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

Updated Sep 24

Twenty stored flat targets with bounded integer line preimages

Ingredient tested

The flat-preimage lemma proves that in at most three directions every bounded integer vector in the line image of a box has an equally bounded integer line preimage. Whether every flat integer vector in the full line image of a four-direction box has a flat integer line preimage remains unproved. The records here are finite examples in that direction. No color-lifting assertion was tested.

What is checked

Run, from the repository root,

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

The checker reads the twenty stored target and preimage pairs and, with the standard library, integer arithmetic and the shared root tools checker, verifies for each that the entries lie in $\{-1,0,1\}$ , that the target is nonzero, that the line combination with the stored coefficients equals the target at every point, and that the target is annihilated by every quotient functional, so that it lies in the line image. It performs no search and finishes in under a second; a failed clause exits nonzero. Points are ordered lexicographically with zero-based coordinates, and line columns by axis and then by fixed coordinates.

Generation and its exact range

The targets are optimization-generated, not uniform random flat vectors, and their generation is not part of the verification. The quotient matrix used to constrain a target $b$ to the line image is

T_{r,x}=\prod_i(1_{x_i=r_i}-1_{x_i=d_i-1}), \qquad 0\le r_i<d_i-1.

Targets came from a mixed-integer solver minimizing a fresh Gaussian linear objective over integer $b\in[-1,1]^X$ with $Tb=0$ ; preimages came from a real feasibility solve followed by an integer feasibility solve with the same equality and bounds.

Probe	Exact finite scope and recorded conclusion
Seed 20260918, continued across both boxes	Eight trials on $[3]^3$ , then eight on $[3]^4$ . All 16 targets had full $\{\pm1\}$ support. Every real and integer preimage solve returned a feasible status.
Seed 20260919	Four trials on $[3]^4$ ; choose 41 allowed point indices without replacement, force the other 40 target coordinates to zero, then generate the Gaussian objective. Supports were 35, 29, 35, 30; all four preimage solves succeeded.
Separate overlapping probe, seed 774	Thirty targets on $[3]^3$ , then twenty on $[3]^4$ , with the same quotient constraints and independent Gaussian objectives. All returned a target and integer preimage with no reported infeasibility. Those preimages were not residual-checked or retained, so these fifty statuses are noncertified solver observations.

In the first two probes each integer call had an eight-second limit and the real feasibility solve had none. The matrix sizes were $A:27\times27$ , $T:8\times27$ and $A:81\times108$ , $T:16\times81$ . The separate probe used a three-second limit on both integer calls and stopped at its fixed caps. Only the twenty pairs of the first two probes are retained, and only they are checked; the fifty noncertified statuses are used in no proof.

Linked from (3)

Evidence Integer height-preserving line preimages in at most three directions Main argument and missing coloring theorem

Graph