Dwivedi, Ripunjay (2025) The Spectral Quotient: A Categorical Resolution to Surplus Structure. [Preprint]
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Abstract
Recent scholarship (e.g., Dewar, Weatherall) challenges the strict elimination of surplus structure, noting that in geometric theories, quotienting is often mathematically 'hostile,' yielding spaces that lack the requisite differential structure (e.g., non-Hausdorff manifolds). We contend, however, that while this skepticism is justified in the geometric domain, it does not apply to the algebraic structures governing Quantum Mechanics.
In this paper, we propose the Spectral Quotient: a categorical reduction mapping abstract syntax into a faithful Operational Cogenerator. We demonstrate that for algebraic theories, the excision of surplus structure is generative rather than destructive.
We validate this constructive reduction across four foundational domains: (1) Symmetry, where Cayley's Theorem grounds groups in permutation; (2) Topology, where the Kolmogorov ($T_0$) quotient enforces empirical distinguishability; (3) Fields, where the Banach-Mazur construction guarantees separability; and (4) Quantum Mechanics, where the Gelfand-Naimark-Segal (GNS) construction derives the Hilbert space as the necessary quotient of the algebra of observables.
Unifying these results under the Yoneda Lemma, we identify the physical object with its presheaf of operational outcomes. This establishes a Constructive Structural Realism, demonstrating that for algebraic and functional theories, the metaphysically reduced theory is mathematically superior to the sophisticated one.
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