Sieve Gap Entropy and Non-Geometric Structure: A Maximum-Entropy Bound and a KL-Divergence Identity

For the periodic gap law P_k of integers surviving sieving by the first k primes (defined via cyclic gaps around one full period m_k = p_1···p_k, to avoid boundary ambiguity), we prove the mean gap is exactly μ_k = m_k/φ(m_k) = ∏i≤k(1−1/p_i)^−1, by an elementary telescoping argument with no independence assumption. Combined with the classical maximum-entropy property of the geometric distribution, this gives an unconditional bound H_k ≤ H_max(μ_k) on the gap distribution's Shannon entropy, and we show the resulting slack D_k := H_max(μ_k) − H_k is exactly the Kullback–Leibler divergence D_KL(P_k ‖ G_μk) to the geometric distribution of the same mean. This turns the sieve's asymptotic entropy behavior into a precise question: does P_k → G_μk in relative entropy? We prove the explicit finite-k inequality D_k ≥ H_max(μ_k) − H_max(μ_k/2) (since 2 is always a sieve prime, every gap is forced even; folding this constraint through the same maximum-entropy argument gives the bound), whose right-hand side tends to 1, and hence liminf D_k ≥ 1 bit as k→∞: the sieve provably retains a permanent information-theoretic obstruction to geometricity. This bound and the parity floor require no periodic sieve structure and transfer as corollaries to the actual prime gap distribution P_X (gaps between real primes up to X), which we study directly. Writing B_k (resp. B_X) for the parity floor and R_k := D_k − B_k (resp. R_X := D_X − B_X) for the residual, the two objects behave differently: for the truncated law P_k, numerics show R_k falling overall (from 0.38 down to 0.017 bits over three orders of magnitude, non-monotonically), consistent with a conjectured R_k → 0; for real primes, direct sieving to X = 1.5×10^9 shows R_X instead plateauing at ≈0.082–0.083 bits across two full orders of magnitude, suggesting a different, nonzero limit R_X → C_prime ≈ 0.082 for the object of actual number-theoretic interest. Interpolating between these two regimes via a coupling exponent α (sieve depth p_k ~ N^α, with α→0 resembling the truncated law and α=1/2 exactly real primes), tested on both sides at matched depth N=10^10, reveals a sharp asymmetry: a real, still-declining gradient for every α < 1/2 (0.08267 at α=0.47 down to 0.08250 at α=0.49), and a plateau numerically indistinguishable from flat at the tested precision, matching the critical value to four significant figures, for every tested α ≥ 1/2. This matches a clean mechanistic explanation: below 1/2 a genuine population of composites dilutes the statistics and shrinks toward the critical point; above 1/2 no composites survive at all, and the construction instead only truncates a vanishing initial segment of primes, which does not affect the tail statistics. This decomposition, its proved floor, and this sharper real-prime finding — not any specific growth-rate constant — are the paper's central contributions. We also report, explicitly as secondary and exploratory, numerical evidence for a growth-law conjecture δH_p ~ C/p (tested by AIC against competing decay rates, not goodness-of-fit alone), a heuristic model for it whose independence assumption we show to be measurably false, a partial residue-class correction, and a tentative finite-range extension to prime constellations.