Training - the optimisation of complex models - is traditionally performed through small, local, iterative updates [D. E. Rumelhart, G. E. Hinton, R. J. Williams, Nature 323, 533-536 (1986)]. Approximating solutions through truncated gradients is a paradigm dating back to Cauchy [A.-L. Cauchy, Comptes Rendus Mathématique 25, 536-538 (1847)] and Newton [I. Newton, The Method of Fluxions and Infinite Series (Henry Woodfall, London, 1736)]. This work introduces the Method of Infinite Descent, a semi-analytic optimisation paradigm that reformulates training as the direct solution to the first-order optimality condition. By analytical resummation of its Taylor expansion, this method yields an exact, algebraic equation for the update step. Realisation of the infinite Taylor tower's cascading resummation is formally derived, and an exploitative algorithm for the direct solve step is proposed. This principle is demonstrated with the herein-introduced AION (Analytic, Infinitely-Optimisable Network) architecture. AION is a model designed expressly to satisfy the algebraic closure required by Infinite Descent. In a simple test problem, AION reaches the optimum in a single descent step. Together, this optimiser-model pair exemplify how analytic structure enables exact, non-iterative convergence. Infinite Descent extends beyond this example, applying to any appropriately closed architecture. This suggests a new class of semi-analytically optimisable models: the \emph{Infinity Class}; sufficient conditions for class membership are discussed. This offers a pathway toward non-iterative learning.

Fermat's Last Theorem (FLT) states that there are no positive integers x, y, and z that satisfy the following Diophantine equation The French lawyer and mathematician Pierre de Fermat made this conjecture in 1637 in the margin of a copy of the book Arithmetica, an Ancient Greek mathematical text written by Diophantus of Alexandria in the 3rd century AD. Fermat famously conjectured he had a proof of Eq. 1, but it was too large to fit in the margin of the book. The English mathematician Andrew Wiles published the first successful proof of the conjecture in 1995, after more than 350 years of effort by some of the greatest mathematicians in history (see this link for more details). We will prove the particular case where n 4, which is the simplest one. However, before that, we need to prove the following simpler auxiliary theorem about Pythagorean triples (x, y, z).