Before my AI pipeline could hunt for new shapes, it had to rebuild a known one

When an AI system says it has found something new in mathematics, the first question is whether its search works at all. My answer was an entrance exam, and it caught a rule in the program that would have quietly thrown away the very solid it was meant to find. No check on the program's output could have seen that, because a solid that is thrown away early leaves no output to check. Later, the same program checked whether that solid was really new, and with a reviewer's help it found the solid printed, unnamed, in a survey from 2016.

I do mathematics with AI agents. Here the agents (Claude, made by Anthropic) wrote and ran all of the program's code. I set the rules it had to pass and decided what got reported. I wrote up what happened in a short paper, Rediscover Before You Discover↗ (a free copy on Zenodo), accepted as a poster at a NeurIPS 2026 workshop (MATH-AI).

The solid it had to rebuild#

A space-filling solid is one whose copies pack together with no gaps, the way cubes do. Nobody has a complete list of them. In April 2026 Mathias Bernhard published one and named it the Josehedron. It has 12 faces: four triangles and eight four-sided faces. He presented it as new, with a careful hedge: "it would be surprising if the Josehedron had never been encountered before."

Solids like this come from a pattern of points. Give each point the region of space that is closer to it than to any other point. For the right pattern, those regions are all copies of one solid, and together they fill space. Bernhard's paper lists the 12 points that produce the Josehedron this way.

I had already worked with this solid: in August 2026 I counted how many different shapes can be built by gluing Josehedron cells together face to face, and those counts are now in the OEIS, a public database of number sequences. The counts and how they were checked have their own page. The animation shows that kind of counting, first with squares, then with Josehedron cells. Its final number is the count of shapes made from 11 cells, counting a shape and its mirror image as one.

Animation, no sound. Squares fly together into the five shapes that can be built from four squares. Bars then rise for sizes 1 to 7, showing how many different shapes of squares there are at each size, counting a shape and its mirror image as one: 1, 1, 2, 5, 12, 35, 108, with the four-square bar highlighted. The squares fade and a single Josehedron cell appears and turns slowly. It has four triangular faces and eight four-sided faces. Next to it a counter climbs to 4,032,748,840, the number of shapes that can be built from 11 Josehedron cells, again counting mirror images as one.

The entrance test#

A search program that has never found anything has no track record. So the program's first job was to find something already known: rebuild the Josehedron from nothing but Bernhard's list of 12 points. It was not given the solid's shape, its face count, or its symmetry. I chose a recent discovery on purpose. A solid named in 2026 is less likely to be sitting already in the software and catalogs used to check the work, or in what the AI models learned from. As it turned out, that held only for the name. The Josehedron was new as a named find, but the shape itself had sat, unnamed, in a 2016 survey all along (more on that below). Recency protected the name, and perhaps the list of points, but not the shape.

Before any computation ran, the pass conditions were written into a dated file and saved to the project's version history, so they could not be quietly changed later. The rule was blunt: if the rebuild failed, nothing else would run. To pass, the program had to produce cells with 12 faces, four triangles and eight four-sided, and its record of which cells touch which had to match the record from my earlier counting work, giving the same shape counts up to six cells. That earlier record is my own work, sharing no code but made by the same person, with the same method and the same family of AI models, so this match is a consistency check, not an independent reproduction. All the geometry was done in exact fractions, with no rounding, which rules out rounding errors but is still my own construction.

As a control, the same program also rebuilt three classic space-fillers from their own point patterns, the cube among them. All three came out right.

What the test caught before it ran#

The program's draft rules included one for awkward geometry: throw out any cell with a corner where more than three faces meet. Eight of the Josehedron's 12 corners have four faces meeting. So the rule would have thrown out the very solid the program was supposed to find, and every later check would have failed with nothing to look at.

Because the Josehedron was the fixed pass condition, that conflict showed up as soon as the rule met the target. In a real search it would have stayed hidden: a shape thrown out early leaves no output, and a check on the outputs cannot see something that was never produced.

The rule was changed before the test first ran, with the change dated and recorded. Cells like that are now flagged and passed to the exact-fraction step to decide. The pass conditions themselves did not change. Two of the paper's reviewers pointed out that this rule was fixed with the target in view, so the pass is not a blind one. That is fair. A blind test needs a target nobody has seen yet, such as the next discovery someone publishes.

With that one recorded change, the program passed: every condition matched.

Then the program checked whether the solid was new#

The same program, unchanged, was then pointed at Bernhard's claim. The place to look was a survey of space-filling solids that M. W. Schmitt published in 2016 as his doctoral dissertation. His computers searched 145 of the 230 possible symmetry types of a repeating pattern (crystallographers call them space groups), at a cost of some 351 years of computer time (spread across many processors), and printed tables of the 3,315 distinct solids they found, one table per symmetry type.

Bernhard's paper does not say which symmetry type his points have. The program worked it out: number 220 in the standard numbering. Schmitt's table for type 220 has one row with the Josehedron's counts of corners, edges and faces (12, 22 and 12). Matching counts do not make two solids the same, so the program rebuilt the solid at Schmitt's printed point. It came out different: six triangles, four four-sided faces and two five-sided faces. So the version of the paper the reviewers read concluded that the Josehedron was not in that table. The final version, the free copy linked above, carries the corrected verdict below.

The reviewer's pointer#

A reviewer pointed to two more tables to check, for symmetry types 198 and 161. Here is why those tables matter. Type 220 is the full set of symmetry moves (shifts, turns and reflections) that map the Josehedron's pattern of points onto itself. Types 198 and 161 are smaller sets of moves, each contained in type 220. Under type 220, each of the 12 points sits on a special spot: four of the moves, counting the one that does nothing, leave it exactly where it is. Schmitt's survey sampled points on a grid, and grid sampling tends to miss special spots like that. Under the two smaller types, no move except doing nothing leaves those points in place, so the same 12 points form an ordinary, generic pattern, the kind a grid survey does sample. So the Josehedron could appear in those types' tables even though the type-220 table missed it.

The program ran on those two tables within hours. The table for type 161 has no row with the right counts. The table for type 198 has one, and the solid at its printed point is the Josehedron: the same faces, joined the same way. The Josehedron's shape has been printed in the 2016 survey all along, without a name. Bernhard's hedge was right. The survey does not connect that row to his solid; the match is from this work.

The computation was right both times. What was wrong in the first version was how many tables it checked, and nothing in my written rules required checking more. That rule is now written down, dated 28 September 2026: if a solid's points sit on special spots, check the table of every smaller symmetry type in which the same points form an ordinary, generic pattern.

The test, for anyone building a discovery tool#

Rediscovering a known result as a sanity check is an old idea. Early AI discovery programs, such as AM and BACON, used it, and in software terms it is a regression test. What the paper adds is the setup: a recent find as the target, the smallest input that should lead to it, pass conditions saved before the run, nothing else allowed to run until it passes, and then the same program, unchanged, auditing the find's claim to be new. Recent systems such as FunSearch and AlphaEvolve check each output they produce; this checks the search itself first, because a candidate it throws away leaves no output to check.

It needs a discovery a computer can rebuild from a small input, an exact way to decide whether two results are the same, and a catalog to compare against. That limits it to fields where sameness can be decided exactly, as it can for the shape of a solid. It helps to have a fact the discoverer's own paper does not state, as the symmetry type was here, because a program that only repeats the paper cannot produce it.

For specialists: the details from the paper

The Josehedron is a plesiohedron (the Voronoi cell of a point relative to its orbit under a space group) with f-vector (12, 22, 12). The regression input was its generating orbit, 12 integer points mod 8 from Bernhard's Table 4, with no cell geometry, space-group label or f-vector. The 230 space groups were frozen into one table of 4,425 operations from spglib 2.7.0, gated by a checker sharing no code with the generator, and cross-checked operation by operation against an independent derivation of the symmetry operations recovered from Schmitt's survey software, on all 229 groups present in that operations data (a separate thing from the 145 groups his search covered and tabulated). Cells are computed from candidates within radius D, with an exact a posteriori check that 4ρ² ≤ D² (ρ the covering radius); type identity is the canonical planar code of the boundary complex (Brinkmann and McKay).

The rebuild matched every pre-specified anchor after one recorded amendment: 12 points per conventional cell, a determinant-256 body-centered cubic sublattice with 6 translation types, and the facet clause. A complete search recovered the honeycomb's 24 point operations modulo lattice, and the derived neighbor tables equal the earlier record's tables under a machine-verified bijection; the OEIS enumerator gives the same fixed and free counts on both through n = 6. The earlier record (the A398957 family) is temporally prior and shares no code, but shares the operator, method and model family, so its independence is at the artifact level only. The controls were the origin orbits of the simple cubic, FCC and BCC lattices (cube, rhombic dodecahedron, truncated octahedron).

The orbit is Wyckoff position 12a of IT(220) = I-43d. A point's site symmetry is the subgroup of operations that fix it, and a Wyckoff position is a class of points with conjugate site symmetry, labeled by multiplicity and a letter (12a: 12 points per conventional cell). Here the site symmetry is -4, of order 4, and the cell's automorphism order is exactly 4. A subgroup of IT(220) in which the 12 points form a single general-position orbit has index 4, the order of the site symmetry; exactly five exist, of the two types the reviewer named. In I2₁3 (IT(199)) the point keeps a 2-fold stabilizer, so that table is not one to diff. The 12 points are a general-position orbit of P2₁3 (IT(198), the proper operations of IT(220) without the body centering, generating point (3/8, 0, 1/4)) and of R3c (IT(161), four conjugate copies, one per body diagonal). By the paper's definition the Josehedron is therefore a Dirichlet stereohedron (a fundamental domain) of P2₁3 and of R3c, though not of IT(220). Schmitt's R3c table has no (12, 22, 12) row. His P2₁3 table has one, at (−455/6984, −655/6984, 545/3492), with frequency 801,918,314 of 1,000,677,997 grid points (80 percent), and that point returns the Josehedron's canonical code. His IT(220) row gives 6 triangles, 4 quadrilaterals and 2 pentagons with trivial automorphism group. Koch's 1973 study of cubic lattice complexes should cover 12a and is unchecked: it could not be obtained.

Before the paper went to reviewers, the project's catalog had flagged the Josehedron's f-vector as printed in the P2₁3 table without comparing that row; no rule then required it. The new rule (compare a special-position cell against the table of every group in which its orbit is in general position) has since run across the census in exact arithmetic, 52 runs: it reproduces the Josehedron as a control and returns 7 cells in the survey (control included), 43 not in the checked rows, 2 not applicable and 0 undetermined. A first cubic sweep, run without reference to the target, re-sighted the Josehedron's type 13 times in four configurations up to isometry; four of those 13 sightings were at Bernhard's own points or their inversion.

A second incident ran into the frozen record: a prompt I wrote cited target values from two unrelated honeycombs, and the agent checked them against the earlier record, refused them and reported the mismatch.

Limits: one domain and one target; the rebuild is given the orbit, so orbit search is untested by it; the code-writing agents could have known the target from training data or the earlier record; and one model family built and audited every component. The observed maximum facet count of a Dirichlet stereohedron is 38 and the proven upper bound is 92, so the survey is a sample. Code and census: github.com/Tyorden/plesiohedron-census↗.

@misc{orden2026reproducebeforeyouextend,
  author       = {Tyler Satchel Orden},
  title        = {Before my AI pipeline could hunt for new shapes, it had to rebuild a known one},
  year         = {2026},
  howpublished = {Cyborg Mathematics},
  url    = {https://cyborgmathematics.com/writing/reproduce-before-you-extend}
}