A Resolution of the Collatz Conjecture

This paper develops a system from first principles. The analysis is built onfour complementary components that together provide a complete description of theodd–to–odd Collatz dynamics:1. A finite residue–phase automaton governing all minimally admissible odd-to-odd Inverse transitions, determined by congruence classes modulo 6 and phasebehavior modulo 3.2. A Normal–State index in which the removed admissible dyadic factors fromodd iterative minimally admissible output yields a canonical affine arithmeticskeleton of the families of 4-adic rails generated by the map k→k+2, orm→m+1.3. A dyadic slice decomposition indexed by the valuation k = ν_2(3n+1), producinga disjoint arithmetic partition of the odd integers with exact weights 2−k.4. A Forward–Inverse locking identity linking the reduced Forward mapT(n)=3m+1/2^ν_2(3m+1)to the admissible inverse mapR(m;k)=2^k m−1/3,ensuring that each odd integer has a uniquely determined admissible ancestry.A principal result of this work is that the affine rails arising from the normal–stateconstruction coincide exactly with the dyadic slices determined by k = ν2(3m + 1).Thus the odd integers admit two independent but equivalent global parametrizations:one affine and one dyadic. This coincidence yields a global arithmetic structure inwhich all odd integers arise from admissible lifts above the anchors {1,5}.Because the residue–phase automaton is finite and the admissible inverse relationinduces a unique Forward parent at each step, the resulting ancestry relation is wellfounded. No nontrivial odd cycles can exist, and no infinite runaway is possible.Consequently, every Forward Collatz trajectory converges to the fixed point 1.