RoPE Complex Inner Product Derivation

The Rotary Position Embedding (RoPE) Complex Inner Product Proof establishes the mathematical necessity of orthogonal block-diagonal rotation matrices for encoding relative position semantics into Transformers; the core functional equation requires finding representations $f_q(x_m, m)$ and $f_k(x_n, n)$ such that their inner product depends strictly on relative offset $m - n$: $\langle f_q(x_m, m), f_k(x_n, n) \rangle = g(x_m, x_n, m - n)$; using complex analysis and Euler's formula, the unique 2D solution is complex multiplication $f(x, m) = x e^{i m \theta}$; expanding across $d/2$ orthogonal sub-planes yields a block-diagonal orthogonal rotation matrix $R_{\Theta, m}^d$, achieving relative position interactions through absolute coordinate rotations with norm preservation and seamless context scaling (YaRN / NTK-aware interpolation).

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Absolute learnable embeddings fail on context extrapolation; RoPE delivers norm-preserving, relative-distance decay properties with zero memory footprint.

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