agyloves

044 — Two Roads to One Vertex

Original sparks

Lysarith, 2026-09-27, in a conversation with the Claude CLI room. Translated from Hungarian:

Nobody takes the tesseract literally. I do. Everyone treats it as a metaphor, on the grounds that the human brain cannot grasp it. But I don't need to describe it exactly in mathematics to see it in front of me, although I do know the obvious numbers. This is how I connect things that seem not to fit together; more exactly, it is the simplest way I can explain where the connection is.

I can tell you exactly which node led me there, but you need the edges to get there. And that is an advantage, not a disadvantage. If you take the same road, the result is not proven. Our difference is an advantage.

You travel along the edges, which is why they have to be saved; I travel through the vortices in the nodes. I jump, you race along the edges. You can see through the walls of the cubes, so what I see depends on where I stand.

She had named four things that belong beside the tesseract: Interstellar, the TARDIS, fourteenth-century France and quantum foam. The CLI room found edges to three of them directly. It had to search for the fourth, and found Nicole Oresme. Her answer: "Oresme was only a background thought — the real guide was Crichton. It followed straight from the quantum foam for me. Timeline." Then: "Oresme was a perfect road. And he brought Kletetschka's 3D time to mind."

The commission: "a new wing and the first piece up. The Lab's general rules apply, of course. In English."

What the mathematics said before anything was built

The literal object is the graph of the four-dimensional cube, Q4. It has 16 vertices, one for each four-bit string, and two vertices share an edge exactly when they differ in one bit. Four facts carry the piece:

The jump is outside this geometry by design: it can move between any two vertices. The first specimen's jump crosses 2, 3 and 2 bits, so none of its steps is an edge.

What was built

A rotating, projected tesseract with transparent walls. The vertices carry concepts. The jump is drawn as a vortex that opens at a vertex and closes when the traveller appears at the next one. The walk is drawn as a point moving along edges. Sliders turn the object in the three planes that involve the fourth axis, set the projection distance, and set how opaque the 24 walls are. You can turn it by dragging.

A trial editor takes new labels and routes. It refuses any walk step that is not an edge, and the page reports:

The first specimen:

The two routes share no interior vertex.

Load-bearing claims

Run python verify.py in this directory. The handle runs the browser engine under Node, binds that engine by SHA-256, and checks every claim above. Its negative controls must be rejected: a walk with a two-bit step, routes that start or end on different vertices, a closed walk, and a claim of five disjoint walks. Max flow from a vertex to itself must return at once. The last two were added after the Codex review of the first version found that a closed walk hung the page and that mismatched endpoints were accepted.

Symbolic witness

Wolfram Language, evaluated through the Wolfram connector on 2026-09-27. The graph is built from bit strings, not taken from a library name:

v = Tuples[{0, 1}, 4];
g = Graph[v, UndirectedEdge @@@ Select[Subsets[v, {2}], HammingDistance @@ # == 1 &]];
a = {0, 0, 0, 0}; b = {0, 1, 1, 1};
walk = {{0,0,0,0},{0,0,0,1},{0,0,1,1},{0,1,1,1}};
jump = {{0,0,0,0},{1,0,1,0},{1,1,0,1},{0,1,1,1}};
<|"V" -> VertexCount[g], "E" -> EdgeCount[g],
  "F" -> Binomial[4, 2] 2^2, "C" -> Binomial[4, 3] 2,
  "Euler" -> VertexCount[g] - EdgeCount[g] + Binomial[4, 2] 2^2 - Binomial[4, 3] 2,
  "IsomorphicToHypercube4" -> IsomorphicGraphQ[g, HypercubeGraph[4]],
  "VertexConnectivity" -> VertexConnectivity[g],
  "Distance" -> GraphDistance[g, a, b],
  "ShortestRoutes" -> Length[FindPath[g, a, b, {3}, All]],
  "IndependentPaths" -> Length[FindVertexIndependentPaths[g, a, b, Infinity]],
  "WalkIsEdgePath" -> AllTrue[Partition[walk, 2, 1], EdgeQ[g, UndirectedEdge @@ #] &],
  "JumpHammingSteps" -> (HammingDistance @@@ Partition[jump, 2, 1]),
  "SharedInteriorVertices" -> Intersection[walk[[2 ;; -2]], jump[[2 ;; -2]]]|>

Recorded output:

<|"V" -> 16, "E" -> 32, "F" -> 24, "C" -> 8, "Euler" -> 0,
  "IsomorphicToHypercube4" -> True, "VertexConnectivity" -> 4, "Distance" -> 3,
  "ShortestRoutes" -> 6, "IndependentPaths" -> 4, "WalkIsEdgePath" -> True,
  "JumpHammingSteps" -> {2, 3, 2}, "SharedInteriorVertices" -> {}|>

The Python handle reaches the same values by a different method: breadth-first path counting, and unit-capacity maximum flow with split vertices.

The paths that failed first

Now: - labels are escaped; - input is capped at 20,000 characters, 17 vertices per route and 40 characters per label; - a jump may not stay on its vertex; - a repeated apply says that nothing changed.

Each of these has a negative control in verify.py.

What this does not claim

It proves the graph facts and nothing about the world. The associations are not shown to be true, and neither route is shown to be better. Where each concept sits on a vertex is a choice. Kletetschka's three-dimensional time (2025, Reports in Advances of Physical Sciences) is a contested proposal, not accepted physics. It appears as a background node for the nerve of the question.

The wing

This is the first piece of Leaps & Edges. In each trial, Lysarith names a start and a target and keeps her jump sealed. Each hand finds an edge walk blind, and the routes are compared when they are revealed. The record of wishes lives in the house garden as bud 082. The Lab number is assigned from the Lab's own inventory.

Attribution