An Empirical Operator Algebra on a Latent Manifold of Polynomial Continued Fractions

We construct a finite-dimensional coefficient space of polynomial continued-fraction (PCF) generators and train a Geometric Operator Autoencoder (GOT v3) to embed these objects into a low-dimensional latent manifold z in R³. We then define empirical operator vectors vT (x) = E (Tx) − E (x) for ten symbolic transformations including index shifts, coefficient scaling, sign flips, and Apéry-like perturbations. Using this representation, we test closure, inverses, involutions, commutativity, commutator norms, cosine similarity, approximate additivity, Jacobi-type identities, Lie-bracket closure, braid relations, and conservation laws at symbolic, latent, and arithmetic levels. The results suggest that smooth operators admit approximate linearisation in latent space, while the shift operator behaves as a disruptive non-commuting generator. We identify a small subset of smooth operators with approximate Lie-algebra-like closure under empirical brackets. These findings are preliminary and checkpoint-dependent, but demonstrate a reproducible framework for studying operator geometry in spaces of mathematical objects. Keywords: continued fractions; operator algebra; latent manifolds; geometric deep learning; automated conjecture generation; Apéry-like perturbations; ζ (3). Scope note. This report does not claim new continued-fraction identities for ζ (3), ζ (5), or ζ (7). ζ (3) -proximity is used as a scalar field over a generated PCF corpus. The main contribution is a reproducible empirical framework for probing approximate operator algebra on a learned latent manifold.