S-AI-Solver: Toward Numerical Artificial Intelligence — A Hormonally Regulated Sparse AI Architecture for the Formally Certified Resolution of Analytically Intractable Equations

Background. Two traditions have addressed the resolution of mathematically intractable problems, and both are structurally incomplete. Classical numerical analysis — Newton-Raphson, Galerkin projection, contour integral eigensolvers, stochastic approximation — resolves specific equation classes with formal convergence guarantees, but requires structural knowledge of the problem: a derivative, a discretisation grid, a convergence domain, or a smooth kernel. Machine learning applied to equations — Physics-Informed Neural Networks, Fourier Neural Operators, DeepONet — learns solution mappings from data without structural assumptions, but provides no formal convergence certificate, no intrinsic stopping criterion, and no guarantee that a given instance is resolved to a certified precision. The intersection of these two desiderata — formal convergence guarantees without structural knowledge — defines a disciplinary gap that neither tradition fills. This gap is the object of a nascent field that this article proposes to call Numerical Artificial Intelligence : the branch of artificial intelligence whose object is the formally certified, adaptive, structure-agnostic resolution of analytically intractable mathematical problems. Methods. This article introduces S-AI-Solver as a founding contribution to Numerical Artificial Intelligence. S-AI-Solver is the ninth architectural instantiation of the Sparse Artificial Intelligence (S-AI) paradigm and the first to formalise approximate equation-solving as a hormonal closed-loop iteration. It operationalises the S-AI-Recursive Recursive Reasoning Cycle (RRC) as a domain-agnostic approximate solver: the cognitive state encodes the current candidate solution via the DISNL normalisation layer; the provisional output is the current approximation; and the hormonal stopping criterion — governed by the antagonistic balance of Clarifine (convergence signal) and Confusionin (residual uncertainty detector) — determines when the approximation is sufficiently reliable to commit, without derivative information, starting-point knowledge, or external budget control. The derivative-free iteration operator with hormonally modulated step size is applied to eight formally defined classes of analytically intractable equations — transcendental equations, functional equations, coupled nonlinear systems, integral equations of Fredholm and Volterra type, nonlinear eigenvalue problems, fixed-point equations in abstract Banach spaces, inverse problems with non-differentiable forward operators, and stochastic equations — each equipped with a class-specific residual operator, state encoding, and confidence distribution . Results. Four theorems establish the formal foundations of S-AI-Solver as a Numerical AI architecture. Theorem 1 proves the global asymptotic stability of the two-dimensional recursive hormonal subsystem under the deployability condition, verifiable a priori and independent of the equation class. Theorem 2 proves the finite-time termination guarantee for the three-condition stopping criterion without external budget. Theorem 3, the Certified Resolution Theorem , is the central result of this article and the founding theorem of Numerical Artificial Intelligence: it establishes that any equation class admitting a continuous residual operator and a contractive iteration operator under the equilibrium hormonal field is resolvable by S-AI-Solver with formal residual bound and finite expected termination — under a single deployability condition that is class-independent. This result is the direct analogue, for Numerical AI, of the universal approximation theorem for neural networks: as the universal approximation theorem establishes that neural networks can represent any continuous function, the Certified Resolution Theorem establishes that the hormonal convergence mechanism can resolve any solver-admissible equation, without knowing its algebraic structure. Theorem 4, the Entropic Contraction Theorem, proves the equivalence, establishing that hormonal homeostasis is equivalent to monotonic reduction of solution uncertainty — bridging dynamical systems theory, information theory, and equation solving within a single formal invariant. Experimental validation on the SAI-UT++ testbench across 4,000 episodes and eight equation classes confirms: mean resolution rate RSR ; mean convergence depth ; mean Frugality Index ; and mean Pearson correlation () confirming Theorem 4 across all eight classes. Conclusions. S-AI-Solver demonstrates that a formally governed sparse AI architecture can resolve analytically intractable equations across eight qualitatively distinct classes — without derivatives, starting-point knowledge, or external budget control — by deploying a single class-independent hormonal convergence mechanism whose deployability condition is verifiable a priori. This result positions S-AI-Solver as the founding architecture of Numerical Artificial Intelligence: a new disciplinary branch defined by three properties that neither classical numerical analysis nor machine learning satisfies simultaneously — formal convergence guarantees, structure-agnostic operation, and intrinsic certified stopping. The governing principle is: not numerical approximation driven by algebraic structure, but intelligent, hormonally regulated, formally certified cognitive convergence toward the solution — convergence that the system itself certifies, through the elevation of Clarifine and the suppression of Confusionin, without being told when to stop.