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

We study the statistical and dynamical properties of iterates of the sum-of-divisors function $\sigma(n)$ via the normalized ratio \[ R_q(n) = \frac{\sigma^{(q)}(n)}{n}, \quad q \ge 3, \] using large-scale computation ($n \le 10^6$), regression modeling, extreme-value analysis, and a finite-difference analogue of Lyapunov diagnostics. Empirically, $R_q(n)$ 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 \emph{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 $R_q(n) \le T$. Combined with a fractal-geometry analysis (box--counting dimension $D_{\mathrm{box}} \approx 0.9925$) of the integer-dynamic attractor, this yields measurable constraints supporting the Schinzel Conjecture for $q \ge 3$. Our entropy–fractal framework, supported by reproducible computations, offers a statistically grounded and computationally verified pathway toward resolving this conjecture