Essential Periods and Monochromatic Triangles in Measurable Layered Colorings

Let \(f:\mathbb{R}\to\{-1,1\}\) be measurable and color the plane by \(\chi(x,y)=f(y)\). Call an equilateral side length essentially avoided when, for each fixed orientation, the set of monochromatic translations has planar measure zero. We prove that essential avoidance of side \(s\) forces the essential period group of \(f\) to be exactly \(\sqrt{3}s\mathbb{Z}\). Consequently at most one equilateral side length is essentially avoided. The classical EGMRSS side-length transfer then gives every nonequilateral three-point configuration a positive-measure set of monochromatic translations in some orientation. This includes collinear triples and survives arbitrary changes on a planar null set.

The argument uses closed essential-period groups and one tilted equilateral triangle. It requires no interval or locally finite boundary assumption on the measurable profile. This is a complete result for essentially layered measurable colorings, related to AIM-COMBINATORICS-0085 in ulamai/UnsolvedMath v1.6.0; it does not solve the conjecture for arbitrary planar two-colorings. The standard stripe example, classical side-length transfer and earlier polygonal results are credited.

Four-page English preprint by Alper Ferudun, Mercury Software GmbH. Version 1.0, 17 October 2026. AI-assisted, self-audited and unrefereed; no independent review, proof-assistant verification or absolute-priority certification is claimed. The accompanying source archive includes portable exact algebra and finite regression checks. CC BY 4.0.