Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Sep 24

A height-two circuit among four-direction axis lines

Ingredient tested

On three-direction slices of the axis-line family, rational dependence implies a ternary subrelation, because the slices are cographic; see the line coloring page. The question tested here is whether that collapse extends to four full axis-line directions. It does not: the retained certificate is a rational circuit of 27 full axis-parallel lines of $[3]^4$ , of rank 26, whose primitive relation has exactly one coefficient of magnitude two and all other coefficients $\pm1$ . Rank and full support prove that the support is a rational circuit, and the primitive coefficient two proves that it is dissociated.

What is checked

Run, from the repository root,

uv run --no-sync python wiki/research/erdos_774/evidence/axis_line_circuit/main.py

The checker reads the fixed certificate, which lists every line and coefficient with zero-based coordinates and axes, the three fixed coordinates of each line in increasing coordinate order. It constructs exactly the $81\times27$ incidence matrix and checks, with the standard library, exact rational elimination and the shared root tools checker, that the 27 lines are distinct, that the coefficients are nonzero with exactly one of magnitude two and greatest common divisor one, that the signed combination vanishes at all 81 points, and that the matrix has rank 26. It performs no search and finishes in under a second; a failed clause exits nonzero.

Discovery and its exact range

The circuit was found by a bounded discovery run that is not part of the verification: exact symbolic rational elimination with random seed 8043 and at most eight random full-column orders for each of the dimensions $(3,3,3,3)$ , $(2,3,3,3)$ and $(3,3,3)$ , inspecting a rational basis and its fundamental circuits for each order and stopping a dimension at the first primitive circuit of height greater than one. No floating-point certificate was accepted.

Exact input	Bounded discovery observation
$[3]^4$ , 81 point rows and 108 line columns	Trial 0 found the retained 27-column rank-26 circuit of height two, with one coefficient two
$[2]\times[3]^3$ , 54 rows and 81 columns	Trial 4 found a 32-column height-two circuit; this extra observation is not a durable proof input
$[3]^3$ , 27 rows and 27 columns	None in eight orders; the algebraic cographic proof, not this search, establishes the exact three-direction statement

The search stopped after answering the ingredient question and did not test chromatic growth.

Extension to larger boxes

The circuit persists when coordinate layers are duplicated. Copy nonparallel lines in the duplicated layer and extend parallel lines through the new layer. Restriction of a new relation to the old rows determines its old coefficients; independence of the copied proper circuit subset determines the remaining ones. Choose layers away from the heavy line's fixed coordinate whenever that line is nonparallel. Thus the coefficient two remains unique, and the example extends to every four-dimensional grid with side lengths at least three. This is an algebraic scope check, not another computation.

Linked from (3)

Evidence Main argument and missing coloring theorem The remaining coloring problem for linear-side line families

Graph