This manuscript explores how a quarter-turn functional equation singles out a quadratic quantity, and how rotation compatibility together with a bounded-scaling argument identifies that quantity with squared Euclidean length.
The work is a proof-oriented manuscript by a sole author and is currently under peer review; the full text is not posted publicly while review is ongoing.
Research question
Can a single functional equation, formulated around a quarter turn, determine a quadratic quantity completely enough that the quantity is forced to be squared Euclidean length?
Approach
The argument isolates the quadratic quantity from the functional equation, uses compatibility with rotations to constrain its behaviour, and closes with a bounded-scaling argument that pins the quantity to squared Euclidean length.
Current status
The manuscript was submitted for peer review in September 2026. The full text is not being posted publicly while review is ongoing.