The Syracuse Conjecture: a categorical resolution by transport of structure in the sense of Bourbaki

This work presents a categorical resolution of the Syracuse conjecture (Collatz conjecture, (3x+1) problem) by transport of structure in the Bourbakian sense. Rather than approaching the conjecture through the enumeration of orbits or the accumulation of numerical verifications, the study transports its arithmetic formulation into a fully specified multi-sorted geometric structure combining Peano arithmetic, Syracuse dynamics, dyadic completion, Haar measure, measure algebra, and a quotient by residual profiles. Geometry is used here neither as an approximation nor as a heuristic representation: it constitutes the chosen fundamental space, linked exactly to the integers and their dynamics by bijection and conjugacy. Within this framework, successive dyadic valuations induce an exact law on cylinders, allowing one to establish at every depth an arithmetic density bound converging exponentially to (1). At the measured level, the macroscopic direction, the crossing of every finite barrier, and affine descent have value (1) in the measure algebra; the event corresponding to the indefinite avoidance of a barrier has exactly Boolean value (0). In parallel, a bi-adic saturation shows that the components obtained from the direct law, its admissible inverse branches, and basal forgetting all possess a maximal profile at every finite depth. They therefore become equal to the terminal component in the profile quotient (C_prof). The categorical transport of the orbital formulation of the Syracuse conjecture is then explicitly identified with this universal terminal relation. The main result is expressed as: S_Syr ⊨ (COL_cat ∧ COL_den) The conclusion therefore relies neither on extrapolation from computed cases, nor on converting a density-(1) property into a pointwise assertion, nor on any added independence hypothesis. Categorical terminal equality, collective density, and informational resolution are established as three distinct levels of one and the same structure. This repository constitutes the English edition of the work. Mathematically equivalent French and Japanese editions are also available and cross-linked through their respective DOIs.