Random polynomials are equations whose coefficients are chosen by chance. As their degree grows, their complex roots often crowd around the unit circle. This paper asks what we see when we magnify the small differences in their distances from that circle.

The coefficients here have uniformly random directions and heavy-tailed sizes. A few unusually large coefficients can retain substantial influence even in a very large polynomial.

The main result is a limiting radial profile that is smooth but still random. Smoothness does not mean that different realizations become alike. At any fixed magnified radius, the limiting fraction of roots inside can be close to any number between zero and one. Its variance stays positive. Averaging many realizations recovers a known deterministic formula, but that average does not describe each realization.

Two new endpoint results explain how this randomness changes with the tail index. When the index approaches zero, the fraction at every finite magnified radius approaches the same uniformly distributed random number. The probability mass escapes towards the two ends of the magnified line; it does not disappear from the original root count.

When the index approaches two instead, the profile approaches a deterministic Gaussian profile, and the variance of each root fraction tends to zero. These statements take the polynomial degree limit first. They do not establish a simultaneous limit in degree and tail index.

The package contains the analytic arguments, source comparisons, a response to the supplied review, and finite consistency checks. Those checks do not verify the infinite-series proof. The result remains an unrefereed candidate, without formal verification or external specialist confirmation.

This briefing accompanies Smooth random radial profiles, released on the eleventh of October, twenty twenty-six. The paper and evidence are linked on Evidence Press. The voice is AI-generated, and this briefing is not additional evidence.