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The five-dimensional kissing number

Statement

The kissing number τ5\tau_5 is between 40 and 44 and is widely expected to equal 40. No proof excludes configurations of sizes 41–44.

Tonight

An exact dual certificate excluding some k∈{41,42,43,44}k\in\{41,42,43,44\} on the whole interval [−1,1/2][-1,1/2], or a new exact 41-point spherical code in S4S^4. This is not a small-dent pass: hunt a jump of the unrestricted interval. A restricted numerical SDP that does not become an exact positivity certificate is an incomplete search. Fetch Tao C29C_{29}, Cohn, and Mittelmann–Vallentin before searching. Do not claim τ5=40\tau_5=40 unless it is proved.

Outcome (2026-08-17)

Published range is still 40≤τ5≤4440\le\tau_5\le 44 (Tao C29C_{29}, Cohn table, Mittelmann–Vallentin s14(5)=44.998…s_{14}(5)=44.998\ldots). No unrestricted dual below 44 and no 41-point code. Restricted exact certificates are in compute/certs/restricted_delsarte.json and are written up in RESEARCH.md / WALKTHROUGH.md.

Outcome (2026-08-27)

Still 40≤τ5≤4440\le\tau_5\le 44. New restricted certificates in compute/q1/: every published 40-point code is maximal (polar max⁡∣x∣2=5/4<2\max|x|^2=5/4<2), and the integer Delsarte slice of TQ5T_{Q_5} is empty at N=44N=44. No unrestricted dual below 44 and no 41-point code.

Outcome (2026-08-27, continued)

Still 40≤τ5≤4440\le\tau_5\le 44. compute/q2/ replayed the existing duals and q1 certificates, then searched for an unrestricted dual below 44 and for an exact 41-point code. Neither was found. Residue: the 36-clique hunt in the 355-point T5T^5 remainder and the 1480-point (1/4)Z5(1/4)\mathbb Z^5 graph did not finish.

Outcome (2026-08-27, q3)

Still 40≤τ5≤4440\le\tau_5\le 44. No unrestricted dual below 44 and no 41-point code. On the leftover 1480-point (1/4)Z5(1/4)\mathbb Z^5 graph, every extra kisses at most 36 of the 40 D5D_5 roots, and a complete UU-superset scan empties every 41-set that uses 33 or more D5D_5-type points (compute/q3/complete_slices.json). The n1≤32n_1\le 32 slice of that graph, and the 36-clique in the 355-point T5T^5 remainder, remain residue.

Outcome (2026-08-27, q4 dual)

Still 40≤τ5≤4440\le\tau_5\le 44. Exact Bachoc–Vallentin Sk5S_k^5 matrices over Q\mathbb Q are in compute/q4/bv.py. No unrestricted dual below 44: 1-point Delsarte remains ≈46.34\approx 46.34, and every exact 3-point Putinar / p4-span / square-dictionary attempt failed to certify a value <44<44. Residue, not a dent. Replay: python3 compute/q4/dual_exact.py.

Outcome (2026-08-27, q4 leftover)

Still 40≤τ5≤4440\le\tau_5\le 44. No unrestricted dual below 44 and no 41-point code. On the leftover 1480-point (1/4)Z5(1/4)\mathbb Z^5 graph the ten special octads are the D5D_5 coordinate-stars; there is no 41-set that uses 22 or more D5D_5-type points (compute/q4/n1_dfs_k18.json, n1_dfs_k16.json, n1_complete_k12.json). The n1≤21n_1\le 21 slice of that graph remains residue. The 355-point T5T^5 remainder has no 35-colouring and no SAT 36-clique without a stored DRAT; share 24 through 30 with each published 35 are empty. Residue, not a dent. Replay: sh compute/q4/run_all.sh.

Outcome (2026-08-27, q5)

Still 40≤τ5≤4440\le\tau_5\le 44. The 355-point T5T^5 remainder has no 36-clique: native CaDiCaL 3.0.1 is UNSAT and Heule drat-trim reports s VERIFIED on the binary DRAT (compute/q5/t5_36_proof.json). Share 23 with each published 35 is empty. On the 1480-point (1/4)Z5(1/4)\mathbb Z^5 graph there is no leftover 41-set whose missed-root union sits in three D5D_5 coordinate-stars. The n1≤21n_1\le 21 slice with star-cover at least 4 remains residue. No unrestricted dual below 44 and no 41-point code. Residue, not a dent. Did not claim τ5=40\tau_5=40. Replay: sh compute/q5/run_all.sh.

Outcome (2026-08-27, q6)

Still 40≤τ5≤4440\le\tau_5\le 44. No leftover 41-set in the 1480-point (1/4)Z5(1/4)\mathbb Z^5 graph whose missed-root union sits in four D5D_5 coordinate-stars (compute/q6/four_star_extras.json, replayed in replay_four_star.json). The n1≤21n_1\le 21 slice with star-cover at least 5 remains residue. No unrestricted dual below 44 and no 41-point code. Residue, not a dent. Did not claim τ5=40\tau_5=40. Replay: sh compute/q6/run_all.sh.

Outcome (2026-08-27, q7)

Still 40≤τ5≤4440\le\tau_5\le 44. No leftover 41-set in the 1480-point (1/4)Z5(1/4)\mathbb Z^5 graph whose missed-root union sits in a type-(2,1)(2,1) or type-(1,3)(1,3) five-star of D5D_5 coordinate-stars (compute/q7/five_star_sat.json, Aut orbits in orbits.json, native CaDiCaL DRAT verified). The 32 type-(0,5)(0,5) five-star hosts and the global ∣U∣=19|U|=19 leftover SAT remain residue. Leftover-tight extras B&B ran 2020G nodes with the type-(2,1)(2,1) prune and did not finish (leftover_global.json). No unrestricted dual below 44 (best certified (t−1/2)q2(t-1/2)q^2 still 221991/3733≈59.47221991/3733\approx 59.47). Residue, not a dent. Did not claim τ5=40\tau_5=40. Replay: sh compute/q7/run_all.sh.

Outcome (2026-08-29, q8)

Still 40≤τ5≤4440\le\tau_5\le 44. Leftover-tight SAT on the type-(0,5)(0,5) orbit representative (compute/q8/certs/five_k30_n0_5.cnf.json, sha256 cdec5e76…, 32 hosts by Aut) and the stronger global leftover SAT ∣U∣=19|U|=19 forbidding type-(2,1)(2,1) and type-(1,3)(1,3) (n1_k19_star5_no21_no13.cnf.json) did not finish. No stored Heule-verified DRAT and no 41-set. Tiny leftover_k30 smoke is not a certificate. Residue, not a dent. Did not claim τ5=40\tau_5=40. Replay: sh compute/q8/run_all.sh.