This work analyses the concept of an invariant in mathematics and introduces a new type of invariant — the structural invariant of the right isosceles triangle (RIT) I = (1 : 1 : √2). It is shown that due to the property of ideal self‑similarity (when divided by the altitude, the RIT yields two smaller RITs similar to the original) this invariant is preserved not only under scaling but also under recursive division and combination into mosaics. Unlike other figures (square, equilateral triangle, circle, arbitrary right triangles), the RIT is the unique (up to similarity) object possessing such recursive closure. The irrationality of √2 prevents degeneration into integer ratios, making the invariant irreducible. The invariant I(RIT) acts as a “genetic code” for numbers, determining primality/compositeness through the distribution of force connections (hypotenuses) in △-mosaics. It is proved that this invariant stands at the top of the hierarchy: it works at all scales, reproduces itself, and constitutes the object itself.
Alexey (KAMAZ) Petrov (Sat,) studied this question.