The Egg, Not the Sphere: A Pair-Correlation Measurement of Structural Deviation in Lattice Sieve Databases

Every cost estimate for lattice sieve algorithms assumes the database vectors are approximately uniform on the unit sphere. We show they are not. We define the pair-correlation function g(π/3, β)—the ratio of observed close-pair fraction to the uniform-on-S^{β-1} prediction—and measure it on the G6K sieve implementation at block sizes β = 40–70. On LLL-reduced lattices, g ranges from 16 to 953 and grows exponentially with β; a separate measurement after BKZ reduction confirms the effect persists (g = 9–125). The sieve amplifies the deviation further (g = 500–43,000 post-sieve). We trace the non-uniformity to a single cause: the Gram–Schmidt norms in the sieve block remain unequal after reduction, producing an "egg-shaped" angular distribution rather than the assumed sphere. We derive an analytical formula for g via the effective dimension d_eff (participation ratio of the Gram–Schmidt norms), yielding log₂ g ≈ 0.2075(β − d_eff) ≈ 0.128β, with the analytical and experimental slopes agreeing to within 3% (though this precision rests on only four data points and should be treated as indicative). Because 0.128/0.2925 ≈ 0.44, the structural deviation accounts for up to ~44% of the BDGL sieve's claimed cost at every security level. Extrapolating to cryptographic dimensions: Kyber-512 has a 48-bit structural gap (BDGL cost 2^117, structural floor 2^69), Kyber-768 a 74-bit gap, Kyber-1024 a 99-bit gap, and Falcon-1024 a gap of 130 bits. The major structural attack improvements found in the prior DRAWDOWN program (60+ rounds, each shaving 2–12 bits) are partial projections of this single geometric effect. The sieve searches an egg, not a sphere.