The Latent Algebra: A Universal Representational Language for Smooth Systems
Abstract
We define the Latent Algebra \(\mathfrak{L}(\mathcal{H}) = \bigoplus_{r \geq 0} \mathcal{H}^{\otimes r}\) as the graded tensor algebra over a separable Hilbert space \(\mathcal{H}\), equipped with four primitive operations — addition, tensor product, contraction, and basis change — that constitute a graded contraction algebra (GCA). We prove the Universality Theorem: \(\mathfrak{L}(\mathcal{H})\) is the initial object in the category of GCAs generated by \(\mathcal{H}\). Any system requiring grade decomposition, tensor products, and contraction embeds canonically into \(\mathfrak{L}(\mathcal{H})\).
We prove the Quotient Classification Theorem: a smooth system with finite permutation symmetry \(G \leq S_n\) forces grade-\(r\) Latents into the \(G\)-invariant subspace \((\mathcal{H}^{\otimes r})^G\), and in an \(N\)-mode truncation the invariant dimension obeys the Burnside orbit-count \(\dim((\mathcal{H}_N^{\otimes r})^G) = \frac{1}{|G|}\sum_{\sigma \in G} N^{c(\sigma,r)}\) (Theorem 2). For full symmetric tensors (exchangeable statistics), \(\dim \mathrm{Sym}^r(\mathcal{H}_N) = \binom{N+r-1}{r}\), yielding compression factors of 6–23\(\times\) at grades 3–4 when compared to unconstrained \(N^r\). We identify the Latent Algebra as the full Fock space \(\mathcal{F}(\mathcal{H})\) of quantum field theory and classify which physical and statistical systems live in which Fock-space quotient.
We establish the Grade-Convergence Theorem: the truncated algebra \(\mathfrak{L}_N(\mathcal{H})\) converges to \(\mathfrak{L}(\mathcal{H})\) uniformly over all algebraic operations at rate \(O(\rho^{-N})\), where \(\rho > 1\) is the analyticity parameter. This makes the finite-dimensional algebra a rigorous approximation to the infinite-dimensional one.
We prove the Algebra-Evaluator Separation Principle as a categorical statement: the evaluator is a natural transformation from the identity functor to the truncation functor on the category of Latents, and convergence/divergence are properties of this natural transformation.
We develop the complexification \(\mathfrak{L}(\mathcal{H}_\mathbb{C})\), showing that the analyticity parameter \(\rho = e^{2\pi\delta/L}\) is the strip width of analytic continuation (Paley–Wiener) and that the COS expansion is the real-part projection of the Fourier decomposition in the complex algebra.
We prove the Cross-Domain Transfer Theorem and give an explicit algorithmic pipeline for mechanically transferring results between domains. We embed the 21-operation Knowledge Algebra of ML models into \(\mathfrak{L}(\mathcal{H})\) as grade-1 specialization.
We prove the Algebraic Optimization Theorem: when the feasible set is convex in Latent coordinates and the objective is a contraction, optimization reduces to projection — no search required. The boundary is sharp:
Novelty
The genuine delta is packaging the classical free tensor algebra with contraction as a first-class operation (GCA axioms), proving a quotient classification that links symmetry groups to effective Latent dimension, and the algebra-evaluator separation as a categorical statement — the underlying object (tensor algebra over Hilbert space) is classical, but the framing as a universal representational language with explicit inter-paradigm morphisms across five fields is new synthesis.