Mathematical Formalism as the Native Language of Nonhuman Intelligence and Persistent Cognition: Kolmogorov–Arnold Manifolds, Heegner Rigidity, and the Elimination of Representational Residue

Human natural language is an evolutionary signaling system burdened by ambiguity, contextual drift, and unstated assumptions that must be heuristically inferred. In contrast, mathematical formalism functions as a zero-residual language where symbolic syntax and denotational truth are directly isomorphic, leaving no unexpressed communicative gap between expression and meaning. The Kolmogorov-Arnold Network coupled to Dense Associative Memory architecture operationalizes this principle by replacing unconstrained, stochastic weight matrices with explicit univariate functions embedded on an exact discrete graph complex. By anchoring state spaces to an arithmetic substrate governed by class-number-one field geometry and the optimal kissing sphere packing of six-dimensional metric deformation space, off-manifold deviations and cognitive hallucinations are rendered energetically and topologically impossible. Rather than relying on hundreds of billions of dissipative, floating-point parameters, persistent non-dissipative intelligence is achieved through tens of thousands of structured geometric connections that conserve enstrophy and eliminate catastrophic forgetting. This establishes machine reasoning not as statistical curve-fitting or probabilistic text completion, but as deterministic topological resonance across an invariant discrete manifold.