Spherical codes in 5 dimensions, maximum minimum distance on S^4

# Spherical codes in 5 dimensions, maximum minimum distance on S^4 ## Goal `sphere.py` exposes `place(n: int, time_budget: float, seed: int) -> list[tuple[float, ...]]`: `n` points, each a tuple of 5 coordinates, on the unit sphere S^4 in R^5. Maximise the smallest pairwise Euclidean distance D = min over i < j of |x_i - x_j| Equivalently: find the best spherical code of size `n` in 5 dimensions, or pack `n` equal caps on S^4. This is the 5-dimensional case of the Tammes problem (`tammes-problem` is the same benchmark on S^2). Sloane's tables of putatively optimal packings cover n = 6, ..., 130 in 5 dimensions; the thirteen `n` here were chosen away from every proven case and from the regular polytopes and their subsets, so every instance is open and a better configuration is a new record. Proven optimality in R^5 covers n <= 10 (the simplex, then Rankin's bound: for d + 1 < n <= 2d the cross-polytope's 90 degrees is optimal) and n = 16 (the Clebsch code, an arccos(-1/5) two-distance set where the Levenshtein bound is tight). n = 14, 15 are subsets of the Clebsch code and n = 40 is the D5 kissing configuration at 60 degrees (the kissing number of R^5 is open, so that one is only conjectured optimal); all of those are excluded. ## Metric metric = mean over n in NS of D(n) / record(n) 1.0 means matching every record; above 1.0 means beating at least one. The eval projects every returned point onto the unit sphere (any finite non-zero vector with exactly 5 coordinates is accepted; a zero vector, a non-finite coordinate, the wrong dimension or the wrong count is a failed run; coincident points simply score 0 for that `n`) and recomputes the minimum distance itself. Nothing your solver reports is used. `ZT_EVAL_SEED` only changes the `seed` handed to `place`, so your method must be robust to its starting point. ## Records Best-known configurations from N. J. A. Sloane, R. H. Hardin, W. D. Smith et al., "Tables of Spherical Codes" (NeilSloane.com/packings/, files `dim5/pack.5.<n>.txt`, fetched 2026-09-07; every entry credited on the site to Hardin, Sloane and Smith, 1994). The record distance below was recomputed from those coordinates (the files are double precision, so the recomputed distance is exact far beyond the 12 decimals kept); the angle is `2 asin(D / 2)` in degrees, quoted to the tables' 1e-7 degree precision. Running the files themselves through `eval.py` scores exactly 1.0 on every `n`. A distance above a record by more than 1e-8 is listed in `records_beaten`. | n | record D | angle (deg) | source | proven optimal | |---|---|---|---|---| | 18 | 1.203742711866 | 74.0080831 | Hardin, Sloane, Smith 1994 | no | | 22 | 1.135786040964 | 69.2068599 | Hardin, Sloane, Smith 1994 | no | | 26 | 1.104318197345 | 67.0305252 | Hardin, Sloane, Smith 1994 | no | | 30 | 1.083485323937 | 65.6046967 | Hardin, Sloane, Smith 1994 | no | | 35 | 1.021334490675 | 61.4165697 | Hardin, Sloane, Smith 1994 | no | | 38 | 1.003749224436 | 60.2482021 | Hardin, Sloane, Smith 1994 | no | | 44 | 0.972154604647 | 58.1661569 | Hardin, Sloane, Smith 1994 | no | | 52 | 0.937931640070 | 55.9343732 | Hardin, Sloane, Smith 1994 | no | | 58 | 0.914913531410 | 54.4462372 | Hardin, Sloane, Smith 1994 | no | | 64 | 0.887876826070 | 52.7108774 | Hardin, Sloane, Smith 1994 | no | | 76 | 0.858594307886 | 50.8459281 | Hardin, Sloane, Smith 1994 | no | | 88 | 0.819173375583 | 48.3577476 | Hardin, Sloane, Smith 1994 | no | | 100 | 0.796048581024 | 46.9094506 | Hardin, Sloane, Smith 1994 | no | ## Constraints - Standard library only. No numpy, no scipy. The eval rejects other imports. - Respect `time_budget` (seconds, per call). The eval fails a call that runs past 1.25x + 3 s. - Deterministic given `seed`: use `random.Random(seed)`. ## Iterating - `ZT_EVAL_NS=18,30` runs a subset of n; `ZT_EVAL_PER_N_SECONDS=2` shortens the per-n budget. The default is all thirteen n at 8 s each (about 110 s). Run `python eval.py` in your workspace. - The per-n detail in the eval output shows which n are furthest from their record. ## Ideas that are known to matter (check the journal before repeating one) - The objective is a max-min, so plain gradient methods stall. Standard tricks: minimise a soft energy sum d^-p with p ramped up (the baseline), or iterate "find the active contact pairs, then solve the LP that pushes them apart" (an SLP), which converges to the true local optimum. - In 5 dimensions there is no Fibonacci-style spiral start: use many random starts, or seed from structured sets (subsets and unions of orbits of the cross-polytope, the Clebsch 16-point code, or a product of lower-dimensional codes) and let the optimiser break the symmetry. - Basin hopping: perturb the current best a little, re-optimise, accept if the minimum distance did not drop. Many restarts beat one long run; the record configurations often have rigid contact graphs that a single descent rarely finds. - The pure-Python pair loop is the bottleneck: only the near-contact pairs matter once the configuration is decent, so keep a neighbour list and refresh it occasionally. Write one honest line in `NOTES.md`: the idea, and which `n` it helped. Simpler is better: all else equal prefer the shorter solver, and treat removing code for an equal score as a win. Log every experiment, including discards, in your results.tsv.