Primes in Square Intervals: A Structural Audit of the Square-Centered Dispersion Route

For J_n = [4n²-n, 4n²+n], x = (2n)², we examine whether the perfect-square center can be exploited by a dispersion argument to detect primes in J_n. Beginning with the primorial pre-sieve at B = log n, we follow the resulting progression counts through Poisson summation and reciprocity and determine exactly what analytic information the route requires.

The audit reveals a sharp separation between genuine square-centered structure and artifacts of the chosen reduction. The discrepancy of the pre-sieved set in an individual progression is already O(n^{o(1)}), so distribution estimates for that set alone cannot supply prime-detection information. The exact second-level expression does retain the square numerator (2n)², but it contains two coupled arithmetic phases and a Ramanujan coefficient c_P(h). We determine the norm and Fourier structure of this coefficient and show, both analytically and by finite-window diagnostics, that its mass-bearing part is frequency-spread rather than concentrated in the form required by current large-sieve estimates. We also show that completion to ordinary Kloosterman sums is an invertible detour, that the natural all-coefficients trilinear estimate is either false or ineffective for the relevant coefficients, and that the square numerator yields no Salié-type evaluation.

The result is therefore a precise boundary for this form of the dispersion method. Any viable dispersion route must either produce a different, mass-bearing coefficient while preserving the square-centered phase, or obtain cancellation directly in the exact two-phase expression after averaging over the prime moduli.