Weighted 1-Hodge Spectral Deflation for Binary Embedding Hashing: Two Orthogonal Eigenvectors via Weighted Line Graph Construction and Hotelling Deflation

Standard spectral graph methods fail at extreme embedding compression because they operate on node signals — and node signals collapse to the same macro-cluster structure across every subspace. This paper breaks that ceiling. We move the spectral computation from nodes to edges, operating directly on the 1-cochains of a weighted Line Graph where connection strength is proportional to the cosine similarity of non-shared endpoints. This single architectural shift — from the 0-Laplacian to the weighted 1-Hodge Laplacian — produces a 10x recall lift over random baselines at 1-bit-per-24D-subspace compression, independently of the local partitioning method (Micro-IVF: 56. 1%, Micro-LSH: 52. 7%). We then introduce Hotelling spectral deflation: after extracting the dominant eigenvector v0 via weighted power iteration, we apply orthogonal projection at every iteration step to extract a second eigenvector v1 ⊥ v0. The two bits carry genuinely independent spectral information — empirically confirmed with an independence score of 0. 625 — pushing the theoretical recall ceiling from ~56% toward 70–75% with a 2-bit Hamming code in 2¹28 space. All methods are hardware-agnostic and substrate-independent. The mathematics applies identically to ARM, x86, FPGA, and ASIC substrates. This work establishes formal prior art on weighted 1-Hodge spectral hashing and Hotelling deflation for binary embedding compression.