Let f: R -> {-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 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.
