Asymptotic stability of spectral entropy for a fixed prime and resolution

Establish that for any fixed prime p and fixed logarithmic resolution M ≥ 2, the spectral entropy H_R(p)—defined by (i) forming the truncated inter-prime distance multiset D_R(p) = { |p − q| : q is prime, q ≠ p, |p − q| ≤ R }, (ii) aggregating D_R(p) into M logarithmic bins between d_min = min D_R(p) and d_max = max D_R(p) to obtain probabilities (p_1,…,p_M), (iii) computing the discrete log-frequency spectrum \hat{μ}(k) = ∑_{j=1}^M p_j exp(−2πi (k−1)(x_j − x_1)/(x_M − x_1)) from the bin-center locations x_j, and (iv) forming weights w_k = |\hat{μ}(k)|/∑_{ℓ=1}^M |\hat{μ}(ℓ)| and entropy H_R(p) = −∑_{k: w_k>0} w_k log w_k—stabilizes as R → ∞ in the sense that there exists a function η(R) → 0 with |H_{R_1}(p) − H_{R_2}(p)| ≤ η(min(R_1,R_2)) for all sufficiently large R_1, R_2.

Background

The paper defines a scale-invariant entropy statistic H for point configurations by log-binning inter-point distances at fixed resolution, taking a discrete Fourier transform on the logarithmic coordinate, and computing the entropy of normalized spectral magnitudes.

For prime numbers, H_R(p) depends on the truncation radius R used to collect distances around a base prime p. The conjectured stability asks that H_R(p) become essentially independent of R for large radii, ensuring the statistic is well-defined for asymptotic comparisons.