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4194 bytes, as of the version from 2026-10-25 08:29 (current). Recorded here, not run.
# Sources and conventions for T479
Snapshot fetched 2026-10-25 UTC. No values have been published or offered for
review by this run. The draft is intended for the separate critique stage.
The family conventions came from numberdb-data proposal #202. Corpus searches
for Tammes and spherical codes, the screening helpers, and four numerical
lookups found no duplicate. `discrete geometry` and `sequence` exist. The
proposed `extremal configurations` tag does not yet exist; it would fit T394,
T449, T446, T477, T478 and this draft. T477 and T478 are unpublished drafts,
so their links are deferred. T394, T449 and T446 were read through the API and
their slugs were read from search results.
The primary numerical source for 7 <= n <= 100 is Henry Cohn's Table of
spherical codes, https://spherical-codes.org/, archived for citation at
https://hdl.handle.net/1721.1/153543. The attached sources.json preserves every
coordinate as a decimal string, its URL and SHA-256, the displayed cosine,
the source's references and optimality mark, and the minimal polynomial when
one is supplied. The raw HTML and plain table are retained in the build's
research artifacts. The HTML includes n=11; the plain file omits that repeated
icosahedral cosine, so its polynomial was read from the HTML.
For 4 <= n <= 6, values and coordinates are from N. J. A. Sloane, with
R. H. Hardin, W. D. Smith and others, Tables of Spherical Codes,
https://neilsloane.com/packings/. The regular tetrahedron and octahedron give
closed forms independently of those numerical coordinates. The source starts
at n=4. The range to 100 gives a contiguous reference for small spherical codes.
It stores the separation of each best known configuration in degrees and its
invertibly equivalent chord length on the unit sphere, not cap radius.
The source's minimal polynomial is a certified algebraic description of an
attainable angle, not necessarily a proof that the configuration is globally
optimal. For 27 source polynomial cases and the three elementary cases, the
generator uses exact rational bisection and certified monotonicity on the
initial interval, then 768-bit ball arithmetic for the angle and chord. The
initial interval is the printed cosine plus or minus 10^-10; uniqueness is
certified by the shifted polynomial derivative. After 400 bisections, the
smallest measured relative accuracy of an algebraic result is 431 bits,
comfortably more than 100 decimal digits. Angles 90 for n=5,6 remain integers.
For other cases the source cosine is assigned an uncertainty of one unit in
its twelfth decimal place and converted to ten significant digits. Each
written decimal is checked to contain that entire transformed input ball.
That source-error model is an agreement check, not a rigorous guarantee on
the unrounded optimizer, and certainly not on the global optimum.
checks.py uses three known-answer controls before reading the data. It
recomputes all pairwise distances and inner products after normalizing each
coordinate vector, checks 97 configurations against the published cosines,
checks both conversions for all 194 entries, and compares regular tetrahedron,
octahedron, square antiprism, n=9, and icosahedron formulas. Numeric code
values overlap the intervals actually stored. The high precision rows have
mathematical enclosures from their algebraic descriptions; the coordinates
independently corroborate only the available numerical digits.
The established optimality references cover 4 <= n <= 14 and n=24. Cohn also
marks n=15 as optimal, citing mt0-svg's 2026 preprint. The repository states
that the Lean result assumes graph enumeration and computational refutation;
this run has neither replayed those certificates nor reviewed that proof.
The table records the preprint's claim separately, without declaring other
cases globally optimal or presuming that every unmarked case remains open.
The older Sloane degree table agrees within one unit in its seventh decimal
place for 96 of the 97 configurations. At n=97, the current Cohn coordinates
have a larger separation. The current coordinates reproduce Cohn's displayed
cosine, so the difference is a different configuration, not rounding.