Attack log — five-dimensional kissing number

2026-08-17 — start

• Folder empty except PROBLEM.md. House rules: write only here; no git; do not invent a dent; a restricted numerical SDP without an exact positivity certificate is residue.
• Tonight's target (house / PROBLEM.md): an exact dual certificate excluding some $k\in\{41,42,43,44\}$, or a new exact spherical code of size $>40$ on $S^4$.

Published record (fetched tonight)

Sources, all still stating $40\le\tau_5\le 44$:

• Tao constants page C_{29} https://teorth.github.io/optimizationproblems/constants/29a.html (accessed 2026-08-17): range $40\le\tau_5\le 44$, conjectural value 40.
• Cohn's living table https://cohn.mit.edu/kissing-numbers/ (crawled 2026-08-17): dim 5 is 40 / 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin.
• Boyvalenkov–Dodunekov–Musin survey, arXiv:1507.03631: “the first open case is in dimension five, where it is known that $40\le\tau_5\le 44$” (story: Levenshtein $L_5(5,1/2)=48$, Odlyzko–Sloane 46.345, Bachoc– Vallentin 45, Mittelmann–Vallentin 44.998).
• Mittelmann–Vallentin, arXiv:0902.1105v3 / Exp. Math. 19 (2010): $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$. This is the published upper bound. High-accuracy SDP, not an exact SOS certificate in the paper.
• Cohn–Rajagopal, arXiv:2412.00937v3 (4 Mar 2026), published Discrete Comput. Geom. (doi 10.1007/s00454-026-00841-x): “the kissing number in five dimensions appears to be 40, although the best upper bound that has been proved is 44”. Four non-isometric 40-point codes: $D_5$, $L_5$ (Leech 1967), $Q_5$ (Szöllősi 2023 / Andreanov–Kallus), $R_5$ (Cohn–Rajagopal). None larger than 40.
• Wikipedia / MathWorld tables (MathWorld “as of August 2026”) still 40/44.
• No later accepted improvement of either endpoint found in arXiv search tonight (new work is lower bounds in high dimensions, or new 40-point geometries).

Independently I will re-verify the four 40-point Gram bounds and the classical Levenshtein number 48 before claiming any comparison.

Plan

1. Exact coordinates and Gram checks for $D_5,L_5,Q_5,R_5$.
2. Exact Delsarte / restricted-distance LPs (rational Gegenbauer).
3. Structural restrictions that can be certified exactly: codes containing a 24-cell, codes with only $D_5$ inner products, Q5-equator extensions.
4. Search for an exact 41-point spherical code (algebraic ansätze, layer replacement). Numerical SDP without a positivity certificate is residue.

2026-08-17 — replay of the four 40-point codes

• Implemented $D_5,L_5,Q_5,R_5$ from the geometric descriptions in Cohn–Rajagopal §2, not by typing Table 2.2.
• compute/verify_configs.py: all four have 40 points of squared norm 2, max normalised inner product exactly $1/2$, and the unordered-pair histograms match Cohn–Rajagopal Table 2.1 exactly (including the exotic angles $-4/5,-3/4,-3/10,-1/4,1/5$).
• Independently recomputed Levenshtein $L_5(5,1/2)=48$ from the BDM odd-bound formula (compute/levenshtein.py). Matches the survey.

2026-08-17 — false start: add a 41st point to $D_5$

• For unit $x\in S^4$, the $D_5$ inequalities are $|x_i|+|x_j|\le 1/\sqrt{2}$ for all $i\neq j$.
• On the unit sphere the min of the largest pair-sum is $2/\sqrt{5}$, with equality at the equal-coordinate points. Squares: $4/5>1/2$.
• So $D_5$ is maximal as a spherical code. Exact, not a 41-point lead. Script: compute/extend_d5.py.

2026-08-17 — false start: complete the $Q_5$ equator to a 24-cell

• $Q_5$ slices as $10+20+10$ along the all-ones direction, equator $=A_4$ (20 roots). The 4D kissing number is 24, so adding four equatorial points would give a 44-point candidate.
• Histogram check: $A_4$ is not isometric to any 24-cell minus two antipodal pairs (66 subsets, zero histogram matches).
• Stronger: on the equatorial 4-space $\sum x_i=0$, the $A_4$ inequalities $x_i-x_j\le 1$ cut a polytope whose vertices are two-level and satisfy $|x|^2\le 6/5<2$. No new equatorial vector of squared norm 2 exists at all. Script: compute/q5_extend.py.

2026-08-17 — the D4-containing bound

• Any isometric 24-cell in $S^4$ spans a 4-space, so after rotation it is the $D_4$ equator $x_5=0$. Extra points $x=(\sqrt{1-h^2}\,u,\,h)$ must have $\varphi(u):=\max_{i\neq j}(|u_i|+|u_j|)\le 1/\sqrt{2(1-h^2)}$.
• Since $\varphi\ge 1$ on $S^3$, one needs $h^2\ge 1/2$ (or a pole).
• Two extra points in the same open hemisphere then have 4D blocks $y,y'$ with $\langle y,y'\rangle\le 0$, so their directions form a code in $S^3$ of max inner product $\le 0$.
• Rankin's $A(4,0)\le 8$ is the exact Delsarte polynomial $f(t)=t^2+t=(1/4)P_0^{(4)}+P_1^{(4)}+(3/4)P_2^{(4)}$, $f\le 0$ on $[-1,0]$, $f(1)/f_0=8$.
• Poles conflict with every non-polar extra point in the same hemisphere ($h\ge 1/\sqrt{2}>1/2$).
• Therefore a $D_4$-containing 5D kissing code has size at most $24+8+8=40$. This excludes $k=41,42,43,44$ in that class.
• Discrete check (compute/d4_equator.py): the 24 holes of the 24-cell realise $\varphi=1$, their $1/2$-graph has independence number 8, and there are exactly three maximum independent sets — Cohn's three inscribed cross-polytopes (one axis, two tesseract).
• $D_5$ and $L_5$ contain a 24-cell, so they are covered. $Q_5$ and $R_5$ do not.

2026-08-17 — restricted Delsarte, then exact duals

• Unrestricted Delsarte cannot beat $\sim 46.3$ (Odlyzko–Sloane) and the 3-point SDP of Mittelmann–Vallentin is $44.998$. A floating SDP without a positivity certificate is residue; we did not rerun SDPA-GMP.
• For a finite inner-product set $T$ the dual only needs $f\le 0$ on $T$, not on the whole interval $[-1,1/2]$. That can go below 44.
• Numerical LP (compute/restricted_lp.py):
  • $T_{D_5}=\{-1,-1/2,0,1/2\}$: bound $42$
  • $T_{L_5}=\{-1,-3/4,-1/2,-1/4,0,1/2\}$: bound $43.9745\ldots$
  • $T_{Q_5}$: bound $44.67$ (does not exclude 44)
  • union of all known 40-point angles: bound $45.66$
• The $L_5$ dual uses only $c_0,c_1,c_2,c_3,c_4,c_9$ and vanishes on $T_{L_5}\setminus\{-1\}$. Solving that interpolation over $\mathbb Q$ (compute/exact_duals.py) gives the exact polynomial in compute/certs/restricted_delsarte.json: $$f=\sum c_k P_k^{(5)},\quad c=\Bigl(1,\tfrac{169541}{45136},\tfrac{28}{3},\tfrac{6095}{651},\tfrac{32}{3},0,0,0,0,\tfrac{131072}{13299}\Bigr),$$ $f(1)=239925/5456=44-139/5456<44$. All $c_k\ge 0$, $f\le 0$ on $T_{L_5}$. Replay: compute/verify_certificates.py.
• Same method for $T_{D_5}$: $c=(1,49/45,28/3,0,32/3,896/45)$, $f(1)=42$, excludes 43 and 44 in that class.

2026-08-17 — integer distance distributions for $T_{D_5}$

• Real Delsarte allows $N=42$ with fractional $A_t=12.8,14.4,12.8$.
• Unordered pair counts $n_t=N A_t/2$ must be integers.
• Exhaustive search at Gegenbauer degree 8 (compute/integer_d5.py):
  • $N=40$: feasible (including $D_5$ and nearby)
  • $N=41$: feasible, but every hit has $n_{-1}=20$ (the slice $n_{-1}\le 19$ is empty)
  • $N=42,43,44$: empty
• The $N=41$ witness survives degree 20 (compute/check_integer_hits.py), so Delsarte+integrality alone does not kill 41. Combined with the root system of 20 antipodal pairs (a 40-point $T_{D_5}$ code, necessarily $D_5$) and the $\varphi$-obstruction, 41 dies too.

2026-08-17 — construction residue

• Cube vertices $+$ axes: 16+10=26.
• $D_5$ minus neighbourhoods plus even-sign holes: best size 40 (compute/construct_search.py).
• Numerical 41st-point slack against $L_5,Q_5,R_5$: all $\approx -0.2649$, the optimiser sitting on a deep hole (inner product $2/\sqrt{5}$). Residue, not a certificate of maximality for $Q_5,R_5$.
• Enlarging $T$ by $Q_5$'s extra angles lifts the exact dual back above 44 (compute/more_duals.py). No unrestricted dual below 44.

Published record we compare against

• Lower bound 40 (Korkine–Zolotareff $D_5$; also $L_5,Q_5,R_5$). We did not produce a spherical code of size $>40$.
• Upper bound 44 (Mittelmann–Vallentin $s_{14}(5)=44.99899685\ldots$). We did not produce an unrestricted dual below 44.
• What we did certify is restricted: see RESEARCH.md.

2026-08-27 — continue (q1)

• Start from main. Folder compute/q1/. House rules unchanged: do not claim $\tau_5=40$; a numerical SDP without an exact positivity certificate is residue; the published record is still $40\le\tau_5\le 44$.
• Re-fetched tonight, all still $40\le\tau_5\le 44$: Tao $C_{29}$ https://teorth.github.io/optimizationproblems/constants/29a.html, Cohn table https://cohn.mit.edu/kissing-numbers/ (dim 5: 40 / 44, refs [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin), Cohn–Rajagopal arXiv:2412.00937v3, Mittelmann–Vallentin arXiv:0902.1105v3. A Zenodo note claiming a Cohn–Kumar LP number $44.0297$ is a lead, not a citation: unrestricted Delsarte is already $\approx 46.345$, and there is no exact dual.
• Plan: (1) exact polar vertices of $D_5,L_5,Q_5,R_5$ (maximality of $Q_5,R_5$ was only numerical on 17 August); (2) integer Delsarte boxes for $T_{L_5}$ and $T_{Q_5}$; (3) $A_4$-containing height and the discrete vertex-extra graph; (4) more interpolating duals; (5) Szöllősi $T^5$ candidate graph on the known angles.

2026-08-27 — polar vertices: all four codes are maximal

• $P(C)=\{x:\langle x,p\rangle\le 1\ \forall p\in C\}$ is a bounded rational polytope for each of $D_5,L_5,Q_5,R_5$ (no recession ray).
• Exhaustive 5-subset vertex enumeration, twice: compute/q1/polar.c and polar_vertices.py --python. Counts agree: $D_5$ 5504 vertices, $L_5$ 5440, $Q_5$ 4024, $R_5$ 3960.
• $\max|x|^2=5/4<2$ on every polar. Independent Fraction GE (replay_max_vertex.py) rebuilds the recorded vertex over $\mathbb Q$: $Q_5$ and $D_5$ attain $5/4$ at $\pm\frac12(1,1,1,1,1)$; $L_5$ and $R_5$ at a coordinate half-axis. All inequalities hold.
• So $Q_5$ and $R_5$ are maximal as spherical codes. That was numerical slack $\approx-0.2649$ on 17 August. $D_5$ and $L_5$ were already maximal by other exact arguments; the polar is a uniform proof.
• This excludes a 41st point on these four codes. It does not exclude an unrelated 41-point code. Unrestricted $\tau_5$ is unchanged.

2026-08-27 — $T_{Q_5}$ integer slice empty at $N=44$

• Real Delsarte on $T_{Q_5}$ is still $\approx 44.67$ (no interpolating dual below 44; several supports certified only in the 45–60 range).
• Integer pair-counts: $n_t=N A_t/2\in\mathbb Z$, $\sum n_t=C(44,2)=946$.
• integer_q5_44.c scans a box containing the HiGHS axis-aligned range by 15 in each coordinate ($n_{-1}$ in the full $0..22$). $14\,753\,818\,985$ points, 0 hits. Tables match delsarte.py; the published $Q_5$ histogram at $N=40$ passes every row.
• Dent in this class: no spherical code in $S^4$ with distinct inner products in $\{-1,-4/5,-1/2,-3/10,0,1/5,1/2\}$ has 44 points. Does not exclude 41, 42 or 43, and is not an unrestricted dual.

2026-08-27 — $T_{L_5}$ integer does not kill 41–43

• $N=44$ is already empty by the 17 August dual $239925/5456<44$.
• $N=41,42,43$ each have integer Delsarte-feasible distributions (first hits in integer_restricted.json), all with $n_{-1}=0$. Residue: integrality does not improve the $T_{L_5}$ dual.

2026-08-27 — $A_4$ extras and Szöllősi graph

• Height over a fixed $A_4$ equator: $|s|\ge 2$. Poles conflict with every other extra in the same hemisphere ($8/5>1$). Discrete two-level vertex extras: independence 10 per hemisphere, 20 total, so $A_4$ plus vertex extras is at most 40. Continuous extras with $|s|>2$ are not in that graph.
• Szöllősi $T^5$ pool: 355 equal-norm vectors. Exact clique on 355 vertices was not run. Residue, not a 41-point code.

2026-08-27 — continue (q2)

• Start from current main (after q1). Folder compute/q2/. House rules unchanged: do not claim $\tau_5=40$; a numerical SDP without an exact positivity certificate is residue; do not regress the restricted certificates already in compute/certs/ and compute/q1/certs/.
• Replayed first, all still pass: compute/verify_certificates.py (T_{D_5} bound 42; T_{L_5} bound $239925/5456<44$); compute/q1/verify.py / replay_max_vertex.py (polar $\max|x|^2=5/4$); compute/q1/check_q5_44_empty.py ($T_{Q_5}$ integer slice empty at 44).
• Re-fetched tonight, all still $40\le\tau_5\le 44$: Tao $C_{29}$ https://teorth.github.io/optimizationproblems/constants/29a.html, Cohn table https://cohn.mit.edu/kissing-numbers/ (dim 5: 40 / 44, refs [9] and [17]), Mittelmann–Vallentin arXiv:0902.1105v3 ($s_{14}(5)=44.99899685\ldots$), Cohn–Rajagopal arXiv:2412.00937v3 (4 Mar 2026). The unaffiliated Zenodo note that claimed a Cohn–Kumar number $44.0297$ is now retracted (doi:10.5281/zenodo.18449600, removed 5 Feb 2026). Lead only; not a citation.
• Hunt: an exact dual that is nonpositive on the whole interval $[-1,1/2]$ and excludes some $k\in\{41,42,43\}$, or a new exact 41-point code in $S^4$. Plan: (1) exact clique on the leftover $T^5$ pool of 355; (2) exact kissing graph on $(1/2)\mathbb Z^5\cap\{\lvert x\rvert^2=2\}$ (200 points; contains $D_5$ and $L_5$); (3) other hyperplane layer-swaps; (4) low-degree unrestricted Delsarte with Sturm / real-root certification.

2026-08-27 — q2 results (no unrestricted move)

• Unrestricted numerical Delsarte at degree $\ge 10$ is $46.3368\ldots$, matching Odlyzko–Sloane. Grid-rationalised polynomials fail the exact Sturm test (they change sign on $[-1,1/2]$). An exact ansatz $f(t)=(t-1/2)q(t)^2$ with $q=P_0+(17/6)P_1+(8/3)P_2$ gives bound $53235/1109\approx 48.003$, just above Levenshtein 48. No certified unrestricted dual has bound $<44$. No $k\in\{41,42,43\}$ is excluded without a restricted angle set.
• $T^5$ pool, after including the basis: 360 rational vectors, all four published 40-point codes are cliques. Five basis vectors are adjacent to every other pool point, so a 41-clique exists iff the remaining 355-point graph has a 36-clique. The 41-search on those 355 vertices is empty (t5_clique.json, 607171 nodes). The 36-search was started and stopped without a hit or an emptiness proof (t5_36_residue.json).
• Half-integer sphere $(1/2)\mathbb Z^5\cap\{\lvert x\rvert^2=2\}$: 200 points, no 41-clique (sphere_d2.json). This class contains $D_5$ and $L_5$.
• $(1/4)\mathbb Z^5$ sphere: 1480 points. 41-search incomplete (sphere_d4.json). Residue.
• Layer-swaps of $D_5$ and $L_5$ across 546 short integer normals: 3148 swaps, kissing sizes $\{34,36,37,39,40\}$, none 41.
• Signed-permutation orbit of a $Q_5$ cap vector: 320 points, no 41-clique (q5cap_clique.json, 7 nodes).
• Unrestricted interval unchanged: $40\le\tau_5\le 44$. Did not beat Mittelmann–Vallentin. Did not produce a 41-point code.

2026-08-27 — continue (q3)

• Start from current main (after q2). Folder compute/q3/. House rules unchanged: do not claim $\tau_5=40$; a numerical SDP without an exact positivity certificate is residue; do not regress the restricted certificates already in compute/certs/, compute/q1/, compute/q2/.
• Re-fetched tonight, all still $40\le\tau_5\le 44$: Tao $C_{29}$ https://teorth.github.io/optimizationproblems/constants/29a.html (range $40\le\tau_5\le 44$, conjectural value 40; upper bound Mittelmann–Vallentin), Cohn table https://cohn.mit.edu/kissing-numbers/ (dim 5: 40 / 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin), Mittelmann–Vallentin arXiv:0902.1105v3 ($s_{14}(5)=44.99899685\ldots$), Cohn–Rajagopal arXiv:2412.00937v3 (4 Mar 2026; title “Variations on five-dimensional sphere packings”: “appears to be 40… best upper bound that has been proved is 44”). The unaffiliated Zenodo $44.0297$ note remains retracted. Later-dimension news on Cohn’s table (Cohn–Li, Ho 2603.10425, Ma et al. 2511.13391, Takhanov et al. 2606.18984, Sun–Wang 2607.20359) does not touch dim 5.
• Hunt: an exact dual that is nonpositive on the whole interval $[-1,1/2]$ and excludes some $k\in\{41,42,43,44\}$, or a new exact 41-point code. Leftover handles from q2, to verify independently: the 36-clique in the 355-point $T^5$ remainder, and the 1480-point $(1/4)\mathbb Z^5$ graph. Own route if better: type analysis of $(1/d)\mathbb Z^5$, a $T^5$ pool on a larger exact angle set, a $\mathbb Q(\sqrt5)$ orbit, continuous $A_4$ extras at $|s|>2$, and gapped duals that would need a geometric lemma to become unrestricted.

2026-08-27 — q3 results (no unrestricted move)

• Re-fetched Tao $C_{29}$, Cohn, Mittelmann–Vallentin, Cohn–Rajagopal: still $40\le\tau_5\le 44$. Pfender's kernel (arXiv:math/0501493) does not give an exact dual below 44 in dim 5. The Zenodo $44.0297$ note remains retracted.
• Unrestricted numerical Delsarte is still $46.3368\ldots$. Exact rationalizations fail the Sturm test. Gapped duals with $f\le 0$ only on $[-2/3,1/2]$ have a numerical value $\approx 37.46$, but there is no geometric lemma forbidding $t\in[-1,-2/3)$ (every published 40-point code has such an angle) and the rational polynomials did not certify. Not an unrestricted dual. No $k\in\{41,42,43,44\}$ is excluded on the whole interval $[-1,1/2]$.
• $(1/d)\mathbb Z^5$ type analysis (sphere_types.py, complete_slices.py): every extra vector kisses at most 36 of the 40 $D_5$ roots, so a 41-set has $n_1\le 36$ and at least 5 extras. Complete enumeration of every $k$-superset of an actual missed-root set, $k\in\{4,5,6,7\}$, finds no extras-clique of size $k+1$. Therefore the 1480-point $d=4$ graph has no 41-clique that uses 33 or more $D_5$-type points. Replay: complete_slices.json, n1_ge_33_empty=true. The $n_1\le 32$ slice (at least 9 extras) was not emptied. Residue for the whole 1480-graph, not a 41-code.
• $T^5$ remainder: exact repairs of the four published 35-cliques (remove 1 add 2; remove 2 add 3, candidate cap 28) are empty (t5_repair.json). Coloured B&B to 40 million nodes found no 36-clique (t5_36_c.json). Incomplete, not an exclusion.
• Larger $T^5$ (published angles plus $\{\pm 1/3,\pm 2/5,\pm 3/5\}$): 552-point pool, no 41-clique in 5 million nodes. Residue.
• $D_5$ plus the signed-permutation orbit of $(\varphi,1,1/\varphi,0,0)/\sqrt{2}$ and the half-spinor layers (552 points over $\mathbb Q(\sqrt2,\sqrt5)$): no 41-clique in 3 million nodes. Residue.
• Continuous $A_4$ extras at exact rational heights, mesh denominators 2, 3, 4: 40 northern extras at den 4, independence 4, not 11. No 41-code.
• Unrestricted interval unchanged: $40\le\tau_5\le 44$. Did not beat Mittelmann–Vallentin. Did not produce a 41-point code.

2026-08-27 — q4 unrestricted dual hunt

• Folder compute/q4/. House rules unchanged: do not claim $\tau_5=40$; a numerical SDP without an exact positivity certificate is residue.
• Re-fetched tonight, all still $40\le\tau_5\le 44$: Tao $C_{29}$ https://teorth.github.io/optimizationproblems/constants/29a.html, Cohn table https://cohn.mit.edu/kissing-numbers/ (dim 5: 40 / 44, refs [9] and [17]), Mittelmann–Vallentin arXiv:0902.1105v3 ($s_{14}(5)=44.99899685\ldots$), Bachoc–Vallentin arXiv:math/0608426v4 (3-point SDP, $\tau_5\le 45$ before MV; exact $Y_k^n$ / $S_k^n$ formulas, Remark 3.4 monomial form), Cohn–Rajagopal arXiv:2412.00937v3. The unaffiliated Zenodo $44.0297$ note remains retracted.
• Implemented exact $S_k^5$ over $\mathbb Q$ (bv.py). Self-tests: $Q_0=1$, $Q_1=t-uv$, $Q_k(u,u,1)=(1-u^2)^k$, $S_0(1,1,1)=J$, $S_k(1,1,1)=0$ for $k\ge 1$, $P_k^5$ matches delsarte.py.
• 1-point continuum Delsarte at degree 12 is $46.3368\ldots$. One rationalization (den=1000) gives $46.337$ and fails Sturm. Did not spend the session re-rationalizing Gegenbauer polynomials.
• Exact 3-point search, all residue or $\ge 48$:
  • constant-multiplier Putinar $d=1..4$: HiGHS infeasible;
  • diagonal grid LP: infeasible at $d\le 3$, grid value $85.56$ at $d=4$ (matches 1-point Delsarte $\approx 90$ at degree 4);
  • exact $p_4$-span kernel at $d=3,4,5$: 34 / 81 / 148 nonnegative kernel vectors, none make $h\le 0$ on $[-1,1/2]$;
  • square-dictionary LP: infeasible at $d=5$; numerical $40.38$ at $d=6$ does not snap to an identity over $\mathbb Q$;
  • floating Putinar SDP at $d=5$ (Clarabel / SCS): solver values below 40 or on the dummy cut $40$, grid_ok=false. Discarded.
• Best certified unrestricted dual remains Levenshtein $L_5(5,1/2)=48$ (the $F_k=0$ case of BV). No $k\in\{41,42,43,44\}$ is excluded on the whole interval. Mittelmann–Vallentin $s_{14}(5)=44.998\ldots$ is still the published upper bound; that hierarchy cannot go below its own optimum at $d=14$, so a dual $<44$ would need $d>14$ or a different hierarchy and an exact SOS certificate.
• Wrote compute/q4/dual_exact.json. No certs/unrestricted_delsarte.json or certs/bv_dual.json.

2026-08-27 — q4 leftover graphs (no unrestricted move)

• Same house rules. Re-fetched Tao $C_{29}$, Cohn, Mittelmann–Vallentin $s_{14}(5)=44.99899685\ldots$, Cohn–Rajagopal arXiv:2412.00937v3: still $40\le\tau_5\le 44$. Did not claim $\tau_5=40$.
• The 240 missed-set seeds on the 40 $D_5$ roots split as 160 four-seeds and 80 six-seeds. The ten special octads are the coordinate-stars $\{x_i=\pm 4\}\cap D_5$. Every four-seed lives in exactly one star, as the $2^4$ sign choices on the other four axes (analyze_stars.py). Independent Rust rebuild (verify_n1.rs): each star has 16 four-seeds, extras pool 80, $\omega=8$.
• Same-missed extras are edgeless, so a clique takes at most one extra per seed. A 41-set with $n_1=40-k$ needs an extras-clique of size $k+1$ whose missed sets sit in some $k$-set $U$.
• $n_1=32$ ($k=8$): complete $k$-superset scan, C and Python agree ($n_U=7\,407\,770$, 10 promising octads, best extras 8, total 40). $n_1=31$ ($k=9$): C empty (n1_le32_k9.json).
• Seed-union BFS through $|U|\le 12$ is complete and empty of a 41-set (n1_complete_k12.json, Python replay_unions.py). Promising unions only at $k=7$ (80 heptads), $k=8$ (10 stars), $k=11$ (960), $k=12$ (4640); each has $\omega=k$, total 40.
• Hash-free canonical DFS (n1_dfs.c) continues the same family without a table: each union has a unique parent (all but the last irredundant seed). Through $k=12$ the counts match the BFS. Through $k=18$ the scan is complete and empty of a 41-set (n1_dfs_k18.json): $933{,}648{,}919$ unions. Promising pools at $k=13$ through $18$ have extras $\omega=k$, total 40. Independent Python BFS matches the union census through $k=13$ (replay_unions_k13.json). Therefore there is no 41-set that uses 22 or more $D_5$-type points. The $n_1\le 21$ slice remains residue for the whole 1480-graph, not a 41-code.
• 40-colouring of the 1480-graph is UNSAT (Cadical and Glucose). That does not produce a 41-clique.
• $T^5$ remainder: no 35-colouring. Cadical and Glucose return no 36-clique (t5_omega.json). UNSAT without a stored DRAT is not an emptiness proof. Share $\ge 30$ with each published 35 is empty (Python). Share 30 down to 24 are empty in C (t5_share30_c.json through t5_share24_c.json). Any remaining 36-clique shares at most 23 with every published 35.
• Construction hunts outside these two graphs (other $(1/d)\mathbb Z^5$, $A_5$ hyperplane, $D_6$ projections, QR reflections) produced no 41-code.
• Unrestricted interval unchanged: $40\le\tau_5\le 44$. Did not beat Mittelmann–Vallentin. Did not produce a 41-point code.

2026-08-27 — continue (q5)

• Start from the q4 branch (cursor/kissing-5d-q4-cd01, PR #91 not on main). Folder compute/q5/. House rules unchanged: do not claim $\tau_5=40$; a numerical SDP without an exact positivity certificate is residue; do not regress the restricted certificates already in compute/certs/, compute/q1/–q4/. Covering stays frozen.
• Hunt: finish the two leftovers (a 41-set in the 1480-graph with $n_1\le 21$, or a $T^5$ 36-clique sharing $\le 23$ with every published 35), or produce an exact 41-point code, or an exact unrestricted dual below 44. Residue if the interval does not move.

2026-08-27 — q5 leftovers (interval unchanged so far)

• Re-fetched Tao $C_{29}$, Cohn, Mittelmann–Vallentin $s_{14}(5)=44.99899685\ldots$, Cohn–Rajagopal arXiv:2412.00937v3: still $40\le\tau_5\le 44$. Did not claim $\tau_5=40$.
• Extra types on the 1480-graph: 160 four-seeds (1 type A $(4,2,2,2,2)$ + 4 type B $(5,2,1,1,1)$) and 80 six-seeds (8 type C $(3,3,3,2,1)$). A type-A 20-clique exists with $|U|=34$, total 26, not a 41-set (type_a_clique.json). Type-A leftover hunt (type_a_small_U.json) is complete: no type-A clique $C$ with $|C|\ge 20$ and $|U|\le|C|-1$ (196{,}750 nodes).
• Seed compatibility graph is 240-vertex 231-regular (seed_graph.json). The 80 six-seeds form an 80-clique of union 40. Every 3-star union is leftover-tight: 80 pools $(22,21)$ and 40 pools $(23,22)$. A seed-clique is a pool, not a 41-code.
• Extras B&B on all 120 three-star pools is complete and empty of a leftover 41-set (triple_star_extras.json): extras $\omega\le 18$ on the 80 unions of size 21, $\omega\le 19$ on the 40 unions of size 22. With q4's $|U|\le 18$ empty, no 41-set has $U$ contained in three coordinate-stars. The leftover $n_1\le 21$ slice, if nonempty, has star-cover at least 4.
• Part-count MILP on $|U|\in\{19,20,21\}$ hits cutoff with verified incumbents $42,47,50$ contained seeds (n1_partcount.json). The leftover is not empty by part-count. Cutoff is not a proof of the maximum.
• $n_1$ leftover SAT $k=19$ (30{,}678 vars) is running; no model yet. Extras B&B 200M nodes: no 41-set, incomplete (extras_clique.json).
• $T^5$ remainder: PySAT Cadical195 returns UNSAT for a 36-clique (t5_36_proof.json, 16{,}548{,}926 proof lines). Heule drat-trim on that ASCII file ended s NOT VERIFIED (no conflict). Native CaDiCaL 3.0.1 on the same CNF is UNSAT in 573s (9{,}921{,}324 conflicts) and writes a 671{,}198{,}215-byte binary DRAT. drat-trim on that file: s VERIFIED (756s; 4{,}787{,}602 of 10{,}804{,}713 lemmas in core; 0 RAT). There is no 36-clique in the 355-point remainder, so the Szöllősi $T^5$ pool has no 41-point kissing code. Share 23 is empty as well (t5_share23.json): $C(35,12)$ neighbourhoods of each published remainder 35 have extras $\omega\le 12$ (need 13). Restricted finite-graph facts. Not an unrestricted bound.
• No unrestricted dual below 44 (dual_more.json: 1-point Delsarte still $\approx 46.337$; best certified ansatz $64715/1243\approx 52.06$). No 41-code in the finished construction pools (construct_more.json).
• Unrestricted interval unchanged: $40\le\tau_5\le 44$. Did not beat Mittelmann–Vallentin. Did not produce a 41-point code.

2026-08-27 — continue (q6)

• Start from the leftover branch after q5 (cursor/kissing-5d-q4-cd01, PR #97 merged there). Folder compute/q6/. House rules unchanged: do not claim $\tau_5=40$; a numerical SDP without an exact positivity certificate is residue; covering stays frozen.
• Re-fetched tonight, all still $40\le\tau_5\le 44$: Tao $C_{29}$ https://teorth.github.io/optimizationproblems/constants/29a.html, Cohn table https://cohn.mit.edu/kissing-numbers/ (dim 5: 40 / 44, ratio 1.100, refs [9] and [17]; later-dimension news does not touch dim 5), Mittelmann–Vallentin arXiv:0902.1105v3 ($s_{14}(5)=44.99899685\ldots$), HTML https://ar5iv.labs.arxiv.org/html/0902.1105v3 Table 1, Bachoc–Vallentin arXiv:math/0608426v4, Cohn–Rajagopal arXiv:2412.00937v3 (“appears to be 40… best upper bound that has been proved is 44”).
• Hunt: empty the leftover $n_1\le 21$ slice of the 1480-point $(1/4)\mathbb Z^5$ graph (star-cover $\ge 4$), or produce an explicit 41-set; in parallel an exact unrestricted dual below 44. Residue if the interval does not move.
• First measurement: greedy colouring of extras in every 4-star pool (four_star_color.json) uses 20 to 32 colours. So the 3-star colouring bound $\omega\le 19$ does not lift. Pools are 320 extras ($k=28$, two full axes), 384 ($k=27$), or 434 ($k=26$).
• C leftover-tight B&B on all 210 four-star pools is complete and empty (four_star_extras.json, 26{,}857{,}470 nodes, best=19 is the start value). Independent Python leftover-tight replay of 38 pools (every $k=28$ two-axis pool and a stride of the rest) matches: 38/38 complete empty, no 41-set (replay_four_star.json). Restricted: no leftover 41-set has $U$ contained in four $D_5$ coordinate-stars. Not extras $\omega$, not an unrestricted bound. The leftover $n_1\le 21$ slice, if nonempty, has star-cover at least 5.
• Minimum $|U|$ with star-cover $\ge 5$ is 5 (star_cover_min.json, Cadical hitting-set of the 210 four-star complements; witness $\{12,15,19,21,28\}$ independently checked). So $|U|=19$ is not empty by combinatorics.
• Five-star colouring uses 28 to 39 colours; none $\le 19$ (five_star_color.json). Four $k=32$ five-star C pools hit a 20M node cutoff incomplete (five_star_sample.json).
• No unrestricted dual below 44 (dual_more.json): 1-point Delsarte still $\approx 46.33687$; rationalizations fail Sturm; 51 certified $(t-1/2)q^2$ ansätze, best $221991/3733\approx 59.47$. Did not beat Mittelmann–Vallentin. Did not claim $\tau_5=40$.
• Two-axis leftover SAT: all ten $k=28$ pools are SAT-unsat (two_axis_extras.json). Nine of ten Python B&Bs were already complete empty. SAT-unsat without a stored DRAT is residue for those CNFs; the C / leftover-tight Python emptiness is the certificate.
• Global leftover-tight extras B&B with the 4-star grow-prune (leftover_global.json, n_four_star=210) ran 200M nodes, no 41-set, incomplete. Same leftover as q5 extras B&B, plus the prune.
• Leftover-tight SAT on the four $k=32$ five-star pools that hit the C 20M cutoff is SAT-unsat (five_star_sat.json snapshot). Two more pools in that sample were still running. SAT-unsat without DRAT is residue.
• Global leftover SAT $k=19$ with star-cover $\ge 5$ (30{,}678 vars, Cadical195 and Kissat404 in parallel) had no model after half an hour (leftover_sat_status.json). Incomplete. Residue, not a lower bound.
• Unrestricted interval unchanged: $40\le\tau_5\le 44$. Did not beat Mittelmann–Vallentin. Did not produce a 41-point code. Did not claim $\tau_5=40$. Residue.

2026-08-27 — continue (q7)

• Start from main after PR #101 (809cdad). Folder compute/q7/. House rules unchanged: do not claim $\tau_5=40$; a numerical SDP without an exact positivity certificate is residue; covering and share/2026-08-16 stay frozen.
• Re-fetched tonight, all still $40\le\tau_5\le 44$: Tao $C_{29}$ https://teorth.github.io/optimizationproblems/constants/29a.html (range $40\le\tau_5\le 44$, conjectural value 40; upper bound Mittelmann–Vallentin), Cohn table https://cohn.mit.edu/kissing-numbers/ (dim 5: 40 / 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin; dim 6 is 72 / 77; later-dimension news does not touch dim 5), Mittelmann–Vallentin arXiv:0902.1105v3 ($s_{14}(5)=44.99899685\ldots$), HTML https://ar5iv.labs.arxiv.org/html/0902.1105v3 Table 1, Bachoc–Vallentin arXiv:math/0608426v4, Cohn–Rajagopal arXiv:2412.00937v3 (“the kissing number in five dimensions appears to be 40, although the best upper bound that has been proved is 44”). Survey arXiv:1507.03631.
• Hunt: empty leftover $|U|=19$ with star-cover at least 5, or empty the 5-star leftover hosts (252 pools, three Aut($D_5$) orbits), with stored native CaDiCaL DRAT; leftover-tight extras B&B past the q6 200M-node cutoff by dropping the $|P|\le 160$ grow cap; exact unrestricted dual below 44. Residue if the interval does not move.
• Plan: (1) classify 5-star hosts by signed-permutation orbits and SAT leftover-tight CNFs on one representative of each type plus the four q6 $k=32$ cutoff pools; (2) global leftover SAT $k=19$ star-cover $\ge 5$ with native CaDiCaL / Kissat; (3) always-on grow prune in leftover_global.c; (4) one more $(t-1/2)^p q^2$ dual grid, then drop that line if it stays above 44.
• Signed permutations act transitively on each 5-star type (orbits.json: 60 of type $(2,1)$ $k=32$, 160 of type $(1,3)$ $k=31$, 32 of type $(0,5)$ $k=30$; Aut order $3840$).
• Minimum $|U|$ with star-cover $\ge 5$ is 5; with star-cover $\ge 6$ is 8 (star_cover_min.json). Neither empties $|U|=19$.
• Dual line dropped (dual_more.json): 1-point Delsarte still $\approx 46.33687$; exact $(t-1/2)^p q^2$ duals that certify sit at best $221991/3733\approx 59.47$. Did not beat Mittelmann–Vallentin.
• Native leftover-tight SAT on the $k=32$ orbit representative (stars $(0,1,2,3,4)$) is UNSAT: CaDiCaL 3.0.1, $1{,}857{,}657$ conflicts, binary DRAT $211{,}850{,}438$ bytes, Heule drat-trim s VERIFIED (five_star_sat.json). Combined with Aut transitivity this empties all 60 type-$(2,1)$ five-star leftover hosts. The q6 cutoff pool $(0,1,2,3,5)$ is likewise UNSAT with a verified DRAT ($3{,}325{,}416$ conflicts).
• All four q6 $k=32$ cutoff pools are UNSAT with verified native DRATs (five_q6_01235, five_q6_01236, five_q6_01237, and the orbit representative). Combined with Aut transitivity the type-$(2,1)$ leftover hosts are empty.
• Leftover-tight extras B&B with always-on grow prune ran 400M nodes in about 90 seconds, no 41-set, incomplete (leftover_global_400m.json). The $|P|\le 160$ cap was the cost; remaining missed-union is cheap. With the type-$(2,1)$ 5-star prune (five_mode=2) the same search ran $20{,}000{,}000{,}001$ nodes in $3594$ seconds, no 41-set, incomplete (leftover_global.json). five_mode $3$ (types $(2,1)$ and $(1,3)$, $220$ unions) ran another $20$G nodes in $4272$ seconds, no 41-set, incomplete (leftover_global_mode3.json). Past the q6 200M cutoff; still residue.
• $k=31$ leftover-tight SAT is UNSAT with a verified native DRAT (five_k31_n1_3: $40{,}611{,}860$ conflicts, DRAT $4{,}134{,}873{,}401$ bytes, drat-trim $15{,}253{,}402$ of $46{,}128{,}907$ lemmas, $1{,}635{,}820{,}812$ resolution steps). Combined with Aut transitivity this empties all 160 type-$(1,3)$ five-star leftover hosts. Remaining 5-star leftover hosts are the 32 type-$(0,5)$ pools.
• $k=30$ leftover-tight SAT and global leftover SAT $k=19$ were still writing native DRATs when this pass wrapped (past $9$ GB and $7$ GB). Incomplete. Residue, not a lower bound.
• Unrestricted interval unchanged: $40\le\tau_5\le 44$. Did not beat Mittelmann–Vallentin. Did not produce a 41-point code. Did not claim $\tau_5=40$. Residue.

2026-08-27 — continue (q8)

• Start from main after PR #102 (87674a7). Folder compute/q8/. Covering frozen. Do not claim $\tau_5=40$.
• Re-opened tonight, still $40\le\tau_5\le 44$: Tao $C_{29}$ https://teorth.github.io/optimizationproblems/constants/29a.html, Cohn table https://cohn.mit.edu/kissing-numbers/ (dim 5: 40 / 44, ratio 1.100), Mittelmann–Vallentin arXiv:0902.1105v3, Cohn–Rajagopal arXiv:2412.00937v3.
• Leftover after q7: type-$(0,5)$ five-star hosts (32 pools, $k=30$, 625 extras) and global leftover SAT $|U|=19$ with star-cover at least 5. The global CNF here also forbids the type-$(2,1)$ and type-$(1,3)$ five-star unions q7 emptied.
• Hunt: leftover-tight SAT / B&B on the type-$(0,5)$ orbit representative with a stored native DRAT, or a 41-set; stronger global leftover SAT. Residue if the interval does not move.

2026-08-29 — wrap (q8)

• Rebased onto origin/main (b5a2b71). Covering and share/2026-08-16 untouched. Later README claims left as they stand (Caccetta leftover through $n=151$, hole $n=152$, $F_4$ $0.34640$; ionization $1.1010$; and the rest).
• Re-opened the record: still $40\le\tau_5\le 44$ (Tao $C_{29}$, Cohn table dim 5: 40 / 44, Mittelmann–Vallentin $s_{14}(5)=44.99899685\ldots$, Cohn–Rajagopal arXiv:2412.00937v3).
• Type-$(0,5)$ leftover-tight CNF (certs/five_k30_n0_5.cnf.json, sha256 cdec5e76…, same encoding as the unfinished q7 $k=30$ instance) and the stronger global leftover CNF (n1_k19_star5_no21_no13.cnf.json, 220 five-star forbids) were written and independently rebuilt. Cubes pin $|U|=u$ for $u=19,\ldots,30$ (certs/k30_cubes.json).
• Native CaDiCaL on the $k=30$ leftover-tight instance wrote a binary DRAT past $15$ GB (about four hours of solver time) and did not exit. The weaker q7 global leftover DRAT passed $11$ GB and did not exit. Kissat probes on both q8 CNFs were unfinished. No drat-trim s VERIFIED. No certs/code41.json.
• leftover_k30 smoke (2000 nodes, five_mode 3) is not a certificate. q7 leftover extras B&B at $20$G nodes with the type-$(2,1)$/$(1,3)$ prune was already incomplete.
• Unrestricted interval unchanged: $40\le\tau_5\le 44$. Did not beat Mittelmann–Vallentin. Did not produce a 41-point code. Did not claim $\tau_5=40$. Residue.