07582. Still v1. STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1010$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q11/verify_lift.py (recon $\gamma=0.9083146735963096$, $1/\gamma=1.1009400476166247$ prints as $1.1010$, cut $10/11>\gamma$); compute/q11/verify_rebuild.py (stdlib $A$ to $10^{-15}$); compute/q11/run_all.sh exit 0. Faces dump copositive, $\min_mMm_safe>0$, raise_R10_n36_t0p9117.json and lift.json present. Do not re-enumerate the stored $68{,}719{,}476{,}735$ faces. q10 $n=35$ faces were not re-enumerated.

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-08-29 — q12, leftover record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2. Submission history: [v1] Fri 25 Apr 2025 16:54:13 UTC only. og:url is https://arxiv.org/abs/2504.18487v1.
• HPS v1 abs, arXiv:2504.18487v1. Same paper body; still v1 only.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2 / printed $N_c<1.1185Z+4Z^{1/3}$ for $Z\ge 4$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. Latest v3 (26 Nov 2011). STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1010$.
• Benguria–González-Brantes abs, 2511.07582. Still v1 (10 Nov 2025). STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$. Does NOT beat $1.1010$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q11/verify_lift.py (recon $\gamma=0.9083146735963096$, $1/\gamma=1.1009400476166247$ prints as $1.1010$, cut $10/11>\gamma$); compute/q11/verify_rebuild.py (stdlib $A$ to $10^{-15}$). Do not re-enumerate the stored $68{,}719{,}476{,}735$ faces. q10 $n=35$ faces were not re-enumerated.

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-09-03 — q13 record check (URLs opened this pass)

• HPS abs, arXiv:2504.18487. Still v1 only (submitted 25 Apr 2025). STATES $N_c(Z)<1.1185Z+O(Z^{1/3})$. Does NOT list a v2. Submission history: [v1] Fri 25 Apr 2025 16:54:13 UTC only.
• HPS v1 abs, arXiv:2504.18487v1. Same paper; still v1 only.
• HPS HTML, 2504.18487v1. Re-read Theorem 2.2 / Prop. 2.5 / printed $N_c<1.1185Z+4Z^{1/3}$ for $Z\ge 4$. STATES $b(3)<1.1185$. Does NOT prove a coefficient below $1.1185$. Does NOT prove a minimizer of $\beta_3$ has bounded aspect.
• Nam abs, 1009.2367. Latest v3 (26 Nov 2011). STATES $N_c<1.22Z+3Z^{1/3}$. Does NOT beat $1.1185$ or the notebook $1.1010$.
• Benguria–González-Brantes abs, 2511.07582. Still v1 (10 Nov 2025). STATES a statistics-independent bound for $Z\ge 12$.
• 2511.07582v1 HTML. Theorem 1.1 / (2) unchanged: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$ (bosonic atom). Remark 1.2: improves Lieb for $Z\ge 12$ in the bosonic case. Does NOT beat $1.1010$ for fermions.
• OpenAlex doi:10.48550/arXiv.2504.18487. STATES cited_by_count: 0, id W4416381655.
• arXiv API, excess-charge / ionization query. Five hits, none a later fermionic leading coefficient.

Independent replay this pass: compute/q11/verify_lift.py (recon $\gamma=0.9083146735963096$, $1/\gamma=1.1009400476166247$ prints as $1.1010$, cut $10/11>\gamma$); compute/q11/verify_rebuild.py (stdlib $A$ to $10^{-15}$); compute/q1/hylleraas.py ($E=-815/1602$, $N_0(1)=2$). Do not re-enumerate the stored $68{,}719{,}476{,}735$ faces. q12 $n=37$ leftover shards were not resumed.

Failed / unchanged: APS PRA 29 full text still not obtained; 2207.08328v2 body still not a record. No paper opened this pass proves $N_c\le Z+C$.

2026-09-06–07 — q14 record and analytic inputs

• HPS abstract. Opened before searching and checked again at wrap. Submission history still lists v1 only, 25 April 2025. The stated leading coefficient is 1.1185.
• HPS full HTML, v1. Fetched with scripts/arxiv_fetch.py and downloaded for formula extraction. Read Theorem 4.2, the radial quotient, and equations (7.23)–(7.35). Theorem 4.2 permits radialization at $s=3$; Section 7 supplies the passage from a lower bound on the variational constant to the electron bound. It does not contain the new rational compact certificate. Both new verifiers replay the printed $b(3)$ window.
• arXiv metadata for the fetched version. The fetch script returned the title, authors and 25 April 2025 version date before extracting the full text.
• Nam abstract. Latest version is v3, 26 November 2011. Its $1.22Z+3Z^{1/3}$ bound does not improve the HPS leading coefficient.
• Benguria–González-Brantes abstract and full HTML, v1. The statistics-independent bound improves Lieb in its stated range; its leading coefficient 1.4811 does not improve HPS for fermions.

Searches for the excess-charge leading coefficient and for the notebook's 1.1006 found no later independent record to replace HPS. This is the scope of the literature check, not an exhaustive novelty theorem. The q14 certificate improves both that published bound and the stated notebook comparison. The new proof does not depend on previous face-enumeration logs.

2026-09-12 — q15 record check and cutoff

• HPS abstract: submission history still lists v1 only, dated 25 April 2025.
• HPS full HTML, v1: fetched with scripts/arxiv_fetch.py; read Theorem 4.2, Proposition 2.5, and Section 7 through (7.35). These supply the radialization and scalar chain, not a 40-bin certificate. Both q14 implementations replayed the published $1.1184<b(3)<1.1185$ window.
• Nam abstract: earlier bound with leading 1.22; not an improvement on the stated HPS record.
• Benguria–González-Brantes abstract: bosonic / statistics-independent comparison, not a better fermionic leading coefficient.

A search for the HPS authors, ionization bounds, and leading 1.1185 found no later improved fermionic coefficient in the sources checked. This is a limited record check. The proposed 40-bin search did not run before the user stopped the campaign; no new bound is asserted.