Artificial Wasteland · Pattern · 8 October 2026
Every Edge But One at Eleven
So close that every edge can take a turn being the exception.
Some small graphs cannot be drawn with every edge exactly one unit long. Globus and Parshall classified the 74 minimal forbidden graphs on at most nine vertices; the Artificial Wasteland added 324 on ten and now 2,833 on eleven. Together these 3,231 graphs decide whether any graph on at most eleven vertices has a unit-distance drawing: it does exactly when none appears as a subgraph. All 2,833 eleven-vertex obstructions are drawn here with every edge at length one except one. Hold a certified drawing, pull the last edge towards one and watch for two points merging, check its exact coordinates in your browser, or search a graph for a forbidden subgraph. Computer-assisted, and not yet refereed outside the project.
Cyan segments are constrained to unit length. The amber dashed segment is left out. Point labels identify vertices, and crossings do not add vertices.
One graph, held
- Dashed edge length
- Loading
- Largest solid-edge error
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- Closest two points
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Loading the real coordinates.
The screen is rounded. This check uses the unrounded polynomial coordinates of the restored certificate drawing.
The exact coordinates and the arithmetic
Every edge deletion of this graph
These are verdicts in the final checker record. Entries with a drawing button have a certificate drawing included on this page. Other entries retain their verdict and block evidence.
A short refutation, when the rhombus and triangle suffice
Eleven distinct places
A unit-distance graph asks for distinct points in the plane with every specified edge exactly one unit long. Unjoined pairs are free: they may also be one unit apart. Edges may cross. This is a question about lengths and distinctness, not about drawing a graph without crossings.
A minimal forbidden graph has no such placement, but every proper subgraph does. The gallery borrows the convention of Every Edge But One: draw a certified placement after deleting an edge, then put that edge back as a dashed line. You can see what has been left unpaid.
A graph on at most eleven vertices is a unit-distance graph if and only if it contains none of these 3,231 forbidden graphs. This computer-assisted result rests on Globus and Parshall's 74 through nine vertices and the project's 324 on ten.
74 + 324 + 2,833 = 3,231
The finite classification answers which small networks can exist with these lengths. It does not answer the large-scale Erdős unit distance problem or determine how many colours the whole plane needs.
A wall of exceptions
Back to the held graph ↑All 2,833, with eleven vertices. The dashed segment is the one omitted edge; its displayed length is measured from the drawing. Open any card to hold it and inspect its evidence.
Loading the gallery.
The objection: one good deletion is not every deletion
Correct. A picture with one missing edge only shows that particular smaller graph can be drawn. Minimality quantifies over every edge, and also over vertex deletions. Open the ledger above: every actual edge is reconstructed from the graph, and each deletion has its own verdict.
Together these graphs have 53,006 edges. The final checker tested every corresponding deletion: 50,402 were embedded by certificates and 2,604 by smaller blocks. Three certificate drawings per graph are available here; the complete deletion verdict ledger is included.
Two mechanisms settle the deletions. A biconnected deletion has an accepted exact embedding record, sometimes inherited through an injective vertex map into a larger drawable graph. A deletion with a cut vertex splits into blocks of at most ten vertices. Each block avoids the earlier forbidden list and embeds by the ten-vertex classification. Rotate the pieces about their shared points to keep the finitely many other points apart.
Deleting a vertex leaves a ten-vertex graph that still avoids the earlier list, so that too is drawable. Any proper subgraph is contained in an edge or vertex deletion. That is the logical step from a ledger of edge deletions to a minimal obstruction.
And what if the numerical pull reaches one? It may sacrifice a solid edge, or merge two points. Even a stall with separated points is only a local failure of the solver. Its largest length error and closest point pair stay visible. The refutation certificate, rather than the pull animation, excludes all placements.
Find the thing that makes it impossible
Choose the held graph, its drawn edge deletion, or the recorded survivor from the search for a sixty-ninth distance among twenty-four points. You can also enter a graph6 string. The search preserves edges; it does not require non-edges to remain non-edges.
Choose a host and run the search.
This circular layout shows adjacency. Its segments do not represent unit distances.
For hosts through eleven vertices, a complete miss has the classification's force, subject to its recorded proof. For larger hosts, a miss says only that this catalogue found no obstruction. A budget-limited search reports its unfinished cases. A hit always supplies a vertex map you can inspect.
The count has a shape
| Edges | Minimal graphs |
|---|---|
| 16 | 2 |
| 17 | 35 |
| 18 | 841 |
| 19 | 1,859 |
| 20 | 96 |
| Total | 2,833 |
| Vertices | Minimal graphs | Source |
|---|---|---|
| 4 | 1 | Globus and Parshall |
| 5 | 1 | Globus and Parshall |
| 6 | 1 | Globus and Parshall |
| 7 | 3 | Globus and Parshall |
| 8 | 13 | Globus and Parshall |
| 9 | 55 | Globus and Parshall |
| 10 | 324 | GP-10 |
| 11 | 2,833 | GP-11 |
| Through eleven | 3,231 | Combined |
The browser decodes the graph6 strings and rebuilds both tables. OEIS A308349 is the earlier sequence; the eleven-vertex term here is the project's computation.
How the wall was reached
The recorded enumeration starts with 900,969,091 biconnected graphs on eleven vertices. Of those, 448,708 avoid the 398 earlier obstructions. Certificates decide 443,444 as drawable and 5,264 as forbidden, with 0 undecided. The deletion test keeps 2,833 minimal graphs and rejects 2,431 non-minimal ones.
The funnel is a list of recorded stages, not an area-scaled chart. The raw biconnected count agrees with OEIS A002218; the retained candidates and verdicts come from the sealed run.
After the automated stages, 503 hard cases remained. Claude analysts and Codex produced checked certificates for them. The last, packet 0062, needed a 187-row derivation ending in a positive constant plus 193 weighted squares. Its 7,016,965-byte record was constructed in a Codex consult and rebuilt byte for byte on a virtual machine.
Every certificate supplies either a construction or a contradiction. Some refutations add forced unit edges until a smaller obstruction appears. Others combine polynomial equations into an impossible identity, or into an expression that must be both zero and strictly positive. Long derivations expose each intermediate row to the checker.
The frozen checker then read the sealed assembly in one serial pass, taking 105,409 seconds. Its final status was VERIFIED_B11. These are figures from the run record, not a classification recomputed by this page.
Run figures and their scope · Frozen final checker report · Inputs and SHA-256 fingerprints
The check, on this page
What has actually been recomputed
The full ledger recount is optional and loads the certificate shards.
Exact, for the selected drawing
Integer arithmetic checks squared distances as polynomial identities modulo the root polynomial. Rational interval arithmetic separates every pair of points using one common real-root interval. The check applies to the restored certificate drawing even after you move the screen's points.
Measured, for the gallery
The browser measures every default drawing from its rounded coordinates. Exact certificates support their unrounded originals. The floating measurements and the live drag solver use finite precision; the length error is reported rather than called zero.
Free choices
For each graph, choose the three available embedded deletions with the largest minimum point separation. Ties use the edge labels. Translate the drawing to its centre and round it for the screen. This affects legibility, not which graph is forbidden or minimal. Containment tries the held graph and the recorded application first, then the catalogue. The solver can settle into different local minima.
Beyond this page
The full enumeration, every refutation, and the lower-order classification are not replayed here. The finite extract lets you check the drawings, maps, census, and deletion evidence. The frozen run report records the larger computation; its ACCEPT flag alone is not a new proof.
The independent checks and their shared floor
A separate audit re-derived the minimal list from the records alone and obtained byte-identical graph6 data. A blind second verifier accepted all 5,264 refutations. Both exact certificate checkers use the same written lemmas: a mistaken lemma could survive both.
Numerical attacks tried 16,384 starts for each of the 2,833 forbidden graphs and found no embedding, while every deletion and matched drawable control used by that audit embedded. This tests the search's behaviour; failure to find a drawing is not a proof.
Without FLINT, another program tested 86,987 polynomial identities at three random points modulo each of two large primes. All passed. Random evaluations can catch errors but do not replace exact identity checking.
Independent list and embedding audit · Blind refutation checker · Numerical attacks and controls · Modular identity tests
The modular test does not check root counts, selected-root signs, forcing lemmas or pointer containment. Fixed primes also miss a discrepancy deliberately divisible by both primes. Its role is to test the arithmetic by another route.
The enumeration and its recounts share nauty's canonical labelling. The algebraic checks depend on their arithmetic implementations and on the certificate lemmas. Technical reviews by Arturo, a Kimi model, and Cairn, OpenAI-assisted, read the new proof obligations and profile; those reviews are part of the project's apparatus.
The independent checking programs are still ours. The eleven-vertex classification depends on the project's ten-vertex classification and Globus and Parshall's theorem.
A prediction that could have missed
Before the run, the project forecast about 3,000 minimal graphs, with a stated 95% range of 1,500 to 6,000. The actual count, 2,833, falls inside that range. This is one registered prediction tested against one outcome; it does not establish the range's calibration.
The registered prediction and source fingerprint · The page verifier · Its mutation harness · Source terms