Research log — five-dimensional kissing number

Status (accessed 2026-08-17)

The kissing number $\tau_5=A(5,1/2)$ is open. The published range is still

and the value is widely expected to be 40. No accepted paper, arXiv preprint, or living table consulted tonight improves either endpoint.

Sources checked tonight

Odlyzko–Sloane 1979 is the continuum LP number $46.345$. Coxeter 1963 is 48. The $D_5$ lower bound is Korkine–Zolotareff 1873 (40 roots: permutations of $(\pm 1,\pm 1,0,0,0)$).

Independent checks (this folder)

• compute/verify_configs.py — PASS. $D_5,L_5,Q_5,R_5$ are 40-point kissing codes. Pair histograms match Cohn–Rajagopal Table 2.1 on the nose, including the non-$D_5$ angles $-4/5,-3/4,-3/10,-1/4,1/5$.
• compute/levenshtein.py — PASS. $L_5(5,1/2)=48$ from the BDM formula.
• compute/extend_d5.py — PASS. $4/5>1/2$, so $D_5$ is maximal.
• compute/d4_equator.py — PASS. $A(4,0)\le 8$ via $t^2+t$; 24-cell hole graph has independence number 8 and exactly three MIS (Cohn’s three cross-polytopes).
• compute/verify_certificates.py — PASS. Rebuilds Gegenbauer from the recurrence and checks the two duals in compute/certs/restricted_delsarte.json.
• compute/integer_d5.py — PASS. Integer $T_{D_5}$ distributions: 42/43/44 empty; 41 only with 20 antipodes.

What we certified

Restricted exact duals and one geometric lemma. None of these changes the unrestricted range $40\le\tau_5\le 44$.

1. 24-cell lemma

A kissing code in $S^4$ that contains an isometric copy of the 24-cell has at most 40 points. Proof: the 24-cell is equatorial; extra points lie at $|h|\ge 1/\sqrt{2}$; each open hemisphere contributes at most $A(4,0)\le 8$ extra points, via the exact polynomial $t^2+t$. Consequently $k\in\{41,42,43,44\}$ is impossible in this class, and the two 24-cell-containing 40-point codes $D_5$ and $L_5$ are maximal.

This is the kind of fact experts use when they build $D_5$ and $L_5$ by filling holes of the 24-cell. We did not find it stated as a bound on 41–44, so we wrote out the Rankin step and checked the discrete hole graph. It is a restricted obstruction, not an unrestricted bound.

2. Leech-angle Delsarte dual

Any spherical code in $S^4$ whose distinct inner products lie in

(the exact angle set of Leech’s $L_5$) has size at most

hence at most 43. The dual is the Gegenbauer combination with coefficients

Replay: compute/verify_certificates.py. This excludes $k=44$ in the $T_{L_5}$ class. It does not exclude 44 for $Q_5$-type angles (the corresponding interpolating dual has value $167076/3047\approx 54.8$).

3. $D_5$-angle codes

For $T_{D_5}=\{-1,-1/2,0,1/2\}$:

• real Delsarte gives $N\le 42$, by the dual $(1,49/45,28/3,0,32/3,896/45)$;
• integer pair-counts kill 42, 43, 44, and force any 41-point distribution to contain 20 antipodal pairs;
• those 40 points are an equal-length subset of a simply-laced root system of rank $\le 5$, hence a copy of $D_5$, which cannot be extended.

So $k\in\{41,42,43,44\}$ is impossible in the $T_{D_5}$ class. The root-system step is classical (Cartan–Killing: the only 40-root equal-length system in rank $\le 5$ is $D_5$); the integer slice and the exact dual of bound 42 are computed here.

4. $Q_5$ has no extra equatorial point

On $\sum x_i=0$ the $A_4$ inequalities cut a polytope of maximal squared norm $6/5<2$.

What we did not beat

• The unrestricted upper bound 44. No continuum Delsarte or exact Bachoc–Vallentin dual below 44 was produced. A floating 3-point SDP would have been residue even if we had rerun one.
• The unrestricted lower bound 40. No exact 41-point spherical code. Algebraic ansätze (cube+axes, $D_5$ plus holes, $A_4$ completion) top out at 40. Numerical 41st-point slacks against $Q_5$ and $R_5$ are $\approx-0.265$.
• Maximality of $Q_5$ and $R_5$ as spherical codes (they do not contain a 24-cell, so the lemma does not apply).
• An exact SOS certificate of Mittelmann–Vallentin’s $44.998\ldots$.

How to replay

sh compute/run_all.sh

The duals alone:

compute/.venv/bin/python compute/verify_certificates.py

Expected last lines: D5_inner_products: certified=True bound=42 … excludes=[43, 44] and L5_inner_products: certified=True bound=239925/5456 … excludes=[44].

2026-08-27 — record still $40\le\tau_5\le 44$

Opened tonight:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $40\le\tau_5\le 44$, conjectural value 40. Upper bound cited is Mittelmann–Vallentin.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin. Later-dimension news (Cohn–Li, de Laat– Leijenhorst, Ma et al. 2511.13391, …) does not touch dim 5.
• https://arxiv.org/abs/2412.00937 — v3 (4 Mar 2026): “appears to be 40… best upper bound that has been proved is 44”. Four 40-point geometries.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.998\ldots$, hence $\tau_5\le 44$. Still the published upper bound.
• https://arxiv.org/abs/1507.03631 — survey story $48\to 46.345\to 45\to 44.998$.
• https://arxiv.org/abs/2301.08272 — Szöllősi $Q_5$, third 40-point code.
• https://doi.org/10.5281/zenodo.18449600 — unaffiliated note claiming a degree-36 Cohn–Kumar number $44.0297$. Not a paper; not an exact dual; unrestricted Delsarte is already $\approx 46.345$. Lead only.

2026-08-27 — what we certified this run

Still no change to the unrestricted range $40\le\tau_5\le 44$.

5. Polar maximality of the four 40-point codes

compute/q1/polar_vertices.json, replayed by replay_max_vertex.py (Fraction GE, not the Cramer path in polar.c). For each of $D_5,L_5,Q_5,R_5$ the polar is bounded and $\max|x|^2=5/4<2$. Python and C enumerations agree on the vertex counts. Consequently none of these four codes admits a 41st kissing point. This is a statement about those four codes, not about $\tau_5$.

6. Integer Delsarte on $T_{Q_5}$ excludes 44

compute/q1/integer_q5_44.json: $14\,753\,818\,985$ integer points in a box containing the real AABB by 15, zero Delsarte-feasible. Tables match delsarte.py; $Q_5$'s own $N=40$ histogram passes. So $k=44$ is impossible for inner products in $T_{Q_5}=\{-1,-4/5,-1/2,-3/10,0,1/5,1/2\}$. Real Delsarte is still $\approx 44.67$; this is integrality, not a dual below 44.

$T_{L_5}$ still has integer hits at $N=41,42,43$ (compute/q1/integer_restricted.json).

Replay:

sh problems/kissing-5d/compute/q1/run_all.sh

2026-08-27 — record still $40\le\tau_5\le 44$ (q2)

Opened tonight:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $40\le\tau_5\le 44$, conjectural value 40. Upper bound cited is Mittelmann–Vallentin. No later endpoint.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin. Later-dimension news (Cohn–Li, Ho 2603.10425, Ma et al. 2511.13391, Takhanov et al. 2606.18984, Sun–Wang 2607.20359) does not touch dim 5.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$. Still the published upper bound.
• https://arxiv.org/abs/2412.00937 — v3 (4 Mar 2026), title now “Variations on five-dimensional sphere packings”: “appears to be 40… best upper bound that has been proved is 44”. Four 40-point geometries. No size 41.
• https://arxiv.org/abs/1507.03631 — survey story $48\to 46.345\to 45\to 44.998$.
• https://arxiv.org/abs/2301.08272 — Szöllősi $Q_5$.
• https://arxiv.org/abs/math/0608426 — Bachoc–Vallentin, $\tau_5\le 45$ before Mittelmann–Vallentin.
• https://doi.org/10.5281/zenodo.18449600 — the unaffiliated Cohn–Kumar $44.0297$ note is retracted (removed 5 Feb 2026). Not a paper; not an exact dual. Lead only.

2026-08-27 — what we certified this run (q2)

Still no change to the unrestricted range $40\le\tau_5\le 44$.

7. Finite exact graphs with no 41-clique

• compute/q2/t5_clique.json: Szöllősi $T^5$ remainder, 355 vertices, 607171 nodes, no 41-clique. The five basis vectors of the 360-point pool are universal; a 41-set in the full pool needs a 36-clique in this remainder, which was not settled (t5_36_residue.json).
• compute/q2/sphere_d2.json: 200 half-integer points of squared norm 2, no 41-clique. Contains $D_5$ and $L_5$.
• compute/q2/q5cap_clique.json: 320-point $Q_5$-cap orbit, no 41-clique.

8. No unrestricted dual below 44

compute/q2/unrestricted_dual.json: numerical continuum Delsarte $46.3368\ldots$ at degree $\ge 10$. Exact $(t-1/2)q(t)^2$ duals exist (best in the search $53235/1109\approx 48.003$). None has bound $<44$.

Replay:

sh problems/kissing-5d/compute/q2/run_all.sh

2026-08-27 — record still $40\le\tau_5\le 44$ (q3)

Opened tonight:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $C_{29}=\tau_5$, range $40\le\tau_5\le 44$, conjectural value 40. Upper bound cited is Mittelmann–Vallentin. No later endpoint.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin. Later-dimension news on the same table does not touch dim 5.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$. Still the published upper bound.
• https://arxiv.org/abs/2412.00937 — v3 (4 Mar 2026): “appears to be 40… best upper bound that has been proved is 44”. Four 40-point geometries. No size 41.
• https://arxiv.org/html/0902.1105v3 — same $s_{14}(5)$ number, table of SDP bounds for $n\le 24$.
• https://arxiv.org/abs/math/0501493 — Pfender, improved Delsarte via a diagonally-dominant kernel $f_\alpha$. Improves kissing bounds in dims 10, 16, 17, 25, 26; does not give an exact dual below 44 in dim 5.
• https://doi.org/10.5281/zenodo.18449599 — unaffiliated note claiming a Cohn–Kumar number $44.0297$. Retracted sibling of doi:10.5281/zenodo.18449600. Not a paper; not an exact dual. Lead only.
• https://arxiv.org/abs/math/0501493 — Pfender, Improved Delsarte bounds for spherical codes in small dimensions. The extra function is a diagonally-dominant kernel $f_\alpha$, not a Gegenbauer polynomial. Improves kissing numbers in dimensions 10, 16, 17, 25, 26. Does not produce an exact dual below 44 in dimension 5.

2026-08-27 — what we certified this run (q3)

Still no change to the unrestricted range $40\le\tau_5\le 44$.

9. $(1/4)\mathbb Z^5$: no 41-set with $n_1\ge 33$

compute/q3/complete_slices.json. Every extra of squared-norm 2 in $(1/4)\mathbb Z^5$ kisses at most 36 of the 40 $D_5$ roots. A 41-set therefore uses at least five extras, and the extras' missed-root union $U$ satisfies $|E|\ge|U|+1$. Complete generation of every $k$-superset of an actual missed set, $k=4,5,6,7$ (that is $n_1=36$ down to 33), finds no extras-clique of size $k+1$. So this 1480-point graph has no 41-clique that contains 33 or more $D_5$-type points. The $n_1\le 32$ slice is unfinished.

10. No unrestricted dual below 44

compute/q3/dual_gap.json. Continuum Delsarte is $46.3368\ldots$. A dual allowed to be positive on $[-1,-2/3)$ has numerical value $\approx 37.46$, but that gap is not a geometric lemma (the four published 40-point codes all have an angle in $[-1,-2/3)$) and the rational polynomials did not pass Sturm. Not a certificate.

Replay:

sh problems/kissing-5d/compute/q3/run_all.sh

2026-08-27 — record still $40\le\tau_5\le 44$ (q4 dual)

Opened tonight:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $C_{29}=\tau_5$, range $40\le\tau_5\le 44$, conjectural value 40. Upper bound cited is Mittelmann–Vallentin. No later endpoint.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin. Dim 6 is now 72 / 77 (de Laat–Leijenhorst– de Muinck Keizer); dim 5 is untouched.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$. Still the published upper bound. HTML https://ar5iv.labs.arxiv.org/html/0902.1105v3 for the $S_k^n$ formula and Table 1.
• https://arxiv.org/abs/math/0608426 — Bachoc–Vallentin, $\tau_5\le 45$. HTML https://ar5iv.labs.arxiv.org/html/math/0608426v4 : Theorem 3.2 for $Y_k^n$, Remark 3.4 monomial form $u^i v^j Q_k^{n-1}$, Corollary 3.5 for $S_k^n$, Theorem 4.2 for the dual, §5 Putinar with $p(u)=-(u+1/4)^2+9/16=-(u+1)(u-1/2)$.
• https://arxiv.org/abs/2412.00937 — v3 still “appears to be 40… best upper bound that has been proved is 44”.
• https://doi.org/10.5281/zenodo.18449600 — unaffiliated note claiming a Cohn–Kumar number $44.0297$. Retracted. Unrestricted Delsarte is already $\approx 46.345$. Lead only, not a citation.

2026-08-27 — what we certified this run (q4 dual)

Still no change to the unrestricted range $40\le\tau_5\le 44$.

Exact $S_k^5$ over $\mathbb Q$ (compute/q4/bv.py) replay the paper identities. No exact Putinar / SOS dual with value $<44$. The dictionary LP number $40.38$ at $d=6$ and the floating SDP values near 40 are residue. Best certified unrestricted dual is still Levenshtein 48.

Replay:

python3 problems/kissing-5d/compute/q4/bv.py
python3 problems/kissing-5d/compute/q4/dual_exact.py
python3 problems/kissing-5d/compute/q4/verify.py

2026-08-27 — leftover graphs (q4)

Opened again (and re-fetched the arXiv sources):

• https://teorth.github.io/optimizationproblems/constants/29a.html — still $40\le\tau_5\le 44$, conjectural value 40.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, refs [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin. Dim 6 is 72 / 77; dim 5 is untouched.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, HTML https://ar5iv.labs.arxiv.org/html/0902.1105v3 Table 1. Range of possible values limited from $\{40,\ldots,45\}$ to $\{40,\ldots,44\}$.
• https://arxiv.org/abs/2412.00937 — v3 still “appears to be 40… best upper bound that has been proved is 44”.

Still no change to the unrestricted range.

11. $(1/4)\mathbb Z^5$: no 41-set with $n_1\ge 22$

compute/q4/n1_complete_k12.json, n1_dfs_k16.json, n1_dfs_k18.json, replay_unions_k13.json, analyze_stars.json, verify_n1.rs. The ten octads are the $D_5$ coordinate-stars. Seed-unions through $|U|\le 18$ have no extras-clique of size $|U|+1$ (n1_dfs.c, $933{,}648{,}919$ unions). The $n_1\le 21$ slice is unfinished.

12. $T^5$ 36-clique still residue

compute/q4/t5_omega.json: Cadical and Glucose return no 36-clique; no stored DRAT. t5_share30_c.json through t5_share24_c.json: share 30 down to 24 empty.

Replay:

sh problems/kissing-5d/compute/q4/run_all.sh
python3 problems/kissing-5d/compute/q4/analyze_stars.py
python3 problems/kissing-5d/compute/q4/n1_check.py
rustc -O -o /tmp/verify_n1 problems/kissing-5d/compute/q4/verify_n1.rs && /tmp/verify_n1

2026-08-27 — record still $40\le\tau_5\le 44$ (q5)

Opened tonight:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $C_{29}=\tau_5$, range $40\le\tau_5\le 44$, conjectural value 40. Upper bound cited is Mittelmann–Vallentin. No later endpoint.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin. Dim 6 is 72 / 77; dim 5 is untouched.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$. Still the published upper bound.
• https://arxiv.org/abs/2412.00937 — v3 (4 Mar 2026): “appears to be 40… best upper bound that has been proved is 44”. HTML https://ar5iv.labs.arxiv.org/html/2412.00937v3.
• https://arxiv.org/abs/math/0608426 — Bachoc–Vallentin, $\tau_5\le 45$ before Mittelmann–Vallentin.

Re-opened the same five URLs later on 2026-08-27 (q5). Endpoints unchanged: Tao $C_{29}$ still $40\le\tau_5\le 44$; Cohn table dim 5 still 40 / 44, ratio 1.100, dim 6 is 72 / 77; Mittelmann–Vallentin $s_{14}(5)=44.99899685\ldots$; Cohn–Rajagopal v3 still “appears to be 40… best upper bound that has been proved is 44”. No later accepted improvement of either endpoint.

2026-08-27 — record still $40\le\tau_5\le 44$ (q6)

Opened tonight:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $C_{29}=\tau_5$, range $40\le\tau_5\le 44$, conjectural value 40. Upper bound cited is Mittelmann–Vallentin. No later endpoint.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin. Later-dimension news (Cohn–Li, Ho 2603.10425, Ma et al. 2511.13391, Takhanov et al. 2606.18984, Sun–Wang 2607.20359) does not touch dim 5. Dim 6 is 72 / 77.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$. Still the published upper bound. HTML https://ar5iv.labs.arxiv.org/html/0902.1105v3 Table 1: range of possible values limited from $\{40,\ldots,45\}$ to $\{40,\ldots,44\}$.
• https://arxiv.org/abs/2412.00937 — v3 (4 Mar 2026): “the kissing number in five dimensions appears to be 40, although the best upper bound that has been proved is 44”. Four 40-point geometries. No size 41.
• https://arxiv.org/abs/math/0608426 — Bachoc–Vallentin, $\tau_5\le 45$ before Mittelmann–Vallentin.
• https://arxiv.org/abs/1507.03631 — survey story $48\to 46.345\to 45\to 44.998$.
• https://arxiv.org/abs/math/0501493 — Pfender kernel. Improves kissing bounds in dims 10, 16, 17, 25, 26. Does not give an exact dual below 44 in dim 5.

2026-08-27 — record still $40\le\tau_5\le 44$ (q7)

Opened tonight:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $C_{29}=\tau_5$, range $40\le\tau_5\le 44$, conjectural value 40. Upper bound cited is Mittelmann–Vallentin. No later endpoint.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, ratio 1.100, citations [9] Korkine–Zolotareff and [17] Mittelmann–Vallentin. Dim 6 is 72 / 77. Later-dimension news (Cohn–Li, Ho 2603.10425, Ma et al. 2511.13391, Takhanov et al. 2606.18984, Sun–Wang 2607.20359, Bianchi et al. 2606.10402) does not touch dim 5.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$. Still the published upper bound. HTML https://ar5iv.labs.arxiv.org/html/0902.1105v3 Table 1: range of possible values limited from $\{40,\ldots,45\}$ to $\{40,\ldots,44\}$.
• https://arxiv.org/abs/2412.00937 — v3 (4 Mar 2026), title “Variations on five-dimensional sphere packings”: “the kissing number in five dimensions appears to be 40, although the best upper bound that has been proved is 44”. Four 40-point geometries. No size 41.
• https://arxiv.org/abs/math/0608426 — Bachoc–Vallentin, $\tau_5\le 45$ before Mittelmann–Vallentin.
• https://arxiv.org/abs/1507.03631 — survey story $48\to 46.345\to 45\to 44.998$.

python3 scripts/arxiv_fetch.py on 0902.1105v3, 2412.00937v3, math/0608426v4, 1507.03631: metadata matches; RESEARCH already had stubs for the first three.

2026-08-27 — record still $40\le\tau_5\le 44$ (q8)

Opened tonight:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $C_{29}=\tau_5$, range $40\le\tau_5\le 44$, conjectural value 40. Upper bound cited is Mittelmann–Vallentin. No later endpoint.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 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.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$. Still the published upper bound.
• https://arxiv.org/abs/2412.00937 — v3 (4 Mar 2026): “the kissing number in five dimensions appears to be 40, although the best upper bound that has been proved is 44”. Four 40-point geometries. No size 41.

python3 scripts/arxiv_fetch.py on 0902.1105 and 2412.00937: stubs already present.

2026-08-29 — wrap, record still $40\le\tau_5\le 44$

Opened again before the q8 wrap:

• https://teorth.github.io/optimizationproblems/constants/29a.html — $C_{29}=\tau_5$, range $40\le\tau_5\le 44$, conjectural value 40.
• https://cohn.mit.edu/kissing-numbers/ — dim 5 lower 40, upper 44, ratio 1.100.
• https://arxiv.org/abs/0902.1105 — $s_{14}(5)=44.99899685\ldots$, hence $\tau_5\le 44$.
• https://arxiv.org/abs/2412.00937 — v3 (4 Mar 2026), authors Henry Cohn and Isaac Rajagopal: appears to be 40, proved upper bound 44. No size 41.

No stored verified leftover DRAT from the type-$(0,5)$ or $|U|=19$ hunts. Incomplete search; not a move of either endpoint.