Entropy, Statistical, and Dynamical Analysis of σ-Iterates: Heavy Tails, Fractal Geometry, and Empirical Evidence on the Schinzel Conjecture for k ≥ 3

Abstract We study the statistical and dynamical properties of iterates of the sum-of-divisors function σ(n) via the normalized ratio \[ R_q(n) = {σ(q)(n)}{n}, q ≥ 3, \] using large-scale computation (n ≤ 10⁶), regression modeling, extreme-value analysis, and a finite-difference analogue of Lyapunov diagnostics. Empirically, Rq(n) is strongly right-skewed and heavy-tailed, with rare large spikes linked to highly composite integers; Lyapunov analysis shows a contraction-dominated local sensitivity consistent with boundedness. Regression on arithmetic predictors (log-scale, divisor count, prime factor indicators) explains much central variation but leaves structured extreme residuals, motivating peaks-over-threshold analysis. We introduce an entropy-based lower-tail criterion linking bounded empirical Shannon entropy to exponential bounds on upper-tail mass and proving that bounded entropy with a vanishing-tail condition forces infinitely many n with Rq(n) ≤ T. Combined with a fractal-geometry analysis (box--counting dimension Dbox ≈ 0.9925) of the integer-dynamic attractor, this yields measurable constraints supporting the Schinzel Conjecture for q ≥ 3. Our entropy–fractal framework, supported by reproducible computations, offers a statistically grounded and computationally verified pathway toward resolving this conjecture