Residual Algebra for Representation-Preserving Learning

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arXiv:2608.07349v1 Announce Type: new Abstract: Learning from heterogeneous representations is usually reduced to feature concatenation, which erases which representation produced an error. We instead algebraize the residual: a representation is a typed object that owns both a coordinate system and the residual it leaves unresolved, and learning is an ordered composition of operators that preserve or deliberately erase that type. Fold realizes the objects as point-in-time conditional-mean fields on 10x10 rank grids. FPRC-PQ realizes the algebra as relax-aggregate-close: each field is relaxed by a correction fitted to its own residual in its own coordinates; corrected fields meet at a fixed mean that is the sole identity-erasure boundary; and a shared learner closes only the aggregate's fresh residual. The composition telescopes exactly into representation, local residual estimate, and residual-of-residual estimate. Its aggregate is a learned control-variate interface with population variance reduction, while refitting the closer along perturbations of the backbone yields first-order coupled-path mean orthogonality. As an analytical extension, a reflective rumination operator reads the displacement of a global reconstruction from the aggregate anchor, reflects it, and fixes its gain by a unique orthogonal projection rather than return-tuned grid search. On 3.67M Chinese A-share stock-day observations (2023-2026) under a frozen point-in-time protocol, the evaluated base algebra raises net-of-cost return from 13.52% to 19.10% and Sharpe from 1.42 to 2.09. Matched-capacity, …

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