Attack log — simon-ionization-excess

Chronological attempts, newest last. A failed attack belongs here too.

2026-08-27 — mint and fetch

• Folder minted with scripts/new-problem.sh simon-ionization-excess. No sibling PR.
• Record fetched before any claim. Wikipedia / MathWorld used only as a map to Simon 2000 #9 and 1984 10(a).
• arXiv: Nam 1009.2367v3, Nam survey 1209.3642v2, Nam 2206.15393v1, HPS 2504.18487v1, Solovej HF math-ph/0012026v3, FHJN 1808.09017. Benguria–González-Brantes–Tubino 2207.08328v2 abs comment says the version has errors; not used as a record.
• Published non-asymptotic record to beat: Hundertmark–Pattakos–Schulz, $N_c<1.1185Z+4Z^{1/3}$ for $Z\ge4$, and Prop. 2.5 remainder $3.90$. Nam $1.22$ and Lieb $2Z+1$ remain the comparison for small $Z$.

2026-08-27 — false start: beat 1.1185 by a new $s$ or a better $\beta_3$

• $b(s)=\max_t(1+t^{s-1})/(1+t^s)$ decreases in $s$, but HPS only prove the theorem for $s\le3$. No $s>3$ argument was found.
• Nam $\beta\in[0.8218,0.8705)$ independently replayed. A second radial trial gives $\beta\le0.87022$, which tightens only the upper end of $\beta$ and cannot improve an ionization upper bound.
• FHJN 1.456 is the LT factor in $\kappa$. A later lead 1.44655 (arXiv:2403.04347) was not replayed and is not used.

2026-08-27 — useful failure: HPS $a_1$ endpoint

• HPS claim the supremum of $a_1(x)$ on $[\beta_3^{-1},5/2]$ is at the left. Independently it is a minimum in the middle and the max is at $x=5/2$, value $3.899495\ldots$. So their printed $3.893$ does not enclose that supremum. Printed $3.90$ still does.
• Prop. 2.5 is only stated for $Z\ge4$. Lieb gives $N/Z<2+1/Z\le9/4$ there, not $5/2$.

2026-08-27 — the click

• Restrict $a_1$ to $[b(3),9/4]$. Then the max is at the left, $a_1<3.892$.
• Same extras $0.0134$, $0.184$, $0.0196$. At $Z=4$ the $Z^{1/3}$ remainder coefficient is $3.978009\ldots<3.9781$, and it decreases in $Z$.
• Prop. 2.4: $a(5/2)=2.952038\ldots<2.953$, beating printed $2.96$.
• Leading $1.1185$ unchanged.

Certified (interval arithmetic in tighten_hps.py; second path verify_remainder.py; C and Rust on $b(2)$, $b(3)$):

2026-08-27 — small $Z$ and heuristics

• Hylleraas $\psi=e^{-\alpha(r_1+r_2)}(1+c r_{12})$, $\alpha=5/6$, $c=1/2$: $E=-815/1602$, $E+1/2=-7/801<0$. Combined with Lieb $N_c<3$, $N_0(1)=2$. Already in Lieb 1984. Replay, not a dent.
• Three-electron search for He$^-$ did not go below the He variational energy. Residue, not a lower bound.
• UHF / helium-like table for $Z=1\ldots10$: $\Delta E$ monotone on that table. Heuristic. Does not prove 1984 10(a).

2026-08-27 — Lean

• lean/HPSCoefficient.lean: $11184/10000<b(3)<11185/10000$ and the $b(2)$ enclosure.
• lean/LiebPair.lean: triangle $(\|x\|+\|y\|)/\|x-y\|\ge1$. Not the full Lieb bound.
• lean/B3NatWitness.lean checks with core Lean, no mathlib.

Replay: problems/simon-ionization-excess/compute/q1/run_all.sh.

2026-08-27 — q2 record replay (later papers)

• HPS 2504.18487 still v1 only (25 Apr 2025). Semantic Scholar citationCount 0; OpenAlex W4416381655 cited_by_count 0.
• Nam 1009.2367v3 and the 2206.15393 write-up of Lieb replayed again. Nam’s β interval cannot beat 1.1185 (1/0.8705 ≈ 1.149).
• Benguria–González-Brantes 2511.07582v1 (10 Nov 2025), HTML Theorem 1.1 / (2): $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge12$, statistics-independent / bosonic. Improves Lieb for bosons at those $Z$. Does not beat 1.1185 for fermions. Replacement for 2207.08328v2, whose abs comment says that version has errors (PDF/HTML 404).
• Corso–Ried 2403.04347v2 is an LT/CLR variational bound (1.44655 vs FHJN 1.456). It sits in the HPS remainder, not in $b(s)$. Not used.
• Lewin CR Physique (accepted 31 Mar 2025, PDF opened): even a huge $C$ for $N_{\max}\le Z+C$ is still unknown.

No later fermionic leading coefficient below 1.1185. No paper proves a $Z$-independent bound on the excess. No unique $N_0(Z)$ for $Z>1$.

2026-08-27 — q2 false start: $s>3$

$b(4)\approx1.083$ would be a real jump if Lemma 4.3 extended. Two-shell opposite dipoles make $I_s(\nu)$ negative for every tested $s>3$. At $s=4$ the quadratic is the exact rational $-1025/2048$. At $s=3$ the dipole form stays nonnegative (Hardy threshold). Remark 2.3 stays a conjecture. Residue: compute/q2/certs/s_gt_3.json.

2026-08-27 — q2 false start: finite $Z$

Published envelopes at $Z=2,3,4,5$: Lieb $N_c<2Z+1$ still excludes the most integers. Nam and both HPS forms sit above $2Z+1$ there. Regular tetrahedron: $\alpha_{4,2}\le\sqrt6/4$, so $\alpha_{4,2}\cdot3<2$ because $54<64$. Pair geometry with the kinetic term dropped cannot exclude $N=4$ at $Z=2$. Lieb-style weights, Nam’s $\alpha_N$ remainder, and Temple/intermediate Hamiltonians also failed. Unsettled $N$ at $Z=2$ is $\{2,3,4\}$. Hydrogen uniqueness not claimed. Residue: compute/q2/certs/smallz.json.

2026-08-27 — q2 useful failure: withdrawn $1.1168$

A tail polynomial $h(D_L,D_R)$ was used to lift a middle-window Rayleigh to every Borel probability, claiming $\beta_3\ge0.895396$ and $\beta_3^{-1}\le1.11682<1.1185$. False: $h(0,1)\approx0.991$ exceeds the HPS power-law $I/D\approx0.921$ on a measure supported in that tail. Also $I_{CC}\ge\beta D_C$ is false. certs/beta3_rad.json is withdrawn.

2026-08-27 — q2 compact class, no lift

The mid-radius face enumeration is correct on a compact aspect. After the $P_{\max}=(q-1)/(q+1)$ reweighting error,

so $1/Q\le1.108741$ in that class (certs/beta3_compact.json, C faces at $n=18$, $0$ singular; Rust $n=16$ rebuild). The numerical power-law minimizer has aspect $\sim3.50$, so it sits in the class. HPS $\beta_3$ is an inf over all radial probabilities, so this does not replace $1.1185$ in Theorem 2.2.

Two-window lift with cross terms at $\min f$ has $p_{12}\approx0.995$ and collapses to $\min f$. Geometric $t_0$-chains stay at $Q\ge0.9379$ in a scan (certs/aspect_try.json); that is not a lower bound. No certificate that every minimizer has bounded aspect.

2026-08-27 — q2 status

Residue. Leading coefficient still $1.1185$ (Hundertmark–Pattakos–Schulz arXiv:2504.18487v1). Remainder dent from q1 unchanged. $N_0(Z)-Z$ bounded still open.

Replay: problems/simon-ionization-excess/compute/q2/run_all.sh.

2026-08-27 — q3 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0. Semantic Scholar 429 this pass.
• Nam 1009.2367v3 and Benguria–González-Brantes 2511.07582v1 unchanged: no fermionic leading coefficient below $1.1185$.
• q1 and q2 replayed (compute/q1/run_all.sh, compute/q2/run_all.sh), both exit 0.

2026-08-27 — q3 leftovers: finite $Z$ and $s>3$

Lieb still gives the best integers at $Z=2,\ldots,6$ ($N_c\le 2Z$). Nam and HPS $s=2$ sit below $2Z+1$ as reals at $Z=6$ and still above $12$. Five octahedron vertices block $N=5$ at $Z=3$ by pair geometry ($128<169$), the same obstruction as the tetrahedron. Two-shell dipoles still make $I_s$ negative for $s>3$ ($-1025/2048$ at $s=4$). Infinite $t_0$-chains and log-radius convolution recover only $\min f$ as a global floor. Residue: compute/q3/work/smallz.md, s_gt_3_toeplitz.md.

2026-08-27 — q3 the click: mass-opt, not existence

The leftover from q2 was “a minimizer has bounded aspect.” An adversarial probe found no $Q<0.8995$ and no mass-stationary $k$-atomic with $Q\le 0.921$ and aspect $\ge 12$, but existence of a global minimizer was still a sketch.

The lift does not need a minimizer. For any finite set of radii, mass-optimisation on the simplex exists. The mass-critical point is stationary for $Q$ and satisfies $V(r_i)=(Q/2)(r_i^2+D)$ at every used atom. The endpoint identities then force $Q>R/(R+1)$ if the used aspect is $R\ge 12$. At $R=12$ that is $Q>12/13>0.899526$. If mass-opt drops an endpoint, the used aspect shrinks into the compact class.

So every finitely atomic radial measure has $Q\ge\gamma_{12}$. A compactly supported radial probability is a $Q$-limit of finite spherical shells. A finite-$D$ measure is a $Q$-limit of compact truncations ($r^2$ is UI for that one measure). Hence $\beta_3\ge\gamma_{12}$ on HPS $D_3$.

The aspect-$\le 4$ number $1.1087$ is not used: the split $\min(\gamma_R,R/(R+1))$ only beats $1.1185$ once $R\ge 9$. The stored $R=12$ row is the first certified class that works. $1.1168$ stays withdrawn.

2026-08-27 — q3 dent

Same HPS §7 chain with $\beta_3\ge 0.899526$:

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py, C and Rust on the $12/13$ algebra, mass-opt scan with no counterexample, stored $R=12$ faces (copositive, $5$ residual skips). q1 remainders $2.953$, $3.892$, $3.9781$ unchanged. $N_0(Z)-Z$ bounded still open.

Replay: problems/simon-ionization-excess/compute/q3/run_all.sh.

2026-08-27 — q4 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0.
• Nam 1009.2367v3 and Benguria–González-Brantes 2511.07582v1 unchanged: no later fermionic leading coefficient.
• q1, q2, and q3 replayed (compute/run_all.sh through q3), exit 0.

2026-08-27 — q4 leftovers: finite $Z$, moments, $P_{\max}$

Lieb still gives the best integers at $Z=2,\ldots,6$. The q3 simplified $1.1118Z+3.966Z^{1/3}$ sits below printed HPS as a real and still above $2Z+1$. Nam Lemma 1 at the Lieb edge and a separated four-electron Hylleraas trial do not exclude a new integer. $N_0(Z)-Z$ cannot move from a leading coefficient $>1$.

Endpoint identities plus Hölder/Chebyshev leave a nonempty abstract $(Q,D)$ region below $\gamma_4$; a representing measure on $[1,4]$ was not found in the scan, and no pen-and-paper $R=4$ lift was obtained. The TV amplitude $1-f_{\min}$ is sharp at vertices ($F_{ii}=1$). A spread lemma is numerical only. Residue: compute/q4/work/smallz.md, moment_notes.md, pmax_notes.md. $1.1168$ stays withdrawn.

2026-08-27 — q4 the click: raise $\varphi$ and $n$, split at $R=10$

The q3 bottleneck was the compact $\gamma$ at $R=12$, not the cut $12/13$. Certified $\varphi=0.9055$ sat $0.0014$ below SLSQP. Larger $n$ shrinks $P=(q-1)/(q+1)$. The split $\min(\gamma_R,R/(R+1))$ beats $1.1118$ as soon as $\gamma_R$ moves up and $R/(R+1)$ still exceeds it. $R=10$ is the first row where a certified $\gamma$ and $10/11$ both sit above $1/1.1118$.

2026-08-27 — q4 dent

Same HPS §7 chain with $\beta_3\ge 0.904414$ (aspect $10$, $n=26$ faces, $\varphi=0.9091$):

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py / verify_rebuild.py, C and Rust on the $10/11$ algebra, mass-opt scan with no counterexample, stored $R=10$ $n=26$ faces (C dump and independent Rust Cramer re-enum, both copositive, $23$ residual skips). The unrestricted leading sits below the aspect-$\le 4$ class-only $1.1087$. $1.1168$ stays withdrawn. q1 remainders unchanged. $N_0(Z)-Z$ bounded still open.

Replay: problems/simon-ionization-excess/compute/q4/run_all.sh.

2026-08-27 — q5 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0.
• Nam 1009.2367v3 and Benguria–González-Brantes 2511.07582v1 unchanged: no later fermionic leading coefficient below the notebook $1.1057$.

2026-08-27 — q5 dead line: $R\le 9$ cut

The mass-opt identities still only give $Q>R/(R+1)$. At $R=9$ that is $0.9$, so $\min(\gamma_9,9/10)\le 0.9$ and the leading is at least $1.1111>1.1057$. Same wall at $R=8$ ($1.125$). Residue: compute/q5/certs/r9_cut.json. A sharper large-aspect cut would reopen this line; none is certified.

2026-08-27 — q5 scan: more bins at $R=10$

SLSQP plus the $P_{\max}$ tax (certs/scan_compact.json):

• $R=10$, $n=26$ (q4 row): predicted $1.105688$.
• $R=10$, $n=27$: $1.105110$.
• $R=10$, $n=28$: $1.104547$.
• $R=10$, $n=30$: $1.103463$ ($2^{30}$ faces).
• $R=9.5$, $n=26$: $1.105316$, but $\gamma$ sits $4\cdot 10^{-5}$ below the cut $9.5/10.5$ — too tight to trust as a first row.

The live line is $R=10$ with $n\ge 27$. Face enumeration is the certificate, not the SLSQP prediction.

2026-08-27 — q5 dent

Same HPS §7 chain with $\beta_3\ge 0.906238$ (aspect $10$, $n=30$ faces, $\varphi=0.9103$):

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py / verify_rebuild.py, C and Rust on the $10/11$ algebra, mass-opt scan with no counterexample, stored $R=10$ $n=30$ faces ($1{,}073{,}741{,}823$, copositive, $420$ residual skips). Intermediate certified row $n=28$ gives printed $1.1046$. $1.1168$ stays withdrawn. q1 remainders unchanged. $N_0(Z)-Z$ bounded still open.

Replay: problems/simon-ionization-excess/compute/q5/run_all.sh.

2026-08-27 — q6 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0.
• Nam 1009.2367v3 and Benguria–González-Brantes 2511.07582v1 unchanged: no later fermionic leading coefficient below the notebook $1.1035$.
• Independent replay of the q5 cert: verify_lift.py recon $\gamma=0.906238$, $1/\gamma=1.103463<1.1057$, cut $10/11>\gamma$. Hydrogen $N_0(1)=2$ replayed. q1 remainders $2.953$, $3.892$, $3.9781$ unchanged.

2026-08-27 — q6 dead line: $R\le 9$ cut

The mass-opt identities still only give $Q>R/(R+1)$. At $R=9$ that is $0.9$, so $\min(\gamma_9,9/10)\le 0.9$ and the leading is at least $1.1111>1.1035$. Same wall at $R=8$ ($1.125$). Residue: compute/q6/certs/r9_cut.json. A sharper large-aspect cut would reopen this line; none is certified.

$s>3$ along Lemma 4.3 is still residue (two-shell $s=4$ rational $-1025/2048$). Lieb still gives the best integers at $Z=2,\ldots,6$.

2026-08-27 — q6 scan: more bins at $R=10$

SLSQP plus the $P_{\max}$ tax (certs/scan_compact.json after the scan is stored):

• $R=10$, $n=30$ (q5 row): predicted $1.103463$.
• $R=10$, $n=31$: $1.102938$.
• $R=10$, $n=32$: $1.102546$ ($2^{32}-1$ faces).
• $R=9.8$, $n=32$: $1.102384$, but $\gamma$ sits $2.8\cdot 10^{-4}$ below the cut $9.8/10.8$ — tighter than the $R=10$ row.

The live line is $R=10$ with $n=32$. Face enumeration is the certificate, not the SLSQP prediction.

2026-08-27 — q6 dent

Same HPS §7 chain with $\beta_3\ge 0.906992$ (aspect $10$, $n=32$ faces, $\varphi=0.9108$):

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py / verify_rebuild.py, C and Rust on the $10/11$ algebra, mass-opt scan with no counterexample, stored $R=10$ $n=32$ faces ($4{,}294{,}967{,}295$, copositive, $1157$ residual skips, $\min m^\top Mm>6\cdot 10^{-4}$). $1/\gamma=1.102546$ prints as $1.1026$. Cut $10/11>\gamma$. $1.1168$ stays withdrawn. q1 remainders unchanged. $N_0(Z)-Z$ bounded still open.

Replay: problems/simon-ionization-excess/compute/q6/run_all.sh.

2026-08-27 — q7 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0. Submission history lists only [v1] Fri 25 Apr 2025.
• Nam 1009.2367v3 and Benguria–González-Brantes 2511.07582v1 unchanged: no later fermionic leading coefficient below the notebook $1.1026$.
• Independent replay of the q6 cert: verify_lift.py recon $\gamma=0.9069918524731921$, $1/\gamma=1.1025457365170288<1.1035$, cut $10/11>\gamma$. Rebuild matches $A$ to $10^{-15}$. Hydrogen $N_0(1)=2$ replayed. q1 remainders $2.953$, $3.892$, $3.9781$ unchanged.

2026-08-27 — q7 dead line: $R\le 9$ cut

The mass-opt identities still only give $Q>R/(R+1)$. At $R=9$ that is $0.9$, so $\min(\gamma_9,9/10)\le 0.9$ and the leading is at least $1.1111>1.1026$. Same wall at $R=8$ ($1.125$) and at $R=9.5$ ($1.10526$). Residue: compute/q7/certs/r9_cut.json. A sharper large-aspect cut would reopen $R\le 9$; none is certified.

$s>3$ along Lemma 4.3 is still residue (two-shell $s=4$ rational $-1025/2048$). Lieb still gives the best integers at $Z=2,\ldots,6$.

$M=A-\gamma\mathrm{Sym}(c,1)/2$ at the predicted $n=33$ target has two negative eigenvalues. Zeroing negative off-diagonals does not make it PSD, so the PSD+NN shortcut is not a certificate. Large principal submatrices stay non-PD.

2026-08-27 — q7 scan: more bins at $R=10$

SLSQP plus the $P_{\max}$ tax (certs/scan_compact.json):

• $R=10$, $n=32$ (q6 row): predicted $1.102546$.
• $R=10$, $n=33$: $1.102041$ ($2^{33}-1$ faces). Cut $10/11>\gamma$.
• $R=10$, $n=34$: $1.101667$.
• $R=10$, $n=35$: $1.101300$ ($2^{35}-1$ faces).
• $R=9.8$, $n=32$: $1.102384$, $\gamma$ still $2.8\cdot 10^{-4}$ below $9.8/10.8$.
• $R=9.8$, $n=33$: predicted $\gamma$ sits $3\cdot 10^{-5}$ above the cut, so the split would bind at $10.8/9.8\approx 1.10204$.

The live line is $R=10$ with $n=33$. Face enumeration is the certificate, not the SLSQP prediction.

2026-08-27 — q7 dent

Same HPS §7 chain with $\beta_3\ge 0.907407$ (aspect $10$, $n=33$ faces, $\varphi=0.9111$):

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py / verify_rebuild.py, C and Rust on the $10/11$ algebra, mass-opt scan with no counterexample, stored $R=10$ $n=33$ faces ($8{,}589{,}934{,}591$, copositive, $2518$ residual skips, $\min m^\top Mm>4\cdot 10^{-4}$). $1/\gamma=1.102041$ prints as $1.1021$. Cut $10/11>\gamma$. $1.1168$ stays withdrawn. q1 remainders unchanged. $N_0(Z)-Z$ bounded still open.

Replay: problems/simon-ionization-excess/compute/q7/run_all.sh.

2026-08-27 — q8 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0. Submission history lists only [v1].
• Nam 1009.2367v3 and Benguria–González-Brantes 2511.07582v1 unchanged: no later fermionic leading coefficient below the notebook $1.1021$.
• Independent replay of the q7 cert: verify_lift.py recon $\gamma=0.9074071558364936$, $1/\gamma=1.1020411218579709<1.1026$, cut $10/11>\gamma$. Rebuild matches $A$ to $10^{-15}$. Hydrogen $N_0(1)=2$ replayed. q1 remainders $2.953$, $3.892$, $3.9781$ unchanged.

2026-08-27 — q8 dead line: $R\le 9$ cut

The mass-opt identities still only give $Q>R/(R+1)$. At $R=9$ that is $0.9$, so $\min(\gamma_9,9/10)\le 0.9$ and the leading is at least $1.1111>1.1021$. Same wall at $R=8$ ($1.125$) and at $R=9.5$ ($1.10526$). Residue: compute/q8/certs/r9_cut.json. Chebyshev $D\cdot M_{-1}\ge 1$ on the endpoint slab does not push the $R\le 9$ cut above $1/1.1021\approx 0.90736$ (certs/sharper_cut.json). At $R=9.5$ that Chebyshev number is $0.91029$, which would sit above the floor; it is a probe, not a wired lift. The cheap live line stays the proven $R=10$ split. A still-sharper large-aspect cut would reopen $R\le 9$; none is certified.

$s>3$ along Lemma 4.3 is still residue (two-shell $s=4$ rational $-1025/2048$). Lieb still gives the best integers at $Z=2,\ldots,6$.

2026-08-27 — q8 scan: unused $\varphi$ and $n=34$

SLSQP plus the $P_{\max}$ tax (certs/scan_compact.json after the scan is stored). The notebook prints by ceiling to the next $10^{-4}$, so $1.102041$ is $1.1021$ and is not a printed dent.

Predicted rows that print below $1.1021$ if faces certify (certs/scan_compact.json):

• $R=10$, $n=33$, target $0.91115$ (same matrix as q7): $1/\gamma\approx 1.10198$, printed $1.1020$. Cut $10/11>\gamma$.
• $R=10$, $n=33$, target $0.9112$: $1/\gamma\approx 1.10192$, printed $1.1020$.
• $R=10$, $n=34$: $1/\gamma\approx 1.10167$, printed $1.1017$ ($2^{34}-1$ faces). Cut $10/11>\gamma$.
• $R=9.9$, $n=34$: printed $1.1017$ with the positivity cut still above $\gamma$.

The cheap live line is the $n=33$ target raise. Face enumeration is the certificate, not the SLSQP prediction.

2026-08-27 — q8 dent

Same HPS §7 chain with $\beta_3\ge 0.907507$ (aspect $10$, $n=33$ faces, $\varphi=0.9112$):

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py / verify_rebuild.py, C and Rust on the $10/11$ algebra, mass-opt scan with no counterexample, stored $R=10$ $n=33$ faces ($8{,}589{,}934{,}591$, copositive, $2455$ residual skips, $\min m^\top Mm>7\cdot 10^{-5}$). $1/\gamma=1.101920$ prints as $1.1020$. Cut $10/11>\gamma$. $1.1168$ stays withdrawn. q1 remainders unchanged. $N_0(Z)-Z$ bounded still open.

Replay: problems/simon-ionization-excess/compute/q8/run_all.sh.

2026-08-27 — q9 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0.
• Nam 1009.2367v3: $N_c<1.22Z+3Z^{1/3}$. Does not beat $1.1020$.
• Benguria–González-Brantes 2511.07582v1: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$, bosonic / statistics-independent. Does not beat $1.1020$ for fermions.
• arXiv API excess-charge query: five hits, none a later fermionic leading.
• Independent replay of the q8 cert: verify_lift.py recon $\gamma=0.9075071558364936$, $1/\gamma=1.1019196857772997$ prints as $1.1020$, cut $10/11>\gamma$. Rebuild matches $A$ to $10^{-15}$. Hydrogen $N_0(1)=2$ replayed.

The mass-opt identities still only give $Q>R/(R+1)$. At $R=9$ that is $0.9$, so $\min(\gamma_9,9/10)\le 0.9$ and the leading is at least $1.1111>1.1020$. Same wall at $R=8$ ($1.125$) and at $R=9.5$ ($1.10526$). Residue: compute/q9/certs/r9_cut.json. Chebyshev $D\cdot M_{-1}\ge 1$ on the endpoint slab does not push the $R\le 9$ cut above $1/1.1020\approx 0.90744$ (certs/sharper_cut.json). At $R=9.5$ that Chebyshev number is $0.91029$, which would sit above the floor; it is a probe, not a wired lift. The cheap live line stays the proven $R=10$ split. A still-sharper large-aspect cut would reopen $R\le 9$; none is certified.

$s>3$ along Lemma 4.3 is still residue (two-shell $s=4$ rational $-1025/2048$). Lieb still gives the best integers at $Z=2,\ldots,6$.

2026-08-27 — q9 scan: $n=34$ at aspect 10

SLSQP plus the $P_{\max}$ tax (certs/scan_compact.json after the scan is stored). The notebook prints by ceiling to the next $10^{-4}$, so $1.101667$ is $1.1017$.

Predicted rows that print below $1.1020$ if faces certify:

• $R=10$, $n=34$, target $0.9113$: $1/\gamma\approx 1.10167$, printed $1.1017$. Cut $10/11>\gamma$. $2^{34}-1$ faces.
• $R=10$, $n=34$, target $0.9114$: $1/\gamma\approx 1.10155$, printed $1.1016$, if $\min\varphi$ still clears $0.9114$ (SLSQP sits at $0.911452$).
• $R=9.9$, $n=34$: also printed $1.1017$ if the positivity cut stays above $\gamma$.
• $R=10$, $n=35$: printed $1.1013$ ($2^{35}-1$ faces).

The cheap live line is $n=34$ at the proven split. Face enumeration is the certificate, not the SLSQP prediction.

2026-08-27 — q9 dent

Same HPS §7 chain with $\beta_3\ge 0.907716$ (aspect $10$, $n=34$ faces, $\varphi=0.9113$):

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py / verify_rebuild.py, C and Rust on the $10/11$ algebra, mass-opt scan with no counterexample, stored $R=10$ $n=34$ faces ($17{,}179{,}869{,}183$, copositive, $4618$ residual skips, $\min m^\top Mm>5\cdot 10^{-4}$). $1/\gamma=1.101667$ prints as $1.1017$. Cut $10/11>\gamma$. $1.1168$ stays withdrawn. q1 remainders unchanged. $N_0(Z)-Z$ bounded still open.

Replay: problems/simon-ionization-excess/compute/q9/run_all.sh.

2026-08-27 — q10 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0.
• Nam 1009.2367v3: $N_c<1.22Z+3Z^{1/3}$. Does not beat $1.1017$.
• Benguria–González-Brantes 2511.07582v1: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$, bosonic / statistics-independent. Does not beat $1.1017$ for fermions.
• arXiv API excess-charge query: five hits, none a later fermionic leading.
• Independent replay of the q9 cert: verify_lift.py recon $\gamma=0.9077156846635223$, $1/\gamma=1.1016665426142618$ prints as $1.1017$, cut $10/11>\gamma$. Rebuild matches $A$ to $10^{-15}$. Hydrogen $N_0(1)=2$ replayed. Do not re-enumerate the stored $17{,}179{,}869{,}183$ faces.

The mass-opt identities still only give $Q>R/(R+1)$. At $R=9$ that is $0.9$, so $\min(\gamma_9,9/10)\le 0.9$ and the leading is at least $1.1111>1.1017$. Same wall at $R=8$ ($1.125$) and at $R=9.5$ ($1.10526$). Residue: compute/q10/certs/r9_cut.json. Chebyshev $D\cdot M_{-1}\ge 1$ on the endpoint slab does not push the $R\le 9$ cut above $1/1.1017\approx 0.90769$ (certs/sharper_cut.json). The cheap live line stays the proven $R=10$ split. A still-sharper large-aspect cut would reopen $R\le 9$; none is certified.

$s>3$ along Lemma 4.3 is still residue (two-shell $s=4$ rational $-1025/2048$). Lieb still gives the best integers at $Z=2,\ldots,6$.

2026-08-27 — q10 leftover: $n=35$ at aspect 10

SLSQP plus the $P_{\max}$ tax (certs/scan_compact.json after the scan is stored). The notebook prints by ceiling to the next $10^{-4}$, so $1.10130$ is $1.1013$.

Predicted rows that print below $1.1017$ if faces certify:

• $R=10$, $n=35$, target $0.9115$: $1/\gamma\approx 1.10130$, printed $1.1013$. Cut $10/11>\gamma$. $2^{35}-1$ faces.
• $R=10$, $n=34$, target $0.9114$: $1/\gamma\approx 1.10155$, printed $1.1016$, if $\min\varphi$ still clears $0.9114$. That is a probe on the existing $n=34$ matrix, not an $n=35$ dent. Changing the target changes $M$, so faces must be re-enumerated.

$R\le 9$ with $Q>R/(R+1)$ cannot beat $1.1017$. Face enumeration is the certificate, not the SLSQP prediction. If faces do not certify, this wrap is residue and the printed leading stays $1.1017$.

2026-08-28 — q10 dent

Same HPS §7 chain with $\beta_3\ge 0.908018$ (aspect $10$, $n=35$ faces, $\varphi=0.9115$):

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py / verify_rebuild.py, C and Rust on the $10/11$ algebra, mass-opt scan with no counterexample, stored $R=10$ $n=35$ faces ($34{,}359{,}738{,}367$, copositive, $8362$ residual skips, $\min m^\top Mm>5\cdot 10^{-4}$, $\min\varphi=0.911674$). $1/\gamma=1.101300$ prints as $1.1013$. Cut $10/11>\gamma$. $1.1168$ stays withdrawn. q1 remainders unchanged. $N_0(Z)-Z$ bounded still open. The $n=34$ target-$0.9114$ probe was not run and is not an $n=35$ dent.

Replay: problems/simon-ionization-excess/compute/q10/run_all.sh.

2026-08-28 — q11 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0.
• Nam 1009.2367: $N_c<1.22Z+3Z^{1/3}$. Does not beat $1.1013$.
• Benguria–González-Brantes 2511.07582v1: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$, bosonic / statistics-independent. Does not beat $1.1013$ for fermions.
• arXiv API excess-charge query: five hits, none a later fermionic leading.
• Independent replay of the q10 cert: verify_lift.py recon $\gamma=0.908018018752533$, $1/\gamma=1.1012997312254167$ prints as $1.1013$, cut $10/11>\gamma$. Rebuild matches $A$ to $10^{-15}$. Do not re-enumerate the stored $34{,}359{,}738{,}367$ faces.

The mass-opt identities still only give $Q>R/(R+1)$. At $R=9$ that is $0.9$, so $\min(\gamma_9,9/10)\le 0.9$ and the leading is at least $1.1111>1.1013$. Same wall at $R=8$ ($1.125$) and at $R=9.5$ ($1.10526$). Residue: compute/q11/certs/r9_cut.json. Chebyshev $D\cdot M_{-1}\ge 1$ on the endpoint slab does not push the $R\le 9$ cut above $1/1.1013\approx 0.908018$ (certs/sharper_cut.json). The cheap live line stays the proven $R=10$ split.

$s>3$ along Lemma 4.3 is still residue (two-shell $s=4$ rational $-1025/2048$). Lieb still gives the best integers at $Z=2,\ldots,6$.

2026-08-28 — q11 leftover: $n=36$ at aspect 10

SLSQP plus the $P_{\max}$ tax (certs/scan_compact.json). The notebook prints by ceiling to the next $10^{-4}$, so a value such as $1.10120$ is $1.1012$.

Predicted rows that print below $1.1013$ if faces certify:

• $R=10$, $n=36$, target $0.9117$: $1/\gamma\approx 1.10094$, printed $1.1010$. Cut $10/11>\gamma$. $2^{36}-1$ faces.
• $R=10$, $n=35$, target $0.9116$: $1/\gamma\approx 1.10118$, printed $1.1012$, if $\min\varphi$ still clears $0.9116$. That is a probe on the existing $n=35$ matrix, not an $n=36$ dent. Changing the target changes $M$, so faces must be re-enumerated.
• $R=10$, $n=34$, target $0.9114$: printed $1.1016$, also a probe, not an $n=36$ dent.

$R\le 9$ with $Q>R/(R+1)$ cannot beat $1.1013$. Face enumeration is the certificate, not the SLSQP prediction. If faces do not certify, this wrap is residue and the printed leading stays $1.1013$.

2026-08-29 — q11 dent

Same HPS §7 chain with $\beta_3\ge 0.908315$ (aspect $10$, $n=36$ faces, $\varphi=0.9117$):

Certified: certs/lift.json, interval §7 in tighten_leading.py, stdlib verify_lift.py / verify_rebuild.py, C and Rust on the $10/11$ algebra, mass-opt scan with no counterexample, stored $R=10$ $n=36$ faces ($68{,}719{,}476{,}735$, copositive, $16296$ residual skips, $\min m^\top Mm>6\cdot 10^{-4}$, $\min\varphi=0.911885$). $1/\gamma=1.100940$ prints as $1.1010$. Cut $10/11>\gamma$. $1.1168$ stays withdrawn. q1 remainders unchanged. $N_0(Z)-Z$ bounded still open. Higher-target probes on the $n=34$ or $n=35$ matrix were not run and are not an $n=36$ dent.

Replay: problems/simon-ionization-excess/compute/q11/run_all.sh.

2026-08-29 — q12 record replay

• HPS 2504.18487 still v1 only (25 Apr 2025). OpenAlex W4416381655 cited_by_count 0. Submission history lists only [v1] Fri 25 Apr 2025 16:54:13 UTC. og:url is .../2504.18487v1.
• Nam 1009.2367: $N_c<1.22Z+3Z^{1/3}$. Does not beat $1.1010$.
• Benguria–González-Brantes 2511.07582v1: $N<1.4811Z+3.1516Z^{1/3}$ for $Z\ge 12$, bosonic / statistics-independent. Does not beat $1.1010$ for fermions.
• arXiv API excess-charge query: five hits, none a later fermionic leading.
• Independent replay of the q11 cert: verify_lift.py recon $\gamma=0.9083146735963096$, $1/\gamma=1.1009400476166247$ prints as $1.1010$, cut $10/11>\gamma$. Rebuild matches $A$ to $10^{-15}$. Do not re-enumerate the stored $68{,}719{,}476{,}735$ faces. q10 $n=35$ stays frozen.