Beyond Attention: True Adaptive World Models via Spherical Kernel Operator

The pursuit of world model based artificial intelligence has predominantly relied on projectinghigh-dimensional observations into parameterized latent spaces, wherein transition dynamics aresubsequently learned. However, this conventional paradigm is mathematically flawed: it merelydisplaces the manifold learning problem into the latent space. When the underlying data distributionshifts, the latent manifold shifts accordingly, forcing the predictive operator to implicitly re-learn thenew topological structure. Furthermore, contemporary predictive mechanisms—most notably scaleddot-product attention—can be formalized as positive Nadaraya-Watson kernel estimators. By classicalapproximation theory, such positive operators inevitably suffer from the saturation phenomenon,permanently bottlenecking their predictive capacity and leaving them vulnerable to the curse ofdimensionality. In this paper, we formulate a mathematically rigorous paradigm for world modelconstruction by redefining the core predictive mechanism. Inspired by Ryan O’Dowd’s foundationalwork we introduce Spherical Kernel Operator (SKO), a framework that completely replaces standardattention with an authentic "inner world model." By projecting the unknown data manifold ontoa unified ambient hypersphere and utilizing a localized sequence of ultraspherical (Gegenbauer)polynomials, SKO performs direct integral reconstruction of the target function. Because this localizedspherical polynomial kernel is not strictly positive, it bypasses the saturation phenomenon, yieldingapproximation error bounds that depend strictly on the intrinsic manifold dimension q, rather than theambient dimension. Furthermore, by formalizing its unnormalized output as an authentic measuresupport estimator, SKO mathematically decouples the true environmental transition dynamics from thebiased observation frequency of the agent. We demonstrate that SKO learns on the manifold withoutexplicitly learning the manifold’s eigendecomposition or atlas—mirroring the unified subjectiveprojection spaces characteristic of biological cognition. Empirical evaluations confirm that SKOsignificantly accelerates convergence and outperforms standard attention baselines in autoregressivelanguage modeling. Ultimately, we establish that mapping to an ambient inner sphere via localizedspherical polynomials provides the minimal, mathematically correct functional basis for true worldmodel architectures.The code is available at https://github.com/VladimerKhasia/SKO