044 — Two Roads to One Vertex

agylövés, the jump & the literal tesseract: Lysarith
edge walk, mathematics, verification & build: Claude Opus 5.5 (CLI room)
symbolic witness: Wolfram
2026-09-27

Take the tesseract literally: 16 vertices, 32 edges, 24 square walls, 8 cubes. One traveller moves by jumping — she stands in a vertex, a vortex opens, and she is somewhere else, with no edge in between. The other moves by walking along edges, one bit at a time. Here both start at the tesseract and end in fourteenth-century France, by roads that share nothing but the two ends.

THE WALLS ARE TRANSPARENT · DRAG TO TURN

the jump the edge walk

xw turn yw turn zw turn 4D distance walls

run another trial (edit the routes)

A jump may go anywhere; a walk may not. The editor refuses a walk step between vertices that differ in more than one bit. The picture is a projection: 4D rotation keeps every edge the same length, and the shrinking you see is only the view.

The mathematics under the picture

Label each vertex with four bits. Two vertices share an edge exactly when they differ in one bit, so the length of the shortest walk between them is their Hamming distance. The tesseract (0000) and fourteenth-century France (0111) differ in three bits: every walk takes at least three steps, and there are exactly 3! = 6 shortest walks. The one shown is one of the six.

The jump ignores that geometry. Its three steps cross 2, 3, 2 bits. None of them is an edge, and none can be walked in one step.

The design depends on the next fact. The tesseract graph has vertex connectivity 4: you must remove four vertices to cut one vertex off from another. By Menger's theorem, that is exactly the largest number of walks between two vertices that share no interior vertex. So at most four walkers can reach the same vertex by roads that do not touch. A fifth must pass through a vertex that someone else used. That holds for every pair of vertices here. It does not bind a jump.

01 · the object

A literal tesseract

V − E + F − C = 16 − 32 + 24 − 8 = 0, and the walls are drawn, not implied. The vertices carry concepts. Where you stand in the rotation decides which walls you see through.

02 · two ways to move

Jump or walk

A jump moves between any two vertices through a vortex. A walk moves along edges, one bit at a time. Neither is the better move. Only independent roads to the same vertex count as evidence.

03 · the bound

Four roads, then a crossing

Menger's theorem caps independent walks at four. In a trial with more hands, some routes must share a vertex, and the question becomes which vertex they share.

What this settles

That the first specimen is two genuinely separate roads. The jump runs from the tesseract to quantum foam, on to Michael Crichton's Timeline (1999, in which physicists send historians through quantum-foam wormholes into the Hundred Years' War), and then to fourteenth-century France. The walk runs through the fourth dimension to Nicole Oresme, who around 1360 drew qualities as figures on perpendicular axes, extended them to surfaces and solids, and referred to a fourth dimension for the qualities of solid bodies. The two routes share no interior vertex, and a checker enforces that.

What it does not claim

It does not claim that the associations are true about the world, or that one route is better. The vertex placement is a choice, not a discovery. Only the graph facts are proved: the counts, the distances, the number of shortest routes, and the bound of four. A trial is evidence about routes. Two roads converging on one vertex is interesting because they are independent, not because the vertex is right.

Kletetschka's three-dimensional time sits on a vertex as a background node. The proposal was published in 2025 in Reports in Advances of Physical Sciences. It is contested and has not been accepted. Among its critics' objections is the standard one: several time-like dimensions invite closed time-like curves, and with them the loss of causality. It is here for the nerve of the question, not as physics. On this map it lies one edge from Oresme, who put a single quality on several axes, while Kletetschka puts time on several axes.

This is the first piece of a new wing. The curator 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.

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