We introduce Spectral Field Memory (SFM), a novel context compression architecture for large language models (LLMs) that treats document collections as spectral fields over geometric manifolds. Rather than feeding raw text into LLM context windows or relying on sparse retrieval heuristics, SFM projects sentence embeddings onto manifold-constrained geometries---including flat (Euclidean), spherical, hyperbolic, product-spherical, and M\"obius spaces---and performs spectral decomposition via the graph Laplacian to extract compressed, semantically-ordered representations. We conduct nine systematic experiments on the HotpotQA benchmark (fullwiki and distractor splits) evaluating compression quality, multi-hop reasoning fidelity, manifold geometry effects, noise robustness, and question-type sensitivity. Our key findings are threefold: (1) ~SFM with K=5 spectral modes achieves 99. 6\% of RAG-lite answer quality at 2 compression, a difference that is statistically non-significant (p > 0. 05) ; (2) ~SFM substantially outperforms BM25 on comparison and bridge-type questions under dense noise conditions, demonstrating superior semantic coherence preservation; and (3) ~the M\"obius manifold forms a distinct spectral compression regime, separating cleanly from an equivalence class formed by flat, spherical, hyperbolic, and product-spherical geometries---a novel geometric finding with implications for non-orientable semantic representations. We further develop six mathematical extensions grounded in quantum mechanics, differential geometry, algebraic topology, and random matrix theory---including a query-conditioned Schr\"odinger operator, Ollivier--Ricci curvature for bridge passage detection, and Marchenko--Pastur law for principled spectral mode selection---charting a path toward a direct LLM--memory interface paradigm in which language models probe context through energy field spaces rather than consuming raw tokens.
Vinit Chavan (Sun,) studied this question.