Existence of entropy deviation limit from the Poisson null model for primes

Establish that for fixed logarithmic resolution M ≥ 2, the limit Δ_∞(p) = lim_{R→∞} (H_R(p) − H_R^{null}) exists, where H_R(p) is the spectral entropy defined from the log-binned truncated inter-prime distances D_R(p) and H_R^{null} is the spectral entropy computed by the same procedure on distances D_R^{null} = {x ∈ Π : 0 < x ≤ R} from a homogeneous Poisson point process Π conditioned to contain 0; moreover, ascertain that if prime gaps deviate persistently from Poisson statistics on the logarithmic scale, then Δ_∞(p) ≠ 0 for infinitely many primes.

Background

The Poisson log-distance model provides a scale-invariant null reference for the entropy statistic, with a well-defined asymptotic value H_∞^{null}(M).

This conjecture asks whether the difference between prime-based entropy and the Poisson-based entropy converges to a limit for each prime and whether persistent non-Poisson structure in prime gaps would force a nonzero limiting deviation for infinitely many primes.