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6429 bytes, as of the version from 2026-10-25 07:54 (current). Recorded here, not run.
# Thomson energies, T478: build evidence
The source snapshot was retrieved on 25 October 2026. The unit sphere,
unit charges, Coulomb constant one, and unordered pair convention are fixed
before computation. The index is the number of charges, in ascending order.
No optimization or proof of global minimality was performed by this run.
## Sources and proof status
- Wayne Deeter, https://deetour.net/oppp/thomson/, index Last-Modified
10-Oct-2026. `sources.json` preserves the printed energies and the complete
original coordinate text for n=4,...,100. `source-manifest.json` records
URLs, retrieval times, HTTP metadata, byte counts and SHA-256 checksums.
- Cambridge Cluster Database,
https://www-wales.ch.cam.ac.uk/~wales/CCD/Thomson/table.html,
index Last-Modified 13-Aug-2026. The page explicitly credits Deeter's
updates. The 91 printed energies for n=10,...,100 are in `sources.json`.
- Richard Evan Schwartz, *The 5 Electron Case of Thomson's Problem*,
https://arxiv.org/html/1001.3702. Theorem 1.1 proves the triangular
bipyramid minimises the Coulomb energy. Section 1.1 gives the unit sphere
and unordered-pair convention and lists the familiar proven configurations.
- Henry Cohn and Abhinav Kumar, *Universally optimal distribution of points
on spheres*, JAMS 20 (2007), 99–148,
https://arxiv.org/html/math/0607446,
DOI 10.1090/S0894-0347-06-00546-7. Theorem 1.2 and Table 1 cover the
octahedron and icosahedron. Their ordered-pair energy is twice ours;
their completely monotonic potential is a function of squared distance.
Coulomb potential corresponds to f(t)=t^(-1/2).
The direct Brown PDF download timed out and the AMS PDF returned 403.
Both papers were subsequently read and preserved as arXiv HTML research
artifacts. No numerical or mathematical choice depends on an inaccessible
source.
## Independent numerical checks
`checks.py` first tests a known antipodal energy 1/2, a triangle energy
sqrt(3), and a deliberately wrong comparison. It then:
1. Checks exact rational output and finite enclosures with the repository's
`check.py`, and checks that the generator names its Sage imports.
2. For all six proven cases, computes independent pair-distance sums using
exact geometric coordinates at 200 digits plus a 64-bit guard. The five
radical formulas use 100 digits plus a 64-bit guard, and retain at least
393 relative accuracy bits. All written 100-digit intervals contain the
independent result. The rational row is exactly 1/2.
3. For each of the 97 published configurations, reads coordinates as exact
decimal rationals and projects every vector radially onto the unit sphere.
It computes pair-distance energies at 60 digits plus 64 bits and independently
uses inner products at 90 digits plus 64 bits. Differences are less than
1e-50 and the enclosures are finite and narrow.
4. Compares every heuristic entry with its coordinate energy and every
Cambridge row with the corresponding recomputation. Cambridge is a second
transcription check, not an independent global minimization experiment,
since it credits Deeter's updates.
5. Checks the Jensen lower bound on every row and radius scaling for every
source configuration.
## Precision decision and resolved disagreement
The source prints 13 significant digits. At n=22, the printed 185.2875361492
and the normalized coordinate energy 185.28753614930758... differ by
1.075821 units in that final place. At n=44, 807.1742630845 and
807.17426308462807... differ by 1.280758 final-place units. The raw coordinate
energies are within the source's last-place intervals in both cases;
normalizing shifts them by respectively 0.095332 and 0.633429 final-place
units. Rounded coordinates are not exactly unit vectors: the worst
squared-norm error over the source range is 1.581310e-12.
Thus keeping the source's final digit would assert more agreement than the
published coordinate precision supports. No optimizer result was chosen over
another. Every heuristic row is rounded to 12 significant digits, the largest
common retained precision that contains every recomputed normalized energy.
The worst discrepancy at that precision is 0.628076 units in the last place.
`source-discrepancies.json` records raw and normalized checks for every row.
The 91 Cambridge comparisons all differ by less than one Cambridge last-place
unit (maximum 0.992705). This does not certify an enclosure of an unknown
global minimum; the table's heuristic rigour states that limitation.
## Selection and size
There are 99 entries for n=2,...,100. This is a compact consecutive reference
range of small standard optimization benchmarks, beginning with the elementary
cases. It includes all six proven configurations cited here. The longest
stored value has 101 characters; the measured entry block is about 7.3 KiB.
The range is not a storage target. Deeter's larger configurations are outside
this first table contribution.
## Corpus and lifecycle
Family issue #202, the local skill, and the live skill were read first.
The stricter local convention was retained where the skills differed.
`already_here`, `already_asked`, title/subject searches, and numerical lookups
found no existing Thomson-family table. T478 was claimed before computation.
Published neighbours T394, T446 and T449 were read, and their actual slugs
were obtained from API search results. T477 is still a draft, so it is not
linked from this table. The `extremal configurations` tag does not exist.
It could apply to T394, T446, T449, T477 and T478; this run uses the existing
`discrete geometry` and `physics` tags and proposes the shared tag for a human.
The artifact policy keeps this table unpublished and not offered for review.
A separate critique and repair precede the controller's offer. The generator
and all necessary data and checks are attached to the database draft.
## Checks after storage
`verify(sample=None)` reported 99/99 matched, zero differing, zero missing,
and zero extra. The independent coordinate/formula checks then passed on all
99 values read back from the database, including their ascending parameter
order. The API canonicalizes the submitted flat entries into a mapping keyed
by n; the attached checker accepts both document shapes. `stored-T478.json`
and `stored-checks.json` preserve this read-back evidence. The website audit
returned an empty findings list and `clean: true`.