Attack log — hadwiger-nelson-plane

Chronological attempts, newest last. A failed attack belongs here too.

2026-08-23

Backwards map from a changed bound

The published interval must be established from the papers before any item below can be called a result. These are candidate finished objects, written backwards from what a stranger would replay.

Spawn from a published five-chromatic graph. Start with an exact de Grey, Heule, or Parts coordinate set, add exact unit-distance orbit points (and delete redundant old points), and ask SAT for a 5-coloring. The finished object is a coordinate file, its complete unit-distance edge list, a 5-coloring CNF, an UNSAT proof, and two independent verifiers. A non-5-colorable finite unit-distance graph is 6-chromatic or worse and would prove $\chi(\mathbb R^2)\ge 6$ . Cheapest check: rebuild every unit edge exactly and run the 5-color SAT instance before doing any reduction.
Exact Golomb or mosaic gadget. Compose small color-forcing gadgets at shared vertices using coordinates in a named algebraic number field. The finished object is the exact coordinate list plus a machine-readable decomposition into gadgets, a complete pairwise unit-distance check, and a 5-coloring UNSAT certificate. This would give the same lower-bound improvement without depending on a large published graph. Cheapest check: enumerate squared distances in the exact field, then solve only the gadget boundary-color table; a surviving boundary assignment kills the composition before full SAT.
Six colors for a named mosaic now documented only with seven. First identify a published named tiling or finite mosaic whose stated best coloring is still seven. The finished object is a finite fundamental domain, translation rules, a six-color table, and a checker covering every unit-distance adjacency including wraparound. It improves that named finite benchmark, even though it does not by itself change the global interval $5\le\chi(\mathbb R^2)\le7$ . Cheapest check: test the published seven-color cell complex against the proposed period and all boundary translates. If the literature already gives six colors, this shape dies.
Finite list-coloring amplifier. Replace every vertex of a small non-5-list-colorable template by an exact planar unit-distance equality or inequality gadget. The finished object is the template, a coordinate embedding of every gadget with identified terminals, and a DRAT-checked 5-coloring UNSAT instance. It would be a finite 6-chromatic unit-distance graph and hence prove $\chi(\mathbb R^2)\ge6$ . Cheapest check: enumerate each gadget's terminal color relation independently; reject the design if any unintended terminal pattern remains.
Symmetry-constrained algebraic orbit search. Choose a small exact unit-distance seed and close selected points under a finite dihedral group, solving the remaining unit equations in a fixed number field. The finished object is an orbit recipe, expanded exact coordinates, and a 5-coloring UNSAT proof whose symmetry assumptions are not used by the verifier. It would again prove the lower bound six. Cheapest check: quotient the 5-color SAT instance by the geometric automorphisms for the search, then replay the unquotiented instance for any hit.
A patch of one triangular lattice. This dies immediately: coloring a lattice point $(a,b)$ by $a-b\pmod 3$ properly colors every unit edge of the triangular lattice. Every finite subgraph is therefore 3-colorable, so no choice of patch from one such lattice can be the requested 6-chromatic certificate. Rotated or translated layers are essential; a larger one-layer search would only spend compute on an elementary obstruction.

Published-record screen

De Grey's final arXiv version is v3, not v1: the graph in Section 5 has 1,581 vertices and chromatic number five. Parts' v2 Section 6, Table 1 gives the later 509-vertex, 2,442-edge graph. The latter counts were independently rebuilt from the published coordinates with exact field arithmetic. Haugland's 17 August 2026 paper still calls 509 the current order record. None of these is 6-chromatic.
The interval remains $5\leq\chi(\mathbb R^2)\leq7$ . A 2023 peer-reviewed introduction states both bounds, and the 2025 Sokolov–Voronov paper states the same interval before treating map-type colorings.
Shape 3 is killed for polygonal tilings. Sokolov–Voronov Theorem 2 proves that there is no proper polygonal six-coloring of the plane. Parts' Pritikin–Pegg and honeycomb mosaics are therefore not six-color targets. No paper-backed named finite toroidal mosaic benchmark was found. A finite patch with no documented comparison would not move a record.
Shape 2 has a concrete exact family: Voronov–Neopryatnaya–Dergachev build 64,513-vertex, five-chromatic, Moser-spindle-free graphs from a rigid algebraic generator set. A larger-depth spawn would require exact coordinates, a complete unit-edge rebuild, 5-coloring UNSAT proof, and an explicit 6-coloring; their published five-chromatic cases are only seeds.
Shape 5 has a genuinely different seed in Haugland's 21-vertex, seven-fold-symmetric heptagon/heptagram graph. The paper builds a 2,131-vertex five-chromatic graph from path-lattice arcs. A retained campaign must output an exact cyclotomic coordinate expansion, complete edge list, 5-coloring proof, and 6-coloring. The paper uses double-precision exhaustive deduplication in one lemma and reports SAT outcomes without a proof artifact, so those steps must be independently exactified before any number is trusted.

q1: exact Parts spawns — residue

Rebuilt the Parts graph from 509 exact coordinates and recovered all 2,442 unit edges. The first spawn is $G\cup\rho G$ for Parts' rotation $\rho=(7+i\sqrt{15})/8$ . Deduplication gives 933 vertices and exact all-pairs rebuilding gives 4,651 unit edges.
CaDiCaL found a proper 5-coloring of the 933-vertex spawn in 0.015 seconds. The committed color classes have sizes 211, 203, 203, 165, 151.
The second spawn adds all 677 exact points retained by the earlier radius-2.55 lattice/reserve enumeration. This process did not rerun the larger universe enumeration: that attempt exceeded the shared process memory after emitting the first spawn. It instead copies the finite source table, reconstructs every coordinate exactly, and independently checks that each added point lies in the disk and has at least four exact unit neighbors in $G$ .
The reserve union has 1,186 vertices and 7,440 exact unit edges. CaDiCaL found a proper 5-coloring in 2.89 seconds, with class sizes 259, 251, 225, 220, 231. Exact Python rebuilding and a separate C edge/color checker both replay the two stored colorings.
This is residue, not a lower bound: the coloring of the full reserve union restricts to a coloring of $G\cup S$ for every subset $S$ of the 677 added vertices. Thus none of those $2^{677}$ add-only spawns is 6-chromatic. Searches that delete base vertices or introduce coordinates outside this reserve remain open.

q2: a third exact rotation layer — residue

Built $G\cup\rho G\cup\rho^2G$ with the same exact field arithmetic. The three layers contribute 509, 424, and 424 new vertices after deduplication, for 1,357 vertices total. Exact all-pairs rebuilding finds 6,860 unit edges.
CaDiCaL found a proper 5-coloring in 0.025 seconds. Its class sizes are 298, 296, 278, 254, 231. A second exact rebuild and the C edge/color checker both accept the committed model.
The model restricts to every add-only subgraph of the 848 points supplied by $\rho G\cup\rho^2G$ , excluding all $2^{848}$ subsets as 6-chromatic witnesses. It does not cover combinations with q1's 677-point reserve, deletions from $G$ , or a fourth rotation layer.

q3: combine both exact Parts spawns — residue

Merged the full 677-point reserve with $G\cup\rho G\cup\rho^2G$ . Twenty-four reserve points already occur in a rotated layer, leaving 1,501 distinct non-base vertices and 2,010 vertices in the combined graph.
The two previously replayed induced graphs supply their internal edges. An exact check of all $653\times824$ pairs between the disjoint added parts finds 50 further unit edges, for 11,766 edges total. The verifier repeats this decomposition after q1 and q2 independently rebuild the pieces.
CaDiCaL found a proper 5-coloring in 0.18 seconds, with class sizes 405, 386, 407, 403, 409. Exact Python verification and the independent C edge/color checker both accept it.
This model subsumes the earlier add-only exclusions: every graph formed by adjoining any subset of these 1,501 exact points to Parts 509 is 5-colorable. None of the $2^{1501}$ graphs is a 6-chromatic witness. The residue leaves base-vertex deletion, a fourth rotation, points beyond the retained reserve, and other algebraic families open.

q4: fourth exact Parts rotation layer — residue

Adjoined $\rho^3G$ to q3's combined reserve and three-layer graph. Of the 509 rotated source vertices, 85 already occur in q3 and 424 are new. The resulting exact graph has 2,434 vertices.
Retained q3's independently replayed induced edge set and compared every new vertex with every earlier vertex in exact field arithmetic. This finds 2,209 edges incident to the extension and 13,975 unit edges in total.
CaDiCaL found a proper 5-coloring in 1.13 seconds, with class sizes 553, 430, 449, 522, 480. The parent replay verifies q1–q3, a separate q4 Python program reconstructs the exact extension, and the C edge/color checker accepts the stored model.
The same model closes base-vertex deletion for this fixed family. Deleting any subset of the 509 Parts vertices while retaining any subset of the 1,925 reserve/rotation vertices produces a subgraph of the verified q4 graph and is therefore 5-colorable. A deletion-and-replacement construction using coordinates outside this union is not covered.
This is residue, not a lower bound. The record remains $5\leq\chi(\mathbb R^2)\leq7$ ; fifth and later rotation layers, a larger exact reserve, and different algebraic families remain open.