A researcher presents an integer-native geometric framework that reduces lattice point enumeration complexity from O(r^N) to O(r^2) by decomposing high-dimensional spaces into orthogonal submanifolds and using discrete integer operations instead of continuous Euclidean methods. Four preprints establish theoretical foundations, asymptotic proofs, and C++ implementations verified against OEIS sequences, with hardware benchmarks demonstrating enumeration of billions of lattice points without floating-point arithmetic.
In high-dimensional word embeddings, cosine similarity measures word relatedness through vector angles. A similarity of 0.656 (49° angle) is considered large because random vectors in 200-dimensional space are nearly perpendicular, making close relationships rare; the canonical word arithmetic example 'king' − 'man' + 'woman' ≈ 'queen' demonstrates this contextual interpretation of similarity.