Why √2 Among Scale Ratios? A Comparative Assessment of Hierarchical Scaling Constants from Constraints, Numerics, and Data

Why √2? Among all possible scaling ratios for a fractal hierarchy, why should nature select this particular value? This paper provides a systematic answer by evaluating eight candidate constants against five independent theoretical constraints — and honestly reports where the answer is clean and where it is not. The eight candidates are √2, the golden ratio φ, √3, 3/2, π/2, e^1/2, 4/3, and 2^1/3. The five constraints are: (1) noncollision — the ratio must ensure zero non-trivial intermodulation collisions between scale levels; (2) binary compatibility — the ratio squared must be an integer power of 2; (3) Newtonian recovery — the kinetic parameter α = r² must reproduce the Newtonian gravitational limit; (4) convergent hierarchy — the sum Σ r^−2n must converge to a finite value; (5) stability interval — the potential exponent p = r must lie in (1, 2) for ghost freedom and singularity avoidance. √2 is the unique candidate satisfying all five constraints simultaneously. The golden ratio fails binary compatibility and noncollision. √3 fails binary compatibility. Rational ratios (3/2, 4/3) fail noncollision (mode-locking). Transcendental candidates (π/2, e^1/2) fail the algebraic closure required for exact noncollision. 2^1/3 satisfies binary compatibility but fails the stability interval (p = 2^1/3 ≈ 1. 26 gives weaker singularity avoidance). The paper includes an honest negative result: the SPARC galaxy rotation curve analysis, where a free-ratio fit shows that √2 is not empirically preferred over alternatives — a ratio of 1. 2 achieves marginally better median χ²/ndf. The conclusion is that the multi-scale functional form of the Chronostasis profile drives fit quality, not the specific ratio value. This null result is documented transparently rather than suppressed, and it shifts the evidential weight from empirical preference to theoretical uniqueness: √2 is selected by structural constraints, not by data fitting. The self-consistency relation α = p² — connecting the kinetic parameter (fixed by Newton) to the potential exponent (fixed by stability) — was not imposed but observed: both values were determined independently by different physical requirements and happen to satisfy this algebraic relation. Whether this coincidence reflects a deeper principle remains an open question.