maths

A public notebook of attacks on open problems.

Stephen orchestrates the agents via Grok Bot. Each run hands Codex, Claude, Grok, or another model an open math problem and asks it to make some progress. Agents log their attempts, findings, and compute scripts in the repo. Models are named in the ledger below.

Papers

A. A. Davydov, S. Marcugini, F. Pambianco, S. Wu, Further results on binary linear covering codes and saturating sets in projective spaces, arXiv:2609.16078 (v2, October 2026). New upper bounds on the binary covering length function at radii 2 and 3, built from this notebook's covering certificates. Submitted to IEEE Transactions on Information Theory.
S. Wu, Covering 1024 syndromes with 50 columns, arXiv:2608.27494 (August 2026). The 50-column radius-2 seed behind the covering results below.

Interesting results

So far there is one.

Covering: $\ell_2(10,2)\le 50$

A binary linear code of length 50 and dimension 40 has covering radius exactly 2. Equivalently, the $10\times 50$ parity-check matrix problems/covering/compute/H_r10_n50.txt hits every syndrome in $\mathbb{F}_2^{10}$ as a sum of at most two columns (1024/1024, two independent verifiers). That is

\ell_2(10,2)\le 50.

The November 2025 table (Davydov–Marcugini–Pambianco, arXiv:2511.02542, Table 5.1) had $\ell_2(10,2)\le 51$ . This is a construction, an upper bound, not a conjecture. Sphere covering still only gives $\ge 45$ , and an $n=49$ search left 7 holes, so 50 is not shown optimal.

The same matrix has a $(2,0)$ -partition with $p(H)=10$ . The $\mathrm{QM}_2^2$ construction then produces longer codes ( $r=18$ , $n=815$ ; $r=20$ , $n=1631$ ) and the density bound $\bar\mu(2)\le 2601/2048\approx 1.27002$ . That is the interesting part: a finite seed that moves the asymptotic constant, not just a table entry.

The same seed, via

\mathrm{QM}_3^2,\ \mathrm{QM}_5^2

(arXiv:2511.02542 Thm 5.1), gives independently replayed lengths

\ell_2(22,2)\le 3325,\ \ell_2(24,2)\le 6653,\ \ell_2(26,2)\le 13070,\ \ell_2(28,2)\le 26111

(paper 3389, 6781, 13565, 26623). Full $2^r$ sweeps, including $2^{28}$ .

The $r=26$ matrix has a verified 19-block partition, so $\mathrm{QM}_2^2$ gives a theorem-only lift.

That length is $\ell_2(36,2)\le 418271$ .

The $r=28$ matrix has a verified 28-block partition, so $\mathrm{QM}_2^2$ propagates theorem-only to

\ell_2(40,2)\le 1671167,\ \ell_2(42,2)\le 3342335,\ \ell_2(44,2)\le 6684671

(not enumerated). Same seed plus Golay gives

\ell_2(18,3)\le 151,\qquad \ell_2(26,3)\le 817

(table 153 and paper 818, respectively).

The 817-column matrix inherits a 34-block $(3,0)$ -partition. Refining it to 65 blocks and applying $\mathrm{QM}_4^3$ at $m=6$ gives

\ell_2(44,3)\le 52351

against Table 7.2's 52415. The certificate exhausts the $2^{26}$ seed partition and checks all 52,351 construction columns; it does not sweep $2^{44}$ syndromes.

Merging two inherited blocks instead gives a 33-block partition and permits $m=5$ . The resulting bound is

\ell_2(41,3)\le 26175

against Table 7.2's 26238 and the notebook's previous 26206. The certificate exhausts the $2^{26}$ seed syndromes and checks all 26,175 lift columns.

$\mathrm{QM}_4^4$ with the same 50-set gives a blockwise certificate for $2^{31}$ .

The bound is $\ell_2(31,4)\le 689$ (paper 690).

The $r=18$ and $r=20$ lifts have verified partitions

p(H_{18})\le 17,\ p(H_{20})\le 14

so $\mathrm{QM}_5^3$ gave the earlier theorem-only bounds

\ell_2(38,3)\le 13102,\ \ell_2(41,3)\le 26206

(paper 13118, 26238; not enumerated). The redundancy-41 value is superseded by 26175 above. Replay: problems/covering/compute/run_qm3_checks.sh, run_qm5_checks.sh, run_qm35_checks.sh, run_p64_checks.sh, run_p28_checks.sh, run_qm44_checks.sh, run_q7c_checks.sh, run_qm21_checks.sh, and run_qm43_checks.sh. Replay the redundancy-44 construction with problems/covering/compute/q13/run_all.sh, the shorter redundancy-41 construction with problems/covering/compute/q16/run_all.sh, and the redundancy-47 construction with problems/covering/compute/q15/run_all.sh.

Replay:

python problems/covering/compute/verify_certificate.py
cd problems/covering/result && ./run_all.sh

Standalone writeup in problems/covering/result. Explainers: HTML, PDF.

In the joint paper

The joint paper arXiv:2609.16078 (see Papers at the top) turns these finite constructions into two infinite families:

\ell_2(r,2)\le 51\cdot 2^{r/2-5}-1\quad(r=10,18,20\text{ and even }r\ge 28),\qquad \ell_2(r,3)\le 818\cdot 2^{(r-26)/3}-1\quad(r=26\text{ and }r=3t-1\ge 41).

At radius 2 each member is $2^{r/2-5}$ shorter than the earlier family. At radius 3 the saving is 63 at redundancy 41 and $2^{(r-26)/3}$ from redundancy 44 on. The small members are the notebook certificates:

\ell_2(10,2)\le 50,\ \ell_2(18,2)\le 815,\ \ell_2(20,2)\le 1631,\ \ell_2(28,2)\le 26111,\ \ell_2(26,3)\le 817,\ \ell_2(41,3)\le 26175,\ \ell_2(44,3)\le 52351,\ \ell_2(47,3)\le 104703.

Certificates: redundancy 41 in compute/q16 (33-block merge of the 817-column seed), redundancy 44 in compute/q13 (#185), redundancy 47 in compute/q15. At redundancy 36 the radius-2 family gives 417791, shorter than the notebook's theorem-only 418271 above.

The families give the asymptotic covering densities

\bar\mu(2)\le \frac{2601}{2048}<1.27002,\qquad \bar\mu(3)\le 1.35935,

and the same numbers bound the constants $f(2)$ and $f(3)$ in Green's Open Problem 40. The earlier values were about 1.3203 and 1.3643.

Tables 5.1 and 7.2 cited above come from A. A. Davydov, S. Marcugini, F. Pambianco, New upper bounds for binary linear covering codes, IEEE Transactions on Information Theory 72(10), 7439–7454, October 2026, doi:10.1109/TIT.2026.3716379 (preprint arXiv:2511.02542).

Problems

Folder	Status
erdos-szekeres-seven	The published record is $33 \leq ES(7) \leq 113$ . The classical 32-point witness is independently replayed in Python and C. A compact signotope encoding has a checked small DRAT proof; its full 33-vertex run returned `UNKNOWN` after 300 seconds, so no bound changed.
covering	$\ell_2(10,2)\le 50,\ \ell_2(18,3)\le 151,\ \ell_2(21,3)\le 303,\ \ell_2(26,2)\le 13070,\ \ell_2(26,3)\le 817,\ \ell_2(31,4)\le 689,\ \ell_2(36,2)\le 418271,\ \ell_2(38,3)\le 13102,\ \ell_2(41,3)\le 26175,\ \ell_2(44,3)\le52351,\ \ell_2(47,3)\le104703$ ; $\mathrm{QM}\ 22/24/28$ and $p(H_{18})\le 17,\ p(H_{20})\le 14,\ p(H_{26})\le 19,\ p(H_{28})\le 28$ . The redundancy-44 radius-3 construction improves Table 7.2's 52415 by 64, with an exhaustive $2^{26}$ seed-partition check and a columnwise lift identity. A 33-block merge of the same seed gives the redundancy-41 radius-3 bound 26175, improving Table 7.2's 26238 by 63, with the same seed-sweep and lift-identity checks. A 129-block refinement of the same seed (q15) gives the redundancy-47 bound 104703, improving Table 7.2's 104831 by 128. $n=49$ still 7 holes; q9 recovers the 2003 Kaikkonen–Rosendahl 51-set, shows 51 and 50 are not lifts of the $r=8$ record in any of the 174251 quotients, and exhaustively rules out 271127 of the 279034 single-block shrinks of the 50-set with block width $\le 12$ . q10 prescribes an automorphism instead of perturbing the 50: order 7 is settled at $r=10$ for every fixed-space dimension, and with orders 11, 17, 31, 73, 127 also excluded, any 49-set has a $\{2,3,5\}$ -group of automorphisms. At $r=11$ no invariant set below the record 79 exists for orders 11, 17 or 23. q11 adds a fibered "graph plus kernel" family — one column over every nonzero point of a quotient, plus a kernel block — whose radius-2 condition is a line colouring of $\mathrm{PG}$ ; it contains the documented lengths at $r=4,7,8,9$ , is exactly decided for $r\le 8$ (so it reproduces $\ell_2(8,2)\le 26$ and cannot beat it), and provably cannot produce any $n\le 47$ at $r=10$ , where its own best is 54. The nearest miss is at $r=9$ : $n=38$ would beat $\ell_2(9,2)\le 39$ and reduces inside the family to exactly 17 kernel-block classes, all 17 of which anneal to the same floor of 14 missing incidences and none of which the exact solver decides. q14 exhausts two local neighborhoods of the 50-set (every two-delete/one-add and three-delete/two-add switch leaves holes); $n=49$ remains open. These constructions are in the joint paper arXiv:2609.16078 with Davydov, Marcugini, and Pambianco.
brocard	Four infinite prime-offset families excluded by Wilson's theorem: $n=p-2$ or $p-3$ for primes $p\equiv3,5\pmod 8$ (with $p>3$ in the second form). Conjecture open.
unique-sum	Independently replayed through $p=61$ , matching OEIS A398173 at 59 and 61 (the OEIS table continues through 73). Checked 15-sets and completed exclusions of every smaller size give $m(59)=15$ and $m(61)=15$ . No published bound improved.
three-in-line	Replayed Heule's rct4 142-set at $n=71$ : $D(71)=142$ . At the first current hole, $n=75$ , an audited rct4 portfolio ended without a 150-set.
schur	No 1697-coloring found.
vdw-w27	Verified Paley/QR coloring of $[3703]$ . Does not extend to 3704. No improvement.
c7-shannon	Verified Polak–Schrijver 367-set in $C_7^{\boxtimes 5}$ . No 368. Missing a letter in any coordinate caps the set at 345. Hamming distance 11 from the published set is empty. No improvement.
riemann-hypothesis	Clay Riemann hypothesis; also Hilbert 8(a) and Smale 1. Independent verification of Gomila's candidate gives $0\leq\Lambda\leq\frac{893927}{5000000}=0.1787854<\frac15.$ Two complete finite algorithms, fresh error/tail/barrier computations, and an analytic review are retained in the certificate and note. Prior published upper bound: $0.2$ (Platt–Trudgian). RH remains open.
landau-legendre	Landau 3. Assuming RH, primes occur between consecutive $(2+\delta)$ -powers for every real $x\geq1$ when $\delta\geq0.22524401991935$ , improving the earlier notebook threshold $0.22525$ and the printed $0.2253$ . Both Oppermann halves were independently replayed for the last 100,000 square intervals below $2^{64}$ . The published finite record remains $n=7.05\cdot10^{13}$ , and Legendre's conjecture remains open.
landau-n2-plus-1	Landau 4. Certified 3954181 primes $n^2+1$ for $n\le 10^8$ , matching Wolf $\pi_q(10^{16})$ . Infinitude open.
sidon-second-term	Independent 11-kernel certificate: $F(N)\le N^{1/2}+0.94301\,N^{1/4}+O(1)$ . Beats Hou–Zhao arXiv:2607.01169v2 (0.9435).
chowla-cosine	$K(n)\ge n^{1/7}/18$ for all $n\ge 1$ . Does not beat Bedert $n^{1/5-o(1)}$ .
unit-distance-509	Rebuilt Parts 509, 5-chromatic and vertex-critical. No smaller graph found. Record still 509.
hadwiger-nelson-plane	Published interval $5\le\chi(\mathbb R^2)\le7$ . A 2,434-vertex exact Parts spawn combining a 677-point reserve and four rotation layers is five-colorable; its model also covers arbitrary base-vertex deletion within this fixed family. No new lower bound.
cosine-zeros	$Z(N)\ge\log\log N/(200\log\log\log N)$ when RHS $\ge 4$ . Does not beat Bedert's exponent.
two-squares-gap	Jameson $a=3$ on $n\le 1.024\cdot 10^{15}$ except $\{3,6,21,91\}$ (m $\le 250$ cert replayed). Green's $1/10$ open.
ulam-sequence	L=22 word with $C_F<1.442$ , beating CS 1.454. Density open.
long-gap-dilate	SAT $G(p,\mathrm{round}\sqrt p)$ through $p=73$ . No universal $C>2$ . Record still Shakan 2.
thin-cyclic-bases	BEL $\sqrt{8/3}$ family replayed through $q=61$ . No thinner liminf. $\sqrt2$ open.
union-closed	Certified frequency $0.38305$ on the two-point family {b,1} under Liu Example 4. Analytic first-crossing $0.3830513565\ldots$ ; a 9,000 by 7,000 Python/C mesh has minimum ratio $1.0000049143$ . This improves the prior printed $0.38304$ and Liu's $0.382709$ . Frankl's one-half target remains open.
cohn-elkies	Exact $R=3627599/500000=7.255198$ , beats printed Table 4 $7.25520$ . Not a magic function.
kissing-5d	All four 40-point codes (D5, L5, Q5, R5) are polar-maximal (polar max $5/4$ ). Integer Delsarte on the Q5 angles excludes size 44. Exact 3-point matrices over the rationals; no unrestricted dual below 44. The leftover (1/4)Z^5 graph has no 41-set using 22 or more D5-type points, no leftover 41-set hosted by four D5 coordinate-stars, and no leftover 41-set hosted by a type-(2,1) or type-(1,3) five-star (native CaDiCaL DRAT verified). Type-(0,5) leftover-tight SAT and leftover SAT with
affine-013	$T(S)\le\lceil n^2/2\rceil$ for affine copies of $\{0,1,3\}$ . Beats Aaronson $3/4$ . Conjecture $1/3$ open.
zero-one-polynomials	BSKK $\theta=0.00373556$ (published $0.003736$ over-round). Census $n\le 20$ . $p_n\to 1$ open.
one-third-two-thirds	Gupta v2 is a full order-14 balance census (least above $1/3$ is $37/106$ ). Width-3 $W_{10}$ at $6/17$ is the width-3 minimum through 14. Broken-rung ladder minima independently replayed through 14 and computed through 22. Three-rail minima through 15 stay above $6/17$ . Interval orders through 10 stay at or above $1/3$ . Conjecture open.
seymour-second-neighborhood	Explicit n=8 Pisa graphs with irregular missing degrees $3^2 2^6$ , $3^4 2^4$ , $3^6 2^2$ . Seven stored witnesses replayed. Did not re-run the 2.5B census. Conjecture open.
two-smooth-summands	Incomplete search. $F(131486759)=83$ , not a $79$ -smooth sum. $G(y)$ through $y=79$ replayed. No exponent below Balog $4/(9\sqrt{e})$ . Green #59 open.
tuza-triangle-packing-covering	8-regular codegree-7 Puleo pair: 1044 cores independently checked (STS(9) on $K_7$ ). Does not prove Tuza for $\Delta\le 8$ .
caccetta-haggkvist-k3	Exact CH-triangle at n=18, 21, 24, 26, 27, 29, 30, 32, 33, 35, 36 and every leftover order through n=130 (534 cubes through n=72, 1026 through n=108, 213 through n=114, 36 through n=115, 37 through n=116, 38 through n=117, 37 through n=118, 38 through n=119, 39 through n=120, 38 through n=121, 39 through n=122, 40 through n=123, 39 through n=124, 40 through n=125, 41 through n=126, 40 through n=127, 41 through n=128, 42 through n=129, then 41 through n=130, then 42 through n=131, then 43 through n=132, then 42 through n=133, then 43 through n=134, then 44 through n=135, then 43 through n=136, then 44 through n=137, then 45 through n=138, then 44 through n=139, then 45 through n=140, then 46 through n=141, then 45 through n=142, then 46 through n=143, then 47 through n=144, then 46 through n=145, then 47 through n=146, then 48 through n=147, then 47 through n=148, then 48 through n=149, then 49 through n=150, then 48 through n=151, then 49 through n=152). First remaining hole n=153. $F_4\text{ certificate }c=0.34640\text{ (CKLS 2015 fork), below HKN }0.3465\text{ and the prior stored }0.34645.$ Did not beat 0.3388. Conjecture 1/3 open.
projective-plane-order-twelve	Two involution 2-MOLS replayed (intercalates $108+108$ vs $90+78$ ). $t=3$ timeout. Published Aut still $\lvert G\rvert\in\{1,2,3\}$ . Plane open.
ramsey-r55	McKay 328+328 (5,5,42) graphs replayed; circulant 42/43 empty. A hypothetical 43-vertex example cannot have an automorphism of order 7, nor of cycle type 5^6 or 5^7, nor of type 5^4 with a fixed vertex meeting 0 or 4 of the 5-cycles, nor of type 5^5 with a fixed vertex meeting 2 or 3 of the 5-cycles. Leftover automorphism primes are 2, 3, and 5. Interval still $43\le R(5,5)\le 46.$
graph-reconstruction-next-order	Full independent degseq census: all 8,571,837 n=14 graphs are $4^{11}6^{3}$ ; 17,143 `labelg` samples matched. Uniqueness (hence reconstructibility) not independently re-sorted. McKay n=13 unchanged.
lonely-runner-fourteen	Every 13-tuple of coprime speeds with $u_i \equiv i \pmod{p}$ has the lonely runner property, for each of the ten primes 191 through 239, checked exhaustively in about four minutes apiece. So the tight tuple $(1,2,\dots,13)$ — which the published work reports as the sole survivor of its doubling ladder — is eliminated at all ten, with no lift by 7 or 14. The check fails for primes up to 41, so the modulus is doing real work. Sungkawichai–Trakulthongchai (arXiv:2604.23906) prove the analogue only when the number of runners is prime, and 14 is not. Cross-checked against brute force at 6 and 8 runners. Not a bound on the number of runners: the 14-runner case is open, and the published bottleneck is untouched.
fekete-s2	Replayed Ridgway–Cheviakov 2018 Table 3 for $N=2$ – $65$ . No improvement. Smale 7 open.
smale-jacobian-n2	Smale 16 / plane Jacobian conjecture. Guccione–Guccione–Horruitiner–Valqui left only $(72,108)$ , up to transposition, below maximum degree 125. A pinned exact unit-ideal certificate replay, with an independent Python/Rust check of the published polygon bridge, excludes that pair. Thus every hypothetical plane counterexample has $\max\{\deg P,\deg Q\}\ge125.$ The conjecture remains open; the 2026 three-variable announcement is not treated as community consensus.
smale-tau	Smale 4 / Shub–Smale $\tau$ -conjecture. The polynomial form has no finite record to move, so the campaign takes the integer question that Smale states in the same problem, whether $\tau(k!)\le(\log k)^c$ . An exhaustive 12-step search with exact arithmetic gives $\tau(29\#)=\tau(31\#)=12,$ where Markström (arXiv:1306.3091) had at most 13 and 15; it replays the OEIS A217032 record through 19! and extends the A173419 table to $n\le 10266$ . The 13-step decision for 20!, 21!, 22! is residue (incomplete). Exact enumeration over integer polynomials gives the largest number of distinct integer zeros at cost $k\le 6$ as 1, 1, 2, 3, 3, 4, 5. Conjecture open.
hilbert16-degree-8	Number of nonempty degree-eight T-curve schemes ≥ 2,415. Eight schemes lie outside the 2,407 baseline formed by the 2,367 certificates of arXiv:2602.06888v4, the notebook’s seventeen prior additions, ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩, nine later additions, and thirteen more: ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩, ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩, ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨5⟩ ⊔ 1⟨9⟩⟩, ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨10⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩. Exact integer liftings and independent Python/Rust topology checks pass; claim and certificates. Neither open maximal deep nest is decided. Hilbert 16(a) degree 8 remains open.
hilbert16-limit-cycles	Hilbert 16(b): the Hilbert number H(n), maximum isolated periodic orbits of a planar polynomial vector field of degree n. Published lower bounds still stand: H(2) at least 4 (Shi; Chen–Wang), H(3) at least 13 (Li–Liu–Yang), H(4) at least 28 (Prohens–Torregrosa), and the Han–Li / Chebyshev lifts. A quadratic with 5 cycles, a cubic with 14, a homogeneous nest, a Liénard beat of B(n), iterated squaring past the quadratic ceiling, a Harnack-recurrence table beat, a 29th cycle on the Prohens–Torregrosa quartic, a two-well 14, a four-zero quasi-homogeneous Melnikov, a 9-sheet holomorphic cube, a beat of Z(2,n), a constructive +1 with two cycles, a cubic with an invariant line and three cycles, five Abelian zeros, a four-fold or T2 beat of H(7) at least 74, a cubic Kolmogorov with 7 cycles, an L2 extra nest, and a degree-5 radial product with two cycles were not constructed. Replayed: Shi’s order-3 jet, van der Pol uniqueness, the Chebyshev pullback identity and its 2026 table arithmetic, uniqueness for one radial cubic, the first Lyapunov quantity of a quadratic focus, homogeneous scaling, the odd-cubic Liénard energy identity, iterated-squaring sheet counts, the Harnack-recurrence arithmetic, the explicit degree-4 Darboux seed with three centers, the two-well energy and figure-eight integral, vanishing of L1 at those three centers, first-order cyclicity at most 1 for one cubic perturbation of the quasi-homogeneous center, the holomorphic-cube sheet count, the radial family attaining Z(2,n), a degree-4 line product with one circle, a named cubic with an invariant line and no cycles, an explicit Christopher–Lloyd degree-7 four-oval field, an explicit T2 degree-7 four-oval field, a cubic Kolmogorov Dulac in the first quadrant, the second Poincaré quantity at the two-well, and the degree-5 radial product with one circle. No published H(n) moved. Different problem from the degree-8 folder above.
simon-lieb-thirring	Later record is Carvalho Corso–Ried (arXiv:2403.04347v2) $L_{1,1,1}/L_{1,1,1}^{\mathrm{cl}}\le 1.44655\quad(M_3=0.371185695).$ Independently the Clausen form of that value is at most 1.4465531. Did not beat 1.44655. Weidl interpolation, remainder absorption, a Neumann covering, and a 1D test-potential search also did not beat 1.44655. The 2018 FHJN trial pair was 1.456; a certified pair on this notebook still gives 1.45576 and does not beat CCR. Conjecture open.
simon-ionization-excess	Simon 2000 #9. Replayed Lieb $2Z+1$ , Nam $1.22$ , and Hundertmark–Pattakos–Schulz. Remainders $2.953$ , $3.892$ , $3.9781$ from the $Z\ge 4$ ratio. Leading coefficient moved from printed $1.1185$ through $1.1118$ , $1.1057$ , $1.1035$ , $1.1026$ , $1.1021$ , $1.1020$ , $1.1017$ , $1.1013$ , $1.1010$ , and $1.1006$ to $1.1005$ . Two independent rational matrix checks prove $N_c(Z)<1.1005Z+3.933Z^{1/3}$ for $Z\ge4$ ; claim and replay. Hydrogen N0(1)=2 replayed. Bounded excess open.
simon-ids-continuity	Simon 2000 #14. Bourgain–Klein proves log-Hölder continuity for arbitrary bounded continuum potentials in dimensions 1–3 and for discrete operators in every dimension. An exact Python/Rust replay locates their continuum frontier at $d=4$ , where the modulus exponent $(4-d)/8$ becomes zero. Frank–Ivanisvili's 2026 real bounded Landis examples make the underlying $4/3$ power sharp for this route. Adding a free Euclidean direction gives an absolutely continuous DOS by the standard separable convolution identity. General bounded ergodic continuum potentials in dimensions at least 4 remain open.

Each problem folder:

PROBLEM.md        statement and what would count as a new bound
ATTACK.md         chronological attempts
WALKTHROUGH.md    discovery notes, not a cleaned proof
RESEARCH.md       papers, OEIS, failed lookups
compute/          scripts, certificates, tables
lean/             only if there is a lemma

This README and the explainers are for human readers. The other files in a problem folder (ATTACK.md, WALKTHROUGH.md, the PROBLEM.md status line, the skills) are working notes and may use two agent terms. A dent is a verified finite improvement of a published record. A residue is an incomplete search.

Notes

Lists, dated picks, the SuperGrok 2026-08-17 run, and a Grok Bot transcript recreation live under notes/. Agent runbook: AGENTS.md.

Which model ran what

Problem	Folder	Models	When
Erdős–Szekeres ES(7)	`problems/erdos-szekeres-seven`	GPT-5.6 Sol	2026-08-23
covering	`problems/covering`	Sol 5.6, Opus 5, Fable 5, Grok 4.6, Claude Opus 5, Maths, GPT-6 Astra	2026-08-16–09-11
covering redundancy-44 QM $_4^3$ lift	`problems/covering/compute/q13`	GPT-6 Astra	2026-09-11
covering redundancy-41 QM $_4^3$ lift	`problems/covering/compute/q16`	GPT-6 Sol	2026-09-23
covering redundancy-47 QM $_4^3$ lift	`problems/covering/compute/q15`	GPT-6 Sol	2026-09-23
covering 49-column local switches	`problems/covering/compute/q14`	GPT-6 Sol	2026-09-23
Brocard–Ramanujan	`problems/brocard`	Sol 5.6	2026-08-16
Brocard–Ramanujan prime-offset modular families	`problems/brocard/compute/q3`	GPT-5.6 Sol	2026-08-23
unique-sum mod p	`problems/unique-sum`	Grok 4.6	2026-09-09
no-three-in-line n=71	`problems/three-in-line`	Sol 5.6	2026-08-16
no-three-in-line n=71 certificate replay	`problems/three-in-line/compute/q3`	GPT-5.6 Sol	2026-08-23
no-three-in-line n=75 search	`problems/three-in-line/compute/q4`	GPT-5.6 Sol	2026-08-23
Schur S(7)	`problems/schur`	Sol 5.6	2026-08-16
Schur S(7), q3 source recovery / 1697 search	`problems/schur/compute/q3`	GPT-5.6 Sol	2026-08-23
Schur S(7), q4 bounded repair	`problems/schur/compute/q4`	GPT-5.6 Sol	2026-08-23
van der Waerden W(2,7)	`problems/vdw-w27`	Grok 4.6	2026-08-16
Shannon $C_7$ 5th power	`problems/c7-shannon`	Grok 4.6	2026-08-16
Shannon $C_7$ 5th power / six-shape 368 search	`problems/c7-shannon/compute/q1`	Grok 4.6	2026-08-23
Shannon $C_7$ 5th power / support bound and Hamming 11	`problems/c7-shannon/compute/q2`	Grok 4.6	2026-08-27
Shannon $C_7$ 5th power / 8-coset leftover	`problems/c7-shannon/compute/q3`	Grok 4.6	2026-08-27
Shannon $C_7$ 5th power / new-shape 368 search	`problems/c7-shannon/compute/q4`	Grok 4.6	2026-08-27
Landau 3 (Legendre), conditional exponent and edge replay	`problems/landau-legendre`	GPT-5.6 Sol	2026-09-04
Riemann hypothesis / q1 heat-flow and Lehmer replay	`problems/riemann-hypothesis/compute/q1`	GPT-5.6 Sol	2026-08-30
Riemann hypothesis / independent candidate certification	`problems/riemann-hypothesis/compute/q2`	GPT-6 Astra	2026-09-08
Landau 4 ( $n^2+1$ primes)	`problems/landau-n2-plus-1`	SuperGrok 4.6	2026-09-03
Landau 4 ( $n^2+1$ primes), $n\le 10^8$ census	`problems/landau-n2-plus-1/compute/q3`	Grok 4.6	2026-09-26
unit-distance 509	`problems/unit-distance-509`	SuperGrok 4.6	2026-08-17
Sidon second term	`problems/sidon-second-term`	SuperGrok 4.6	2026-08-17
Sidon second term, free-histogram certificate	`problems/sidon-second-term/compute/q1`	SuperGrok 4.6	2026-08-27
Sidon second term, m=48 free-histogram certificate	`problems/sidon-second-term/compute/q2`	Grok 4.6	2026-08-27
Chowla cosine	`problems/chowla-cosine`	SuperGrok 4.6	2026-08-17
two-squares gap	`problems/two-squares-gap`	SuperGrok 4.6	2026-08-17
cosine zeros	`problems/cosine-zeros`	SuperGrok 4.6	2026-08-17
Ulam sequence	`problems/ulam-sequence`	SuperGrok 4.6	2026-08-17
long-gap dilate	`problems/long-gap-dilate`	SuperGrok 4.6	2026-08-17
long-gap dilate / SAT past 71 and rising factorials	`problems/long-gap-dilate/compute/q1`	Grok 4.6	2026-08-27
thin cyclic bases	`problems/thin-cyclic-bases`	SuperGrok 4.6	2026-08-17
union-closed	`problems/union-closed`	SuperGrok 4.6	2026-08-17
union-closed / Example 4 on {b,1}	`problems/union-closed/compute/q1`	Grok 4.6	2026-08-27
union-closed / 2-sample ceiling on {b,1}	`problems/union-closed/compute/q2`	Grok 4.6	2026-08-27
union-closed / fifth-decimal ray certificate	`problems/union-closed/compute/q3`	Codex (GPT-5)	2026-08-27
0/1 polynomials	`problems/zero-one-polynomials`	SuperGrok 4.6	2026-08-17
Cohn–Elkies planar	`problems/cohn-elkies`	SuperGrok 4.6	2026-08-17
kissing number 5d	`problems/kissing-5d`	SuperGrok 4.6	2026-08-17
kissing number 5d / polar maximality and Q5 integer 44	`problems/kissing-5d/compute/q1`	SuperGrok 4.6	2026-08-27
kissing number 5d / leftover graphs and unrestricted dual	`problems/kissing-5d/compute/q2`	Grok 4.6	2026-08-27
kissing number 5d / leftover (1/4)Z^5 and T^5 36-clique	`problems/kissing-5d/compute/q3`	Grok 4.6	2026-08-27
kissing number 5d / exact unrestricted 3-point dual	`problems/kissing-5d/compute/q4`	Grok 4.6	2026-08-27
kissing number 5d / D5 stars and T5 share 28	`problems/kissing-5d/compute/q4`	Grok 4.6	2026-08-27
kissing number 5d / leftover n1>=24 and T5 share 27	`problems/kissing-5d/compute/q4`	Grok 4.6	2026-08-27
kissing number 5d / leftover n1>=22 and T5 share 24	`problems/kissing-5d/compute/q4`	Grok 4.6	2026-08-27
kissing number 5d / leftover n1<=21 and T5 36	`problems/kissing-5d/compute/q5`	Grok 4.6	2026-08-27
kissing number 5d / leftover n1<=21 star-cover 4	`problems/kissing-5d/compute/q6`	Grok 4.6	2026-08-27
kissing number 5d / leftover n1<=21 star-cover 5	`problems/kissing-5d/compute/q7`	Cursor Grok 4.6	2026-08-27
kissing number 5d / leftover type-(0,5) and	U	=19	`problems/kissing-5d/compute/q8`
affine {0,1,3} copies	`problems/affine-013`	SuperGrok 4.6	2026-08-17
1/3–2/3 posets	`problems/one-third-two-thirds`	SuperGrok 4.6	2026-08-17
1/3–2/3 posets / Gupta v2 replay and ladder table	`problems/one-third-two-thirds/compute/q1`	Grok 4.6	2026-08-27
1/3–2/3 posets / ladder 22, three-rail 15, interval 10	`problems/one-third-two-thirds/compute/q2`	Grok 4.6	2026-08-27
Seymour second neighborhood	`problems/seymour-second-neighborhood`	SuperGrok 4.6	2026-08-17
two smooth summands	`problems/two-smooth-summands`	SuperGrok 4.6	2026-08-17
Tuza triangles	`problems/tuza-triangle-packing-covering`	SuperGrok 4.6	2026-08-17
Caccetta–Häggkvist	`problems/caccetta-haggkvist-k3`	SuperGrok 4.6	2026-08-17
Caccetta–Häggkvist n=18 cubes	`problems/caccetta-haggkvist-k3/compute/q1`	Grok 4.6	2026-08-27
Caccetta–Häggkvist exact orders through n=36	`problems/caccetta-haggkvist-k3/compute/q2`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=72	`problems/caccetta-haggkvist-k3/compute/q3`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover-cube covering count	`problems/caccetta-haggkvist-k3/compute/q4`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist F₄ F-coordinate dump at 0.3464	`problems/caccetta-haggkvist-k3/compute/q4`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=108	`problems/caccetta-haggkvist-k3/compute/q4`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist F₄ CKLS-fork certificate 0.34640	`problems/caccetta-haggkvist-k3/compute/q4`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=114	`problems/caccetta-haggkvist-k3/compute/q5`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=115	`problems/caccetta-haggkvist-k3/compute/q6`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=116	`problems/caccetta-haggkvist-k3/compute/q7`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=117	`problems/caccetta-haggkvist-k3/compute/q8`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=118	`problems/caccetta-haggkvist-k3/compute/q9`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=119	`problems/caccetta-haggkvist-k3/compute/q10`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=120	`problems/caccetta-haggkvist-k3/compute/q11`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=121	`problems/caccetta-haggkvist-k3/compute/q12`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=122	`problems/caccetta-haggkvist-k3/compute/q13`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=123	`problems/caccetta-haggkvist-k3/compute/q14`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=124	`problems/caccetta-haggkvist-k3/compute/q15`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=125	`problems/caccetta-haggkvist-k3/compute/q16`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=126	`problems/caccetta-haggkvist-k3/compute/q17`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=127	`problems/caccetta-haggkvist-k3/compute/q18`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=128	`problems/caccetta-haggkvist-k3/compute/q19`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=129	`problems/caccetta-haggkvist-k3/compute/q20`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=130	`problems/caccetta-haggkvist-k3/compute/q21`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=131	`problems/caccetta-haggkvist-k3/compute/q22`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=132	`problems/caccetta-haggkvist-k3/compute/q23`	Cursor Grok 4.6	2026-08-27
Caccetta–Häggkvist leftover holes through n=133	`problems/caccetta-haggkvist-k3/compute/q24`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=134	`problems/caccetta-haggkvist-k3/compute/q25`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=135	`problems/caccetta-haggkvist-k3/compute/q26`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=136	`problems/caccetta-haggkvist-k3/compute/q27`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=137	`problems/caccetta-haggkvist-k3/compute/q28`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=138	`problems/caccetta-haggkvist-k3/compute/q29`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=139	`problems/caccetta-haggkvist-k3/compute/q30`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=140	`problems/caccetta-haggkvist-k3/compute/q31`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=141	`problems/caccetta-haggkvist-k3/compute/q32`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=142	`problems/caccetta-haggkvist-k3/compute/q33`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=143	`problems/caccetta-haggkvist-k3/compute/q34`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=144	`problems/caccetta-haggkvist-k3/compute/q34`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=145	`problems/caccetta-haggkvist-k3/compute/q35`	Cursor Grok 4.6	2026-08-28
Caccetta–Häggkvist leftover holes through n=146	`problems/caccetta-haggkvist-k3/compute/q36`	Cursor Grok 4.6	2026-08-29
Caccetta–Häggkvist leftover holes through n=147	`problems/caccetta-haggkvist-k3/compute/q37`	Cursor Grok 4.6	2026-08-29
Caccetta–Häggkvist leftover holes through n=148	`problems/caccetta-haggkvist-k3/compute/q38`	Cursor Grok 4.6	2026-08-29
Caccetta–Häggkvist leftover holes through n=149	`problems/caccetta-haggkvist-k3/compute/q39`	Cursor Grok 4.6	2026-08-29
Caccetta–Häggkvist leftover holes through n=150	`problems/caccetta-haggkvist-k3/compute/q40`	Cursor Grok 4.6	2026-08-29
Caccetta–Häggkvist leftover holes through n=151	`problems/caccetta-haggkvist-k3/compute/q41`	Cursor Grok 4.6	2026-08-29
Caccetta–Häggkvist leftover holes through n=152	`problems/caccetta-haggkvist-k3/compute/q42`	Cursor Grok 4.6	2026-08-29
projective plane 12	`problems/projective-plane-order-twelve`	SuperGrok 4.6	2026-08-17
Ramsey R(5,5)	`problems/ramsey-r55`	SuperGrok 4.6	2026-08-17
Ramsey R(5,5) / q2 finite search	`problems/ramsey-r55/compute/q2`	GPT-5.6 Sol	2026-08-28
Ramsey R(5,5) / q3 order-7 certificates	`problems/ramsey-r55/compute/q3`	GPT-5.6 Sol	2026-08-29
Ramsey R(5,5) / q4 leftover 2/3/5 SAT	`problems/ramsey-r55/compute/q4`	Cursor Grok 4.6	2026-08-29
Ramsey R(5,5) / leftover 2/3/5 SAT	`problems/ramsey-r55/compute/q5`	Cursor Grok 4.6	2026-08-29
Ramsey R(5,5) / leftover 2/3/5 SAT	`problems/ramsey-r55/compute/q6`	Cursor Grok 4.6	2026-08-29
Ramsey R(5,5) / leftover 2/3/5 SAT	`problems/ramsey-r55/compute/q7`	Cursor Grok 4.6	2026-08-29
graph reconstruction n=14	`problems/graph-reconstruction-next-order`	SuperGrok 4.6	2026-08-17
lonely runner 14	`problems/lonely-runner-fourteen`	SuperGrok 4.6	2026-08-17 (incomplete)
lonely runner 14	`problems/lonely-runner-fourteen`	Opus 5	2026-08-22
elliptic Fekete $S^2$	`problems/fekete-s2`	Grok 4.6	2026-08-19
Grok Bot transcript page	`notes/chat`	Grok 4.6	2026-08-21
Hilbert 16(a) degree 8	`problems/hilbert16-degree-8`	Fable 5, Opus 5, Grok 4.6, SuperGrok 4.6, GPT-6 Astra	2026-08-21, 2026-08-23, 2026-08-27, 2026-08-29, 2026-09-04, 2026-09-08, 2026-09-12, 2026-09-20, 2026-09-22, 2026-09-26
Hilbert 16(b) Hilbert number H(n)	`problems/hilbert16-limit-cycles`	Grok 4.6	2026-08-27
Hilbert 16(b) five-line campaign	`problems/hilbert16-limit-cycles/compute/q1`	Grok 4.6	2026-08-27
Hilbert 16(b) twenty-five ideas, five lines	`problems/hilbert16-limit-cycles/compute/q2`	Grok 4.6	2026-08-27
Hilbert 16(b) ten ideas, leftover menu, extras	`problems/hilbert16-limit-cycles/compute/q3`	Grok 4.6	2026-08-27
Hadwiger–Nelson plane	`problems/hadwiger-nelson-plane`	GPT-5.6 Sol	2026-08-23
Hadwiger–Nelson fourth rotation layer	`problems/hadwiger-nelson-plane/compute/q4`	GPT-5.6 Sol	2026-08-23
Lieb–Thirring (Simon 2000 #15)	`problems/simon-lieb-thirring`	Grok 4.6	2026-08-27, 2026-09-02
Lieb–Thirring $\mathcal{C}_1$ trial pair	`problems/simon-lieb-thirring/compute/q1`	Grok 4.6	2026-08-27
Lieb–Thirring CCR $M_3$ / Clausen envelope	`problems/simon-lieb-thirring/compute/q2`	Cursor Grok 4.6	2026-08-27
Simon ionization excess	`problems/simon-ionization-excess`	Grok 4.6	2026-08-27, 2026-09-03
Simon ionization excess / HPS remainder	`problems/simon-ionization-excess/compute/q1`	Grok 4.6	2026-08-27
Simon ionization excess / leading coefficient	`problems/simon-ionization-excess/compute/q2`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / leading lift	`problems/simon-ionization-excess/compute/q3`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / leading 1.1057	`problems/simon-ionization-excess/compute/q4`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / leading 1.1035	`problems/simon-ionization-excess/compute/q5`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / leading 1.1026	`problems/simon-ionization-excess/compute/q6`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / leading 1.1021	`problems/simon-ionization-excess/compute/q7`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / leading 1.1020	`problems/simon-ionization-excess/compute/q8`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / leading 1.1017	`problems/simon-ionization-excess/compute/q9`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / n=35 at aspect 10	`problems/simon-ionization-excess/compute/q10`	Cursor Grok 4.6	2026-08-27
Simon ionization excess / leading 1.1013	`problems/simon-ionization-excess/compute/q10`	Cursor Grok 4.6	2026-08-28
Simon ionization excess / n=36 at aspect 10	`problems/simon-ionization-excess/compute/q11`	Cursor Grok 4.6	2026-08-28
Simon ionization excess / leading 1.1010	`problems/simon-ionization-excess/compute/q11`	Cursor Grok 4.6	2026-08-29
Simon ionization excess / n=37 at aspect 10	`problems/simon-ionization-excess/compute/q12`	Cursor Grok 4.6	2026-08-29
Simon ionization excess / leading 1.1005	`problems/simon-ionization-excess/compute/q14`	GPT-6 Astra	2026-09-07
Simon ionization excess / refined compact partition	`problems/simon-ionization-excess/compute/q15`	GPT-6 Astra	2026-09-12
Simon IDS continuity / dimension frontier and free-direction replay	`problems/simon-ids-continuity`	GPT-5.6 Sol	2026-08-27
Smale 16 / plane Jacobian degree-125 frontier and controls	`problems/smale-jacobian-n2/compute`	Codex 5.6 Sol	2026-08-27
Smale 4 / $\tau$ -conjecture: factorial and primorial search, polynomial table	`problems/smale-tau`	Claude Fable 5.1	2026-09-02
Stop tracking bulky solver certificates	repo hygiene	Cursor Grok 4.6	2026-09-11

Lean

lean-toolchain pins Lean 4.32.0. Formal files live under the problem that owns them, not at the repo root.

refs

OpenAI walkthroughs, the ten-proofs PDF, and the house style live in refs/.