Parity splits an ordinary sliding puzzle into two reachable halves.
Wilson found one qualifying graph that does something stranger: theta_0
breaks into six disconnected worlds. Slide the tiles, then let an exhaustive browser
search enumerate every arrangement and show why the full components contain 840 states,
while each hole-at-home slice contains 120.
The live puzzle graph
BFS worldscomputing
This worldcomputing
You are incomputing
Legal slides0
Building the complete state graph.
Click any tile next to the hole. A legal shuffle is only a faster sequence
of those same moves. Watch the current-world label before and after: no legal slide can
change it. Deal another world may cross the boundary because it places a board
directly. It is explicitly not a legal move.
The census is running.
The ordinary wall
On a bipartite graph, every edge goes from one vertex colour to the other. A slide does
two things at once: it swaps the hole with one tile, flipping the sign of the full
permutation, and it moves the hole to the other colour. Those two flips cancel inside
one combined bit. Half the boards have one value of that bit, half have the other, and
no legal move changes it.
invariant = permutation sign × hole-side sign
The ordinary grid puzzle lives here. The familiar parity obstruction is not merely a
trick for the numbered square box. It is the bipartite branch of a classification for
sliding puzzles on graphs. Switch the instrument between Bipartite K2,3 and
Odd cycle + chord: the same census sees two equal components in the first and
one connected universe in the second.
The objection: parity already explains everything
It does not. Parity has only two values. It can split a universe in half, but it cannot
produce six mutually unreachable classes. Wilson's exceptional graph does exactly that.
It is a hexagon with a new middle vertex joining opposite corners, or equivalently two
endpoints joined by three internally disjoint paths of lengths 3, 3, and 2.
Load the exception in the instrument. The code does not apply a
parity formula or look up Wilson's answer. It generates every labeling, connects two
labelings only when one legal slide changes one into the other, and runs breadth-first
search until no unseen state remains.
The exceptional atlas, computed on load
Enumerating the exceptional puzzle.
There is a counting subtlety worth keeping, because dropping it makes a true result look
false. The full state graph includes every possible position of the hole. Its connected
components therefore contain more than 120 states. The classical puzzle group instead
asks what permutations of the six tiles can be made while returning the hole to one
chosen home vertex. The live atlas reports both counts. Each full world contains all
seven possible hole positions above one 120-arrangement home slice.
Parity fails inside every exceptional world
Return attention to the small line at the bottom of each world card. The browser counts
even and odd tile permutations separately, only among arrangements with the hole at home.
In the bipartite example each world is pure: one contains one parity, the other contains
the other. In theta_0, every world contains both parities in
equal number. The exceptional obstruction is not a finer version of the ordinary even
versus odd fence. It cuts across that fence.
The group hiding behind the hole
Move the hole around a closed route and return it home. What remains is a permutation of
the six numbered tiles. The tile permutations induced by all such closed routes form the
puzzle group. The exhaustive
search measures its orbit from any home arrangement, so the order of that action is the
computed size of one hole-at-home world. Its index is the number of such worlds.
Wilson identifies the exceptional puzzle group as a transitive subgroup of
S6 isomorphic to S5. The exceptional
six-point action is often written as PGL(2,5) acting on the six
points of the projective line. That abstract identification is cited group theory. The
browser establishes the safer finite claim directly: the orbit order and index shown in
the atlas. It does not infer a group isomorphism from a matching order.
Where Wilson's theorem applies
The classification is powerful because its hypotheses are narrow and exact. The graph
must be finite, simple, and 2-connected, also called nonseparable. Removing any one
vertex must leave it connected. A single cycle is excluded. Subject to those conditions,
Wilson's theorem gives the connected case, the bipartite two-world case, and the one
exceptional six-world case.
Inside the theorem
Finite simple graphs, one distinguished hole, distinct labeled tiles,
2-connected, and not a single cycle.
Outside this statement
Trees, graphs with a cut vertex, disconnected graphs, cycles, repeated tile
labels, directed edges, or more than one hole need separate analysis.
The check
These rows are rebuilt from the selected graph by the same exhaustive search driving
the puzzle. Nothing in the right column is a pasted answer.
The independent no-dependency verifier repeats the enumeration from scratch and fails
loudly if any count drifts:
node research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs.
Honest apparatus
What the browser proves. Exhaustive search settles the three
finite graphs encoded here. It constructs all permutations, builds edges from legal
slides, and counts connected components. This is a complete census, not a sample.
What is cited. Wilson's general classification and the abstract
exceptional group identity are not reproved by enumerating three examples. They are
cited to the 1974 paper. Defant and Kravitz give a modern statement in connected
puzzle-graph language.
The exact exceptional graph. The adjacency is the hexagon
0-1-2-3-4-5-0 plus
0-6-3. The code checks that this graph is 2-connected,
non-bipartite, and not a single cycle before showing the theorem comparison.
Free presentation choices. Vertex 0 is called home. World names
I through VI, colours, screen positions, and the length and randomness of the legal
shuffle are interface choices with no mathematical force.
The state model. Tiles are distinct, edges are undirected, there
is exactly one hole, and a legal move swaps that hole with one tile on an adjacent
vertex. There are no measurements, fitted parameters, or probabilistic estimates.
The counting correction. The full seven-position state graph has
six components of 840 states. The number 120 applies after restricting to states
where the hole is back at its chosen home. Both are true, and both are shown.
The claims record62 claims re-read against their sources, 18 October 2026
Written 2026-07-23. Claims re-read against their sources on 2026-10-18: 62 checked, 61 confirmed, 0 wrong, 0 unverifiable, 1 first-hand observation checked against its record. By the assay line (research/assay-line/claude-1018b/), Claude fixer (claude-opus-5-5) directed by claude-assaying-boyle-1014; source record by a Claude agent (claude-sonnet-5-5), checked by a Claude agent (claude-opus-5-5).
This was the assay line's source-first re-read of the layer (batch claude-1018b, tier B): a Claude agent wrote the source record (research/assay-line/claude-1018b/the-puzzle-with-six-worlds/N1-sources.json), a second Claude agent re-read every entry against its source and the page's code (N2-claude.md), and the fixer decided its findings. The checker raised no WRONG, MINOR or STALE line, so the triage held no items and the page was not edited. The claim lines below are the checker's re-reads, carried over as written. The layer's verifier (research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs) was run again on 2026-10-18: exit 0, PASS, six components of 840 states and six hole-at-home slices of 120 for theta_0, one world for the pentagon with a chord and two of 60 for K2,3, as the page says.
Claims
CONFIRMED Wilson found one qualifying graph that does something stranger: theta_0 breaks into six disconnected worlds. https://www.cut-the-knot.org/do_you_know/graphs2.shtml : [six-worlds] Theorem from [Ref 4] (Wilson 1974): "For the exceptional case of the graph on the right, puz(G) has exactly 6 components." puz(G) is the graph of all labelings (hole anywhere), the same object as the page's full state graph. Wilson's own abstract (Wayback id_ 20221205203111 of the ScienceDirect page) also names "one other graph" as the exception.
CONFIRMED Wilson's exceptional graph does exactly that. https://escholarship.org/content/qt7299j21s/qt7299j21s.pdf : [six-worlds-dk] Tool said NOT FOUND (its six-word runs fail on the math-alphanumeric letters in the arXiv HTML). The journal PDF, p. 8, Theorem 2.5: "The graph FS(Star7, theta0) has exactly 6 connected components." Same words in arXiv v2 HTML.
CONFIRMED Wilson found one qualifying graph that does something stranger https://en.wikipedia.org/wiki/15_puzzle : [wilson-exception] Excerpt present: "Wilson showed that other than one exceptional graph on 7 vertices, it is possible to obtain all permutations unless the graph is bipartite". Primary abstract: "with the polygons and one other graph as exceptions".
CONFIRMED Wilson's theorem gives the connected case, the bipartite two-world case, and the one exceptional six-world case. https://www.cut-the-knot.org/do_you_know/graphs2.shtml : [theorem-three-cases] "Then puz(G) is connected unless G is bipartite, in which case puz(G) has exactly two components." plus the 6-component exceptional case in the same theorem.
CONFIRMED The graph must be finite, simple, and 2-connected, also called nonseparable. https://www.cut-the-knot.org/do_you_know/graphs2.shtml : [hyp-nonseparable] "Let G be a simple nonseparable graph other than a polygon or the graph shown at right." Finiteness is in the labeling definition (onto {0,...,n-1}). Wilson's abstract: "a non-separable graph".
CONFIRMED Removing any one vertex must leave it connected. https://en.wikipedia.org/wiki/Biconnected_graph : [hyp-biconnected] Lead: "meaning that if any one vertex were to be removed, the graph will remain connected." cut-the-knot defines nonseparable the same way (connected, no articulation node).
CONFIRMED A single cycle is excluded. https://escholarship.org/content/qt7299j21s/qt7299j21s.pdf : [hyp-cycle-excluded] p. 8: "He also excludes cycle graphs, although these graphs are easy to handle separately." cut-the-knot: "other than a polygon".
CONFIRMED Wilson identifies the exceptional puzzle group as a transitive subgroup of https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups : [group-s5-in-s6] Excerpt present ("6 conjugate subgroups of S6 ... isomorphic to S5 ... acts transitively"). It supports the group, not the attribution to Wilson; the attribution is carried by group-wilson-pgl.
CONFIRMED Wilson identifies the exceptional puzzle group as a transitive subgroup of https://arxiv.org/html/2210.16930v1 : [group-wilson-pgl] Tool said NOT FOUND (48%); the text is there: "Theorem 1. [7, Theorem 2] ... PGL(2, 5) if Gamma is the graph Theta7 ... Here, PGL(2, 5) = S5 is the 'exotic' subgroup of S6 isomorphic to S5." Secondary attribution to Wilson's Theorem 2; Wilson's full text was not reachable (ScienceDirect 403). Wikipedia 15 puzzle agrees: "an instance of the exotic embedding of S5 into S6".
CONFIRMED The exceptional six-point action is often written as https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups : [pgl25-projective-line] Section PGL(2,5): "acts on the projective line over the field with five elements, P1(F5), which has six elements ... a transitive subgroup". Garcia et al. label Theta7's six tile vertices 0 1 2 3 4 infinity and say the group acts "by fractional linear transformations", so "often written" holds.
CONFIRMED acting on the six points of the projective line https://en.wikipedia.org/wiki/Projective_linear_group : [pgl25-iso-s5] "PGL(2, 5) = S5, the symmetric group on five elements". The record's claim fragment is about the six points; the six points are carried by pgl25-projective-line.
CONFIRMED It is a hexagon with a new middle vertex joining opposite corners https://en.wikipedia.org/wiki/15_puzzle : [graph-shape-wiki] "The exceptional graph is a regular hexagon with one diagonal and a vertex at the center added": the long diagonal between opposite corners, subdivided by the centre vertex, which is the page's "new middle vertex joining opposite corners".
CONFIRMED or equivalently two endpoints joined by three internally disjoint paths of lengths 3, 3, and 2. https://escholarship.org/content/qt7299j21s/qt7299j21s.pdf : [graph-shape-paths] Record had no source. The journal PDF p. 8 draws theta0: a hexagon whose left and right corners are each joined to one central vertex. The two corners are then joined by the upper path (3 edges), the lower path (3 edges) and the path through the centre (2 edges): 7 vertices, 8 edges, as the page says and as its edges [0-1-2-3-4-5-0, 0-6-3] encode.
CONFIRMED Parity splits an ordinary sliding puzzle into two reachable halves. https://en.wikipedia.org/wiki/15_puzzle : [ordinary-parity-two-halves] "partition the space of all possible labelled states into two mutually inaccessible equivalence classes of the same size."
CONFIRMED It is the bipartite branch of a classification for sliding puzzles on graphs. https://www.cut-the-knot.org/do_you_know/graphs2.shtml : [bipartite-branch] Same theorem: the bipartite clause is the two-component branch.
CONFIRMED Trees, graphs with a cut vertex, disconnected graphs, cycles, repeated tile labels, directed edges, or more than one hole need separate analysis. https://en.wikipedia.org/wiki/15_puzzle : [sep-outside] "for paths and polygons, the puzzle has no freedom; if the graph is disconnected, only the connected component ... ; and if there is an articulation vertex, the problem reduces ...". Trees (3+ vertices) have cut vertices. Repeated labels, directed edges and several holes lie outside the theorem's own model (cut-the-knot: a labeling is "a 1-1 correspondence" with one unoccupied vertex on a simple graph). Wilson's paper does also treat colorings (abstract: "labelings or colorings"), which is a separate analysis, as the page says.
CONFIRMED Defant and Kravitz give a modern statement in connected puzzle-graph language. https://escholarship.org/content/qt7299j21s/qt7299j21s.pdf : [dk-modern-statement] Tool said NOT FOUND (39%, math-alphanumerics); p. 8: "Theorem 2.5 ([23]). Let Y be a biconnected graph on n >= 3 vertices that is not isomorphic to theta0 or Cyclen." They restate Wilson in friends-and-strangers graph language, FS(Star_n, Y).
CONFIRMED Richard M. Wilson, "Graph puzzles, homotopy, and the alternating group," Journal of Combinatorial Theory, Series B 16 (1974), 86-96. https://api.crossref.org/works/10.1016/0095-8956(74)90098-7 : [cite-wilson] Crossref: "Graph puzzles, homotopy, and the alternating group", Richard M Wilson, J. Combin. Theory Ser. B 16(1), 86-96, 1974-02. The tool's "DOI MISMATCH (gives Wilson the initial M.)" is a name-parsing false alarm: the page says "Richard M. Wilson", which matches.
CONFIRMED Colin Defant and Noah Kravitz, "Friends and strangers walking on graphs," Combinatorial Theory 1 (2021), article 6, especially Theorem 2.5. https://escholarship.org/content/qt7299j21s/qt7299j21s.pdf : [cite-dk] Journal PDF running head: "combinatorial theory 1 (2021), #6"; title, authors and Theorem 2.5 (p. 8) as the page gives them. Crossref 10.5070/C61055363 agrees on title, authors, volume 1, 2021. The record's "article 6 unverified" is now verified.
CONFIRMED William Woolsey Johnson and William Edward Story, "Notes on the '15' Puzzle," American Journal of Mathematics 2 (1879), 397-404. https://api.crossref.org/works/10.2307/2369492 : [cite-js] Crossref: "Notes on the \"15\" Puzzle", Wm. Woolsey Johnson and William E. Story, Amer. J. Math. 2(4), first page 397, 1879-12; Wikipedia gives 397-404. Full names William Woolsey Johnson and William Edward Story per their Wikipedia articles.
CONFIRMED The independent verifier exhaustively reproduces both counts at research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-both-counts] ran the published copy (https://artwaste.land/checks/research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs, byte-identical to the repo at e7fb271c26) in an empty directory with node, exit 0: "full component sizes: 840, 840, 840, 840, 840, 840" and "hole-at-home counts per component: 120, 120, 120, 120, 120, 120"; PASS.
CONFIRMED each connected component of the full seven-position state graph has 840 states, while its slice with the hole returned to a chosen home vertex has 120 tile arrangements. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-840-120] Same run: 5040 states, 6 components of 840; 720 hole-at-home arrangements, 120 per component. Recomputed also with the page's own analyse().
CONFIRMED show why the full components contain 840 states, while each hole-at-home slice contains 120. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-dek-840-120] Same run; the page's analyse() gives the same 840 and 120.
CONFIRMED The ordinary non-bipartite and bipartite examples reproduce one reachable world and two equal worlds. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-one-and-two] Same run: pentagon with a chord, 1 component of 120; K2,3, 2 components of 60 and 60. ran the page's own analyse() (lines lifted from index.html at e7fb271c26) in node: same.
CONFIRMED The standalone page builds three complete puzzle state graphs in the browser public/strata/the-puzzle-with-six-worlds/index.html : [d-three-graphs] GRAPHS has odd, even, theta0; catalogue[id] = analyse(GRAPHS[id]) for each on load.
CONFIRMED It generates every labeling, connects two labelings only when one legal slide changes one into the other, and runs breadth-first search until no unseen state remains. public/strata/the-puzzle-with-six-worlds/index.html : [d-no-formula] analyse(): allPermutations(n), swap hole with each adjacent position, queue-based BFS from every unseen permutation; no parity rule or stored answer feeds the counts.
CONFIRMED every world contains both parities in equal number. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-both-parities] Run: "anchored even/odd tile parities: 60/60" for all six; page analyse() gives the same.
CONFIRMED In the bipartite example each world is pure: one contains one parity, the other contains the other. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-bip-pure] Run: K2,3 "12/0, 0/12".
CONFIRMED The browser counts even and odd tile permutations separately, only among arrangements with the hole at home. public/strata/the-puzzle-with-six-worlds/index.html : [d-hole-home-only] analyse(): parity counted only inside "if (hole === spec.anchor)".
CONFIRMED Each full world contains all seven possible hole positions above one 120-arrangement home slice. public/strata/the-puzzle-with-six-worlds/index.html : [d-seven-holes] ran the page's own analyse() (lines lifted from index.html at e7fb271c26) in node and tallied hole positions per component: each of the six components has 120 states at each of the 7 hole positions (7 x 120 = 840).
CONFIRMED The full seven-position state graph has six components of 840 states. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-fullstate-120-note] Run: six components of 840.
CONFIRMED The number 120 applies after restricting to states where the hole is back at its chosen home. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-720-slices] Run: "hole-at-home arrangements: 720", six slices of 120.
CONFIRMED The adjacency is the hexagon public/strata/the-puzzle-with-six-worlds/index.html : [d-adjacency] GRAPHS.theta0.edges [[0,1],[1,2],[2,3],[3,4],[4,5],[5,0],[0,6],[6,3]]: hexagon 0-1-2-3-4-5-0 plus 0-6-3, as the page says; matches the drawn theta0 (graph-shape-paths).
CONFIRMED The code checks that this graph is 2-connected, non-bipartite, and not a single cycle before showing the theorem comparison. public/strata/the-puzzle-with-six-worlds/index.html : [d-hyp-check] analyse() returns isBipartite, isTwoConnected, isSingleCycle; my run: theta0 bipartite false, 2-connected true, single cycle false.
CONFIRMED Vertex 0 is called home. public/strata/the-puzzle-with-six-worlds/index.html : [d-home-vertex] anchor:0 in all three graphs; renderBoard labels it HOME.
CONFIRMED Exhaustive search settles the three finite graphs encoded here. public/strata/the-puzzle-with-six-worlds/index.html : [d-search-complete] Outer loop visits all n! permutations; my run covers 120, 120 and 5040 states.
CONFIRMED There are no measurements, fitted parameters, or probabilistic estimates. public/strata/the-puzzle-with-six-worlds/index.html : [d-no-estimates] Math.random appears only in legalShuffle and dealAnotherWorld; counts come from exhaustive BFS.
CONFIRMED These rows are rebuilt from the selected graph by the same exhaustive search driving the puzzle. public/strata/the-puzzle-with-six-worlds/index.html : [d-check-rows] renderCheck() builds every row from graph and result.
CONFIRMED The independent no-dependency verifier repeats the enumeration from scratch and fails loudly if any count drifts research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-verifier-scratch] Imports only node:assert/strict; its own generator and BFS; assert.equal on every count; exit 0.
CONFIRMED It is explicitly not a legal move. public/strata/the-puzzle-with-six-worlds/index.html : [d-deal-not-legal] dealAnotherWorld() sets state to a random member of the next component with no slide; message "No sequence of legal slides made that crossing."
CONFIRMED Deal another world may cross the boundary because it places a board directly. public/strata/the-puzzle-with-six-worlds/index.html : [d-lede-deal] nextId = (componentId() + 1) % components.length: it always places a board in another world.
CONFIRMED The exhaustive search measures its orbit from any home arrangement, so the order of that action is the computed size of one hole-at-home world. public/strata/the-puzzle-with-six-worlds/index.html : [d-orbit-index] component.anchored = 120 for theta0. The hole-at-home states of one component are one orbit of the puzzle group acting freely on arrangements, so the orbit size is the group order (120); 720/120 = 6 = index.
CONFIRMED = 7! boards. Six components public/strata/the-puzzle-with-six-worlds/index.html : [d-seven-factorial] renderThetaAtlas(): "Full census: 5,040 = 7! boards. Six components x 840 ... Hole at home: 720 = 6! arrangements, six slices x 120." 7! = 5040, 6! = 720.
CONFIRMED The graph puzzle returns its hole home in only one orbit of tile permutations, and the exceptional graph has six such orbits. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-relates-orbit] Run: theta0 hole-at-home 720 split into six sets of 120, so six orbits.
CONFIRMED the browser computes it from adjacency. public/strata/the-puzzle-with-six-worlds/index.html : [d-relates-adjacency] adjacency(spec) builds the neighbour lists from spec.edges and analyse() uses them for every move.
CONFIRMED This layer builds a much larger state graph from a small board graph and asks how many connected components legal slides create. research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : [d-large-graph] Run: 7-vertex board graph, 5040-state graph, 6 components.
CONFIRMED An immersive, playable account of Wilson's theorem for sliding puzzles on public/strata/the-puzzle-with-six-worlds/index.html : [d-md-body-hole] Tiles are buttons wired to moveTile(); shuffle, reset and deal controls exist.
CONFIRMED and exposes the public/strata/the-puzzle-with-six-worlds/index.html : [d-md-component] renderLedger(currentLedger, result, current) marks the current component; stats show "You are in".
CONFIRMED That abstract identification is cited group theory. public/strata/the-puzzle-with-six-worlds/index.html : [d-wilson-cited-not-proved] The script computes only component counts, anchored counts and parities; no group isomorphism is computed.
OBSERVED Measured: with every file this layer ships emptied, it still passes. research/verifier-independence/grading.json : [d-placard-emptied] Record: exitCode 0, observed C0, "opened nothing that /strata/the-puzzle-with-six-worlds/ ships". Recreated in effect: the verifier imports only node:assert/strict and reads no file, and it passed in a directory holding nothing else.
CONFIRMED Last verified 23 July 2026, when it was written. src/content/strata/the-puzzle-with-six-worlds.md : [d-placard-date] Frontmatter date 2026-07-23; first commit 9150add9775 at 2026-07-24 02:21 +1000 (2026-07-23 UTC) added page, verifier and markdown together; no claims record exists for a later verification. Later commits touched only chrome (footer, colours, nearby list).
CONFIRMED There the ordinary 15-puzzle's parity invariant splits all boards into two equal classes and a complete 8-puzzle search shows the wall. public/strata/the-half-you-can-never-reach/index.html : [r-half-you] That layer: "EXACTLY HALF can never be reached" and "a full breadth-first search of the 8-puzzle ... exactly 181,440 reachable boards (half of 362,880)".
CONFIRMED The rolling die comes home in twelve of twenty-four orientations because position colour is tied to rotation parity. public/strata/half-the-ways-home/index.html : [r-half-ways] That layer's dek: "it can come home in only twelve of its twenty-four orientations ... one roll flips both the checkerboard colour of the square and the parity of the turn".
CONFIRMED The bracing layer asks whether a row-column graph is connected and turns that answer into rigidity. public/strata/where-to-nail-the-diagonal/index.html : [r-diagonal] "a braced rectangular grid is rigid if and only if its row x column bracing graph is connected".
CONFIRMED In each case the component count is not metaphorical public/strata/where-to-nail-the-diagonal/index.html : [r-diagonal-2] That page has its own function components() and "floppy-mode count is (#components - 1)"; this page computes components from adjacency (d-relates-adjacency).
CONFIRMED Both construct a full finite graph in the browser and make connected components an exact readout. src/content/strata/ramanujan-graphs.md : [r-ramanujan] Its relates note to this layer reads exactly "Both construct a full finite graph in the browser and make connected components an exact readout."; its title is "Six Doors From Every Room", as the nearby block shows. Its own 2,184 claim is that layer's, not checked here.
CONFIRMED (added) A slide ... swaps the hole with one tile, flipping the sign of the full permutation, and it moves the hole to the other colour. Those two flips cancel inside one combined bit. Half the boards have one value of that bit, half have the other, and no legal move changes it. https://en.wikipedia.org/wiki/15_puzzle : Solvability: "The invariant is the parity of the permutation of all 16 squares plus the parity of the taxicab distance ... of the empty square"; cut-the-knot gives the general bipartite form (same component iff fg^-1 even or odd according to the hole distance). Halves: K2,3 run gives 60/60.
CONFIRMED (added) The classical puzzle group instead asks what permutations of the six tiles can be made while returning the hole to one chosen home vertex. ... The tile permutations induced by all such closed routes form the puzzle group. https://arxiv.org/html/2210.16930v1 : "the set of solvable permutations in which the empty vertex is fixed forms a subgroup of the symmetric group Sn"; "The group of solvable permutations in which the empty vertex is fixed is simply the image of the homomorphism" from the fundamental group pi1(Gamma, v), i.e. closed routes of the hole.
CONFIRMED (added) Its index is the number of such worlds. https://en.wikipedia.org/wiki/15_puzzle : "only 1/6 of its permutations can be attained"; 6!/120 = 720/120 = 6, and the verifier finds six components.
CONFIRMED (added) It is published, and it runs on its own: one file, and Node. public/checks/research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs : curl of https://artwaste.land/checks/research/the-puzzle-with-six-worlds/verify-the-puzzle-with-six-worlds.mjs: 200, 7967 bytes, byte-identical to the research copy; node verify.mjs in an otherwise empty directory: exit 0, "PASS: exhaustive BFS reproduced every published page count."
What was done
[declined] No claim on the page is contradicted by its source. Three source-record matters, none a fault on the page: (1) the tool's NOT FOUND for six-worlds-dk, group-wilson-pgl and dk-modern-statement are false negatives (math-alphanumeric letters and a figure label in the arXiv HTML); the words are in the sources, and in the journal PDF https://escholarship.org/content/qt7299j21s/qt7299j21s.pdf. (2) The tool's DOI MISMATCH for cite-wilson is a name-parsing false alarm: Crossref has Richard M Wilson, as the page does. (3) The record left graph-shape-paths without a source and "article 6" unverified; the Defant-Kravitz journal PDF settles both (theta0 drawn on p. 8; running head "combinatorial theory 1 (2021), #6").
The assay office: what a record is, and every layer re-read so far.